Abstract: In the maximum cover problem, we are given a collection of
sets over a ground set of elements and a positive integer w, and we are
asked to compute a collection of at most w sets whose union contains the
maximum number of elements from the ground set. This is a fundamental
combinatorial optimization problem with applications to resource allo-
cation. We study the simplest APX-hard variant of the problem where
all sets are of size at most 3 and we present a 6=5-approximation algo-
rithm, improving the previously best known approximation guarantee.
Our algorithm is based on the idea of ¯rst computing a large packing of
disjoint sets of size 3 and then augmenting it by performing simple local
improvements.

Abstract: In the maximum cover problem, we are given a collection of
sets over a ground set of elements and a positive integer w, and we are
asked to compute a collection of at most w sets whose union contains the
maximum number of elements from the ground set. This is a fundamental
combinatorial optimization problem with applications to resource allo-
cation. We study the simplest APX-hard variant of the problem where
all sets are of size at most 3 and we present a 6=5-approximation algo-
rithm, improving the previously best known approximation guarantee.
Our algorithm is based on the idea of ¯rst computing a large packing of
disjoint sets of size 3 and then augmenting it by performing simple local
improvements.

Abstract: .We present a new methodology for computing approximate
Nash equilibria for two-person non-cooperative games based upon certain
extensions and specializations of an existing optimization approach pre-
viously used for the derivation of xed approximations for this problem.
In particular, the general two-person problem is reduced to an inde-
nite quadratic programming problem of special structure involving the
n x n adjacency matrix of an induced simple graph specied by the in-
put data of the game, where n is the number of players' strategies.

Abstract: Motivated by the problem of supporting energy-efficient broadcasting in ad hoc wireless networks, we study the Minimum
Energy Consumption Broadcast Subgraph (MECBS) problem. We present the first logarithmic approximation algorithm for the
problem which uses an interesting reduction to Node-Weighted Connected Dominating Set.

Abstract: In this paper, we propose an efficient non-linear task workload prediction mechanism incorporated with a fair scheduling algorithm
for task allocation and resource management in Grid computing. Workload prediction is accomplished in a Grid middleware approach
using a non-linear model expressed as a series of finite known functional components using concepts of functional analysis. The coefficient
of functional components are obtained using a training set of appropriate samples, the pairs of which are estimated based on
a runtime estimation model relied on a least squares approximation scheme. The advantages of the proposed non-linear task workload
prediction scheme is that (i) it is not constrained by analysis of source code (analytical methods), which is practically impossible to be
implemented in complicated real-life applications or (ii) it does not exploit the variations of the workload statistics as the statistical
approaches does. The predicted task workload is then exploited by a novel scheduling algorithm, enabling a fair Quality of Service oriented
resource management so that some tasks are not favored against others. The algorithm is based on estimating the adjusted fair
completion times of the tasks for task order selection and on an earliest completion time strategy for the grid resource assignment. Experimental
results and comparisons with traditional scheduling approaches as implemented in the framework of European Union funded
research projects GRIA and GRIDLAB grid infrastructures have revealed the outperformance of the proposed method.

Abstract: In this work we study the important problem of colouring squares of planar graphs (SQPG). We design and implement two new algorithms that colour in a different way SQPG. We call these algorithms MDsatur and RC. We have also implemented and experimentally evaluated the performance of most of the known approximation colouring algorithms for SQPG [14, 6, 4, 10]. We compare the quality of the colourings achieved by these algorithms, with the colourings obtained by our algorithms and with the results obtained from two well-known greedy colouring heuristics. The heuristics are mainly used for comparison reasons and unexpectedly give very good results. Our algorithm MDsatur outperforms the known algorithms as shown by the extensive experiments we have carried out.
The planar graph instances whose squares are used in our experiments are “non-extremal” graphs obtained by LEDA and hard colourable graph instances that we construct.
The most interesting conclusions of our experimental study are:
1) all colouring algorithms considered here have almost optimal performance on the squares of “non-extremal” planar graphs. 2) all known colouring algorithms especially designed for colouring SQPG, give significantly better results, even on hard to colour graphs, when the vertices of the input graph are randomly named. On the other hand, the performance of our algorithm, MDsatur, becomes worse in this case, however it still has the best performance compared to the others. MDsatur colours the tested graphs with 1.1 OPT colours in most of the cases, even on hard instances, where OPT denotes the number of colours in an optimal colouring. 3) we construct worst case instances for the algorithm of Fotakis el al. [6], which show that its theoretical analysis is tight.

Abstract: In this paper we present a new approximation algorithm for
the Minimum Energy Broadcast Routing (MEBR) problem in ad hoc
wireless networks that has exponentially better approximation factor
than the well-known Minimum Spanning Tree (MST) heuristic. Namely,
for any instance where a minimum spanning tree of the set of stations
is guaranteed to cost at most ½ times the cost of an optimal solution
for MEBR, we prove that our algorithm achieves an approximation ra-
tio bounded by 2 ln ½ ¡ 2 ln 2 + 2. This result is particularly relevant for
its consequences on Euclidean instances where we signi¯cantly improve
previous results.

Abstract: In this paper we propose a new methodology for determining
approximate Nash equilibria of non-cooperative bimatrix games and,
based on that, we provide an efficient algorithm that computes 0.3393-
approximate equilibria, the best approximation till now. The methodology
is based on the formulation of an appropriate function of pairs of
mixed strategies reflecting the maximum deviation of the players¢ payoffs
from the best payoff each player could achieve given the strategy
chosen by the other. We then seek to minimize such a function using
descent procedures. As it is unlikely to be able to find global minima
in polynomial time, given the recently proven intractability of the problem,
we concentrate on the computation of stationary points and prove
that they can be approximated arbitrarily close in polynomial time and
that they have the above mentioned approximation property. Our result
provides the best till now for polynomially computable -approximate
Nash equilibria of bimatrix games. Furthermore, our methodology for
computing approximate Nash equilibria has not been used by others.

Abstract: Urban road networks are represented as directed graphs, accompanied by a metric which assigns cost functions (rather than scalars) to the arcs, e.g. representing time-dependent arc-traversal-times. In this work, we present oracles for providing time-dependent min-cost route plans, and conduct their experimental evaluation on a real-world data set (city of Berlin). Our oracles are based on precomputing all landmark-to-vertex shortest travel-time functions, for properly selected landmark sets. The core of this preprocessing phase is based on a novel, quite efficient and simple oneto-all approximation method for creating approximations of shortest travel-time functions. We then propose three query algorithms, including a PTAS, to efficiently provide mincost route plan responses to arbitrary queries. Apart from the purely algorithmic challenges, we deal also with several
implementation details concerning the digestion of raw traffic data, and we provide heuristic improvements of both the preprocessing phase and the query algorithms. We conduct an extensive, comparative experimental study with all query algorithms and six landmark sets. Our results are quite encouraging, achieving remarkable speedups (at least by two orders of magnitude) and quite small approximation guarantees, over the time-dependent variant of Dijkstra¢s algorithm.

Abstract: We present new combinatorial approximation algorithms for k-set cover. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend them by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k ≥ 6. The analysis technique could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program.

Abstract: We present new combinatorial approximation algorithms for
k-set cover. Previous approaches are based on extending the greedy al-
gorithm by e±ciently handling small sets. The new algorithms further
extend them by utilizing the natural idea of computing large packings
of elements into sets of large size. Our results improve the previously
best approximation bounds for the k-set cover problem for all values
of k ¸ 6. The analysis technique could be of independent interest; the
upper bound on the approximation factor is obtained by bounding the
objective value of a factor-revealing linear program.

Abstract: We study the following Constrained Bipartite Edge Coloring problem: We are given a bipartite graph G=(U,V,E) of maximum degree I with n vertices, in which some of the edges have been legally colored with c colors. We wish to complete the coloring of the edges of G minimizing the total number of colors used. The problem has been proved to be NP-hard even for bipartite graphs of maximum degree three. Two special cases of the problem have been previously considered and tight upper and ower bounds on the optimal number of colors were proved. The upper bounds led to 3/2-approximation algorithms for both problems. In this paper we present a randomized (1.37+o(1))-approximation algorithm for the general problem in the case where max{l,c} = {\`u}(ln n). Our techniques are motivated by recent works on the Circular Arc Coloring problem and are essentially different and simpler than the existing ones.

Abstract: Motivated by the wavelength assignment problem in WDM optical networks, we study path coloring problems in graphs. Given a set of paths P on a graph G, the path coloring problem is to color the paths of P so that no two paths traversing the same edge of G are assigned the same color and the total number of colors used is minimized. The problem has been proved to be NP-hard even for trees and rings.
Using optimal solutions to fractional path coloring, a natural relaxation of path coloring, on which we apply a randomized rounding technique combined with existing coloring algorithms, we obtain new upper bounds on the minimum number of colors sufficient to color any set of paths on any graph. The upper bounds are either existential or constructive.
The existential upper bounds significantly improve existing ones provided that the cost of the optimal fractional path coloring is sufficiently large and the dilation of the set of paths is small. Our algorithmic results include improved approximation algorithms for path coloring in rings and in bidirected trees. Our results extend to variations of the original path coloring problem arizing in multifiber WDM optical networks.

Abstract: We give a deterministic polynomial-time algorithm which for any given average degree d and asymptotically almost all random graphs G in G(n, m = [d/2 n]) outputs a cut of G whose ratio (in cardinality) with the maximum cut is at least 0.952. We remind the reader that it is known that unless P=NP, for no constant {\aa}>0 is there a Max-Cut approximation algorithm that for all inputs achieves an approximation ratio of (16/17) +{\aa} (16/17 < 0.94118).

Abstract: We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312--316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned 'fitness' value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r>0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when r≥1) and of extinction (for all r>0).

Abstract: We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some parameter k, e.g., committee elections. We are interested in the minimax approval voting rule in which the outcome represents a compromise among the voters, in the sense that the maximum distance between the preference of any voter and the outcome is as small as possible. This voting rule has two main drawbacks. First, computing an outcome that minimizes the maximum distance is computationally hard. Furthermore, any algorithm that always returns such an outcome provides
incentives to voters to misreport their true preferences.
In order to circumvent these drawbacks, we consider approximation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given instance. Such algorithms can be considered as alternative voting rules. We present a polynomial-time 2-approximation algorithm that uses a natural linear programming relaxation for the underlying optimization problem and deterministically
rounds the fractional solution in order to compute the outcome; this result improves upon the previously best known algorithm that has an approximation ratio of 3. We are furthermore interested in approximation algorithms that are resistant to manipulation by (coalitions of) voters, i.e., algorithms that do not motivate voters to misreport their true preferences in order to improve their distance from the outcome. We complement previous results in the literature with new upper and lower bounds on strategyproof and group-strategyproof algorithms.

Abstract: The study of the path coloring problem is motivated by the allocation of optical bandwidth to communication requests in all-optical networks that utilize Wavelength Division Multiplexing (WDM). WDM technology establishes communication between pairs of network nodes by establishing transmitter-receiver paths and assigning wavelengths to each path so that no two paths going through the same fiber link use the same wavelength. Optical bandwidth is the number of distinct wavelengths. Since state-of-the-art technology allows for a limited number of wavelengths, the engineering problem to be solved is to establish communication minimizing the total number of wavelengths used. This is known as the wavelength routing problem. In the case where the underlying network is a tree, it is equivalent to the path coloring problem.
We survey recent advances on the path coloring problem in both undirected and bidirected trees. We present hardness results and lower bounds for the general problem covering also the special case of sets of symmetric paths (corresponding to the important case of symmetric communication). We give an overview of the main ideas of deterministic greedy algorithms and point out their limitations. For bidirected trees, we present recent results about the use of randomization for path coloring and outline approximation algorithms that find path colorings by exploiting fractional path colorings. Also, we discuss upper and lower bounds on the performance of on-line algorithms.

Abstract: We examine the problem of assigning n independent jobs to m unrelated parallel machines, so that each job is processed without interruption on one of the machines, and at any time, every machine processes at most one job. We focus on the case where m is a fixed constant, and present a new rounding approach that yields approximation schemes for multi-objective minimum makespan scheduling with a fixed number of linear cost constraints. The same approach gives approximation schemes for covering problems like maximizing the minimum load on any machine, and for assigning specific or equal loads to the machines.

Abstract: This paper addresses stability issues in incremental induction of decision trees. Stability problems arise when an induction algorithm must revise a decision tree very often and oscillations between similar concepts decrease learning speed. We review a heuristic that solves this problem and subsequently employ asymptotic analysis to approximate the basic parameters related to the estimation of computational effort in incremental learning of decision trees. We then use these approximations to simplify the heuristic, we deliver insight into its amortizing behavior and argue how they can also speed-up its execution and enhance its applicability, also providing experimental evidence to support these claims.

Abstract: We present three new coordination mechanisms for schedul-
ing n sel¯sh jobs on m unrelated machines. A coordination
mechanism aims to mitigate the impact of sel¯shness of jobs
on the e±ciency of schedules by de¯ning a local schedul-
ing policy on each machine. The scheduling policies induce
a game among the jobs and each job prefers to be sched-
uled on a machine so that its completion time is minimum
given the assignments of the other jobs. We consider the
maximum completion time among all jobs as the measure
of the e±ciency of schedules. The approximation ratio of
a coordination mechanism quanti¯es the e±ciency of pure
Nash equilibria (price of anarchy) of the induced game. Our
mechanisms are deterministic, local, and preemptive in the
sense that the scheduling policy does not necessarily process
the jobs in an uninterrupted way and may introduce some
idle time. Our ¯rst coordination mechanism has approxima-
tion ratio O(logm) and always guarantees that the induced
game has pure Nash equilibria to which the system con-
verges in at most n rounds. This result improves a recent
bound of O(log2 m) due to Azar, Jain, and Mirrokni and,
similarly to their mechanism, our mechanism uses a global
ordering of the jobs according to their distinct IDs. Next
we study the intriguing scenario where jobs are anonymous,
i.e., they have no IDs. In this case, coordination mechanisms
can only distinguish between jobs that have diffeerent load
characteristics. Our second mechanism handles anonymous
jobs and has approximation ratio O
¡ logm
log logm
¢
although the
game induced is not a potential game and, hence, the exis-
tence of pure Nash equilibria is not guaranteed by potential
function arguments. However, it provides evidence that the
known lower bounds for non-preemptive coordination mech-
anisms could be beaten using preemptive scheduling poli-
cies. Our third coordination mechanism also handles anony-
mous jobs and has a nice \cost-revealing" potential func-
tion. Besides in proving the existence of equilibria, we use
this potential function in order to upper-bound the price of stability of the induced game by O(logm), the price of an-
archy by O(log2 m), and the convergence time to O(log2 m)-
approximate assignments by a polynomial number of best-
response moves. Our third coordination mechanism is the
¯rst that handles anonymous jobs and simultaneously guar-
antees that the induced game is a potential game and has
bounded price of anarchy.

Abstract: We call radiation at a point of a wireless network the total amount of electromagnetic quantity (energy or power density) the point is exposed to. The impact of radiation can be high and we believe it is worth studying and control; towards radiation aware wireless networking we take (for the first time in the study of this aspect) a distributed computing, algorithmic approach. We exemplify this line of research by focusing on sensor networks, studying the minimum radiation path problem of finding the lowest radiation trajectory of a person moving from a source to a destination point in the network region. For this problem, we sketch the main ideas behind a linear program that can provide a tight approximation of the optimal solution, and then we discuss three heuristics that can lead to low radiation paths. We also plan to investigate the impact of diverse node mobility to the heuristics' performance.

Abstract: Intuitively, Braess's paradox states that destroying a part
of a network may improve the common latency of selsh
ows at Nash
equilibrium. Such a paradox is a pervasive phenomenon in real-world
networks. Any administrator, who wants to improve equilibrium delays
in selsh networks, is facing some basic questions: (i) Is the network
paradox-ridden? (ii) How can we delete some edges to optimize equilibrium
ow delays? (iii) How can we modify edge latencies to optimize
equilibrium
ow delays?
Unfortunately, such questions lead to NP-hard problems in general. In
this work, we impose some natural restrictions on our networks, e.g.
we assume strictly increasing linear latencies. Our target is to formulate
ecient algorithms for the three questions above.We manage to provide:
{ A polynomial-time algorithm that decides if a network is paradoxridden,
when latencies are linear and strictly increasing.
{ A reduction of the problem of deciding if a network with arbitrary
linear latencies is paradox-ridden to the problem of generating all
optimal basic feasible solutions of a Linear Program that describes
the optimal trac allocations to the edges with constant latency.
{ An algorithm for nding a subnetwork that is almost optimal wrt
equilibrium latency. Our algorithm is subexponential when the number
of paths is polynomial and each path is of polylogarithmic length.
{ A polynomial-time algorithm for the problem of nding the best
subnetwork, which outperforms any known approximation algorithm
for the case of strictly increasing linear latencies.
{ A polynomial-time method that turns the optimal
ow into a Nash
ow by deleting the edges not used by the optimal
ow, and performing
minimal modications to the latencies of the remaining ones.
Our results provide a deeper understanding of the computational complexity
of recognizing the Braess's paradox most severe manifestations,
and our techniques show novel ways of using the probabilistic method
and of exploiting convex separable quadratic programs.

Abstract: Intuitively, Braess’s paradox states that destroying a part of a network may improve the common latency of selfish flows at Nash equilibrium. Such a paradox is a pervasive phenomenon in real-world networks. Any administrator who wants to improve equilibrium delays in selfish networks, is facing some basic questions:
– Is the network paradox-ridden?
– How can we delete some edges to optimize equilibrium flow delays?
– How can we modify edge latencies to optimize equilibrium flow delays?
Unfortunately, such questions lead to View the MathML sourceNP-hard problems in general. In this work, we impose some natural restrictions on our networks, e.g. we assume strictly increasing linear latencies. Our target is to formulate efficient algorithms for the three questions above. We manage to provide:
– A polynomial-time algorithm that decides if a network is paradox-ridden, when latencies are linear and strictly increasing.
– A reduction of the problem of deciding if a network with (arbitrary) linear latencies is paradox-ridden to the problem of generating all optimal basic feasible solutions of a Linear Program that describes the optimal traffic allocations to the edges with constant latency.
– An algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency. Our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length.
– A polynomial-time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies.
– A polynomial-time method that turns the optimal flow into a Nash flow by deleting the edges not used by the optimal flow, and performing minimal modifications on the latencies of the remaining ones.
Our results provide a deeper understanding of the computational complexity of recognizing the most severe manifestations of Braess’s paradox, and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs.

Abstract: We consider the problem of computing minimum congestion,
fault-tolerant, redundant assignments of messages to faulty parallel de-
livery channels. In particular, we are given a set M of faulty channels,
each having an integer capacity ci and failing independently with proba-
bility fi. We are also given a set of messages to be delivered over M, and
a fault-tolerance constraint (1), and we seek a redundant assignment
that minimizes congestion Cong(), i.e. the maximum channel load,
subject to the constraint that, with probability no less than (1 ), all
the messages have a copy on at least one active channel. We present a
4-approximation algorithm for identical capacity channels and arbitrary
message sizes, and a 2l ln(jMj=)
ln(1=fmax)m-approximation algorithm for related
capacity channels and unit size messages.
Both algorithms are based on computing a collection of disjoint chan-
nel subsets such that, with probability no less than (1 ), at least one
channel is active in each subset. The objective is to maximize the sum of
the minimum subset capacities. Since the exact version of this problem
is NP-complete, we present a 2-approximation algorithm for identical
capacities, and a (8 + o(1))-approximation algorithm for arbitrary ca-
pacities.

Abstract: We study the problem of localizing and tracking multiple moving targets in wireless sensor
networks, from a network design perspective i.e. towards estimating the least possible number
of sensors to be deployed, their positions and operation chatacteristics needed to perform the
tracking task. To avoid an expensive massive deployment, we try to take advantage of
possible coverage ovelaps over space and time, by introducing a novel combinatorial model
that captures such overlaps.
Under this model, we abstract the tracking network design problem by a combinatorial
problem of covering a universe of elements by at least three sets (to ensure that each point in
the network area is covered at any time by at least three sensors, and thus being localized). We
then design and analyze an efficient approximate method for sensor placement and operation,
that with high probability and in polynomial expected time achieves a (log n) approximation
ratio to the optimal solution. Our network design solution can be combined with alternative
collaborative processing methods, to suitably fit different tracking scenaria.

Abstract: We study the problem of localizing and tracking multiple moving targets in wireless sensor networks, from a network design perspective i.e. towards estimating the least possible number of sensors to be deployed, their positions and operation characteristics needed to perform the tracking task. To avoid an expensive massive deployment, we try to take advantage of possible coverage overlaps over space and time, by introducing a novel combinatorial model that captures such overlaps.
Under this model, we abstract the tracking network design problem by a combinatorial problem of covering a universe of elements by at least three sets (to ensure that each point in the network area is covered at any time by at least three sensors, and thus being localized). We then design and analyze an efficient approximate method for sensor placement and operation, that with high probability and in polynomial expected time achieves a {\`E}(logn) approximation ratio to the optimal solution. Our network design solution can be combined with alternative collaborative processing methods, to suitably fit different tracking scenarios.

Abstract: In this work we study the problem of scheduling tasks with dependencies in multiprocessor architectures where processors have different speeds.
We present the preemptive algorithm "Save-Energy" that given a schedule of tasks it post processes it to improve the energy efficiency without any deterioration of the makespan. In terms of time efficiency, we show that preemptive scheduling in an asymmetric system can achieve the same or better optimal makespan than in a symmetric system. Motivited by real multiprocessor systems, we investigate architectures that exhibit limited asymmetry: there are two essentially different speeds. Interestingly, this special case has not been studied in the field of parallel computing and scheduling theory; only the general case was studied where processors have K essentially different speeds. We present the non-preemptive algorithm "Remnants'' that achieves almost optimal makespan. We provide a refined analysis of a recent scheduling method. Based on this analysis, we specialize the scheduling policy and provide an algorithm of (3 + o(1)) expected approximation factor. Note that this improves the previous best factor (6 for two speeds). We believe that our work will convince researchers to revisit this well studied scheduling problem for these simple, yet realistic, asymmetric multiprocessor architectures.

Abstract: A crucial issue in wireless networks is to support efficiently communication patterns that are typical in traditional (wired) networks. These include broadcasting, multicasting, and gossiping (all-to-all communication). In this work we study such problems in static ad hoc networks. Since, in ad hoc networks, energy is a scarce resource, the important engineering question to be solved is to guarantee a desired communication pattern minimizing the total energy consumption. Motivated by this question, we study a series of wireless network design problems and present new approximation algorithms and inapproximability results.

Abstract: We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + epsi, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical.
Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + epsi. Furthermore, we prove that for any arbitrary integer k > 0, it is -hard to decide whether or not any given allocation of users to links can be transformed into a pure Nash equilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is -hard to approximate the worst social cost within a multiplicative factor . The quantity is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.

Abstract: In this work we experimentally study the min order Radiocoloring problem (RCP) on Chordal, Split and Permutation graphs, which are three basic families of perfect graphs. This problem asks to find an assignment using the minimum number of colors to the vertices of a given graph G, so that each pair of vertices which are at distance at most two apart in G have different colors. RCP is an NP-Complete problem on chordal and split graphs [4]. For each of the three families, there are upper bounds or/and approximation algorithms known for minimum number of colors needed to radiocolor such a graph [4,10].
We design and implement radiocoloring heuristics for graphs of above families, which are based on the greedy heuristic. Also, for each one of the above families, we investigate whether there exists graph instances requiring a number of colors in order to be radiocolored, close to the best known upper bound for the family. Towards this goal, we present a number generators that produce graphs of the above families that require either (i) a large number of colors (compared to the best upper bound), in order to be radiocolored, called ldquoextremalrdquo graphs or (ii) a small number of colors, called ldquonon-extremalrdquoinstances. The experimental evaluation showed that random generated graph instances are in the most of the cases ldquonon-extremalrdquo graphs. Also, that greedy like heuristics performs very well in the most of the cases, especially for ldquonon-extremalrdquo graphs.

Abstract: We study geometric versions of the min-size k-clustering
problem, a clustering problem which generalizes clustering to minimize
the sum of cluster radii and has important applications. We prove that
the problem can be solved in polynomial time when the points to be clus-
tered are located on a line. For Euclidean spaces of higher dimensions,
we show that the problem is NP-hard and present polynomial time ap-
proximation schemes. The latter result yields an improved approximation
algorithm for the related problem of k-clustering to minimize the sum of
cluster diameters.

Abstract: This paper presents results from the IST Phosphorus project that studies and implements an optical Grid test-bed. A significant part of this project addresses scheduling and routing algorithms and dimensioning problems of optical grids. Given the high costs involved in setting up actual hardware implementations, simulations are a viable alternative. In this paper we present an initial study which proposes models that reflect real-world grid application traffic characteristics, appropriate for simulation purposes. We detail several such models and the corresponding process to extract the model parameters from real grid log traces, and verify that synthetically generated jobs provide a realistic approximation of the real-world grid job submission process.

Abstract: In this paper we examine the problem of searching for some information item in the nodes of a fully
interconnected computer network, where each node contains information relevant to some topic
as well as links to other network nodes that also contain information, not necessarily related to
locally kept information. These links are used to facilitate the Internet users and mobile software
agents that try to locate specific pieces of information. However, the links do not necessarily point
to nodes containing information of interest to the user or relevant to the aims of the mobile agent.
Thus an element of uncertainty is introduced. For example, when an Internet user or some search
agent lands on a particular network node, they see a set of links that point to information that is,
supposedly, relevant to the current search. Therefore, we can assume that a link points to relevant
information with some unknown probability p that, in general, is related to the number of nodes
in the network (intuitively, as the network grows, this probability tends to zero since adding more
nodes to the network renders some extant links less accurate or obsolete). Consequently, since there
is uncertainty as to whether the links contained in a node?s Web page are correct or not, a search
algorithm cannot rely on following the links systematically since it may end up spending too much
time visiting nodes that contain irrelevant information. In this work, we will describe and analyze
a search algorithm that is only allowed to transfer a fixed amount of memory along communication
links as it visits the network nodes. The algorithm is, however, allowed to use one bit of memory at
each node as an ?already visited? flag. In this way the algorithm has its memory distributed to the
network nodes, avoiding overloading the network links as it moves from node to node searching for
the information. We work on fully interconnected networks for simplicity reasons and, moreover,
because according to some recent experimental evidence, such networks can be considered to be a
good approximation of the current structure of the World Wide Web.

Abstract: We consider the problem of computing minimum congestion, fault-tolerant, redundant assignments of messages to faulty, parallel delivery channels. In particular, we are given a set K of faulty channels, each having an integer capacity ci and failing independently with probability fi. We are also given a set M of messages to be delivered over K, and a fault-tolerance constraint (1 - {\aa}), and we seek a redundant assignment {\"o} that minimizes congestion Cong({\"o}), i.e. the maximum channel load, subject to the constraint that, with probability no less than (1 - e), all the messages have a copy on at least one active channel. We present a polynomial-time 4-approximation algorithm for identical capacity channels and arbitrary message sizes, and a 2[ln(|K|/{\aa})/ln(1/fmax)]-approximation algorithm for related capacity channels and unit size messages. Both algorithms are based on computing a collection (K1,., K{\'i}} of disjoint channel subsets such that, with probability no less than (1 - {\aa}), at least one channel is active in each subset. The objective is to maximize the sum of the minimum subset capacities. Since the exact version of this problem is NP-complete, we provide a 2-approximation algorithm for identical capacities, and a polynomial-time (8+o(1))-approximation algorithm for arbitrary capacities.

Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. The Radiocoloring (RC) of a graph G(V,E) is an assignment function Φ: V → IN such that ¦Φ(u)-Φ(v)≥ 2, when u; v are neighbors in G, and ¦Φ(u)-Φ(v)≥1 when the minimum distance of u; v in G is two. The discrete number and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (¦V¦ = n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with λ colors, in the case λ ≥ 4λ + 50.

Abstract: A constraint network is arc consistent if any value of any of its variables is compatible with at
least one value of any other variable. The Arc Consistency Problem (ACP) consists in filtering out values of
the variables of a given network to obtain one that is arc consistent, without eliminating any solution. ACP is
known to be inherently sequential, or P-complete, so in this paper we examine some weaker versions of it and
their parallel complexity. We propose several natural approximation schemes for ACP and show that they are also
P-complete. In an attempt to overcome these negative results, we turn our attention to the problem of filtering
out values from the variables so that each value in the resulting network is compatible with at least one value of
not necessarily all, but a constant fraction of the other variables. We call such a network partially arc consistent.
We give a parallel algorithm that, for any constraint network, outputs a partially arc consistent subnetwork of it in
sublinear (O.pn log n/) parallel time using O.n2/ processors. This is the first (to our knowledge) sublinear-time
parallel algorithm with polynomially many processors that guarantees that in the resulting network every value is
compatible with at least one value in at least a constant fraction of the remaining variables. Finally, we generalize
the notion of partiality to the k-consistency problem.

Abstract: The voting rules proposed by Dodgson and Young are both
designed to nd the alternative closest to being a Condorcet
winner, according to two dierent notions of proximity; the
score of a given alternative is known to be hard to compute
under either rule.
In this paper, we put forward two algorithms for ap-
proximating the Dodgson score: an LP-based randomized
rounding algorithm and a deterministic greedy algorithm,
both of which yield an O(logm) approximation ratio, where
m is the number of alternatives; we observe that this result
is asymptotically optimal, and further prove that our greedy
algorithm is optimal up to a factor of 2, unless problems in
NP have quasi-polynomial time algorithms. Although the
greedy algorithm is computationally superior, we argue that
the randomized rounding algorithm has an advantage from
a social choice point of view.
Further, we demonstrate that computing any reasonable
approximation of the ranking produced by Dodgson's rule
is NP-hard. This result provides a complexity-theoretic
explanation of sharp discrepancies that have been observed
in the Social Choice Theory literature when comparing
Dodgson elections with simpler voting rules.
Finally, we show that the problem of calculating the
Young score is NP-hard to approximate by any factor. This
leads to an inapproximability result for the Young ranking.

Abstract: We study the performance of approximate Nash equilibria for congestion
games with polynomial latency functions. We consider how much the price of anarchy
worsens and how much the price of stability improves as a function of the
approximation factor . We give tight bounds for the price of anarchy of atomic and
non-atomic congestion games and for the price of stability of non-atomic congestion
games. For the price of stability of atomic congestion games we give non-tight
bounds for linear latencies. Our results not only encompass and generalize the existing
results of exact equilibria to -Nash equilibria, but they also provide a unified
approach which reveals the common threads of the atomic and non-atomic price of
anarchy results. By expanding the spectrum, we also cast the existing results in a new
light.

Abstract: In this paper we present an implementation and performance evaluation of a descent algorithm that was proposed in \cite{tsaspi} for the computation of approximate Nash equilibria of non-cooperative bi-matrix games. This algorithm, which achieves the best polynomially computable \epsilon-approximate equilibria till now, is applied here to several problem instances designed so as to avoid the existence of easy solutions. Its performance is analyzed in terms of quality of approximation and speed of convergence. The results demonstrate significantly better performance than the theoretical worst case bounds, both for the quality of approximation and for the speed of convergence. This motivates further investigation into the intrinsic characteristics of descent algorithms applied to bi-matrix games. We discuss these issues and provide some insights about possible variations and extensions of the algorithmic concept that could lead to further understanding of the complexity of computing equilibria. We also prove here a new significantly better bound on the number of loops required for convergence of the descent algorithm.

Abstract: We focus on the problem of computing an epsilon (Porson)-Nash equilibrium of a bimatrix game, when epsilon (Porson) is an absolute constant. We present a simple algorithm for computing a View the MathML source-Nash equilibrium for any bimatrix game in strongly polynomial time and we next show how to extend this algorithm so as to obtain a (potentially stronger) parameterized approximation. Namely, we present an algorithm that computes a View the MathML source-Nash equilibrium, where {\"e} is the minimum, among all Nash equilibria, expected payoff of either player. The suggested algorithm runs in time polynomial in the number of strategies available to the players.

Abstract: We focus on the problem of computing an -Nash equilibrium
of a bimatrix game, when is an absolute constant. We present a simple
algorithm for computing a 3 -Nash equilibrium for any bimatrix game in
4
strongly polynomial time and we next show how to extend this algorithm
so as to obtain a (potentially stronger) parameterized approximation.
Namely, we present an algorithm that computes a 2+{\"e} -Nash equilibrium,
4
where {\"e} is the minimum, among all Nash equilibria, expected payoff of
either player. The suggested algorithm runs in time polynomial in the
number of strategies available to the players.

Abstract: We focus on the problem of computing an ²-Nash equilibrium
of a bimatrix game, when ² is an absolute constant. We present a simple
algorithm for computing a 3
4 -Nash equilibrium for any bimatrix game in
strongly polynomial time and we next show how to extend this algorithm
so as to obtain a (potentially stronger) parameterized approximation.
Namely, we present an algorithm that computes a 2+¸
4 -Nash equilibrium,
where ¸ is the minimum, among all Nash equilibria, expected payoff of
either player. The suggested algorithm runs in time polynomial in the
number of strategies available to the players.

Abstract: We consider in this paper the problem of scheduling a set of independent
parallel tasks (jobs) with respect to two criteria, namely,
the makespan (time of the last finishing job) and the minsum (average
completion time). There exist several algorithms with a good
performance guaranty for one of these criteria. We are interested
here in studying the optimization of both criteria simultaneously.
The numerical values are given for the moldable task model, where
the execution time of a task depends on the number of processors
alloted to it. The main result of this paper is to derive explicitly
a family of algorithms guaranteed for both the minsum and the
makespan. The performance guaranty of these algorithms is better
than the best algorithms known so far. The Guaranty curve
of the family is the set of all points (x; y) such that there is an
algorithm with guarantees x on makespan and y on the minsum.
When the ratio on the minsum increases, the curve tends to the
best ratio known for the makespan for moldable tasks (3=2). One
extremal point of the curves is a (3;6)-approximation algorithm.
Finally a randomized version is given, which improves this results
to (3;4.08).

Abstract: In this paper we study the problem of assigning transmission ranges to the nodes of a multihop
packet radio network so as to minimize the total power consumed under the constraint
that adequate power is provided to the nodes to ensure that the network is strongly connected
(i.e., each node can communicate along some path in the network to every other node). Such
assignment of transmission ranges is called complete. We also consider the problem of achieving
strongly connected bounded diameter networks.
For the case of n + 1 colinear points at unit distance apart (the unit chain) we give a tight
asymptotic bound for the minimum cost of a range assignment of diameter h when h is a xed
constant and when h>(1 + ) log n, for some constant > 0. When the distances between the
colinear points are arbitrary, we give an O(n4) time dynamic programming algorithm for nding
a minimum cost complete range assignment.
For points in three dimensions we show that the problem of deciding whether a complete
range assignment of a given cost exists, is NP-hard. For the same problem we give an O(n2)
time approximation algorithm which provides a complete range assignment with cost within a
factor of two of the minimum. The complexity of this problem in two dimensions remains open,
while the approximation algorithm works in this case as well.

Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function View the MathML source such that |{\"O}(u)-{\"O}(v)|greater-or-equal, slanted2, when u,v are neighbors in G, and |{\"O}(u)-{\"O}(v)|greater-or-equal, slanted1 when the distance of u,v in G is two. The number of discrete frequencies and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the radiocoloring problem for general graphs is hard to approximate (unless NP=ZPP) within a factor of n1/2-{\aa} (for any View the MathML source), where n is the number of vertices of the graph. However, when restricted to some special cases of graphs, the problem becomes easier. We prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(n{\"A}) time algorithm (|V|=n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where {\"A} the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with {\"e} colors, in the case where {\"e}greater-or-equal, slanted4{\"A}+50.

Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function View the MathML source such that |{\"O}(u)-{\"O}(v)|greater-or-equal, slanted2, when u,v are neighbors in G, and |{\"O}(u)-{\"O}(v)|greater-or-equal, slanted1 when the distance of u,v in G is two. The number of discrete frequencies and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the radiocoloring problem for general graphs is hard to approximate (unless NP=ZPP) within a factor of n1/2-{\aa} (for any View the MathML source), where n is the number of vertices of the graph. However, when restricted to some special cases of graphs, the problem becomes easier. We prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(n{\"A}) time algorithm (|V|=n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where {\"A} the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with {\"e} colors, in the case where {\"e}greater-or-equal, slanted4{\"A}+50.

Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function View the MathML source such that |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted2, when u,v are neighbors in G, and |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted1 when the distance of u,v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP).
In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodic graphs usually model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they can model very large networks produced by the repetition of a small graph.
A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph Gi(Vi,Ei). The edge set of G is derived by connecting the vertices of each iteration Gi to some of the vertices of the next iteration Gi+1, the same for all Gi. We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest.
We give two basic results:
• We prove that the min span RCP is PSPACE-complete for periodic planar graphs.
• We provide an O(n({\"A}(Gi)+{\'o})) time algorithm (where|Vi|=n, {\"A}(Gi) is the maximum degree of the graph Gi and {\'o} is the number of edges connecting each Gi to Gi+1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to View the MathML source as {\"A}(Gi)+{\'o} tends to infinity.
We remark that, any approximation algorithm for the min span RCP of a finite planar graph G, that achieves a span of at most {\'a}{\"A}(G)+constant, for any {\'a} and where {\"A}(G) is the maximum degree of G, can be used as a subroutine in our algorithm to produce an approximation for min span RCP of asymptotic ratio {\'a} for periodic planar graphs.

Abstract: The population protocol model (PP) proposed by Angluin et al. [2] describes sensor networks consisting of passively mobile finite-state agents. The agents sense their environment and communicate in pairs to carry out some computation on the sensed values. The mediated population protocol model (MPP) [13] extended the PP model by communication links equipped with a constant size buffer. The MPP model was proved in [13] to be stronger than the PP model. However, its most important contribution is that it provides us with the ability to devise optimizing protocols, approximation protocols and protocols that decide properties of the communication graph on which they run. The latter case, suggests a simplified model, the GDM model, that was formally defined and studied in [11]. GDM is a special case of MPP that captures MPP's ability to decide properties of the communication graph. Here we survey recent advances in the area initiated by the proposal of the PP model and at the same time we provide new protocols, novel ideas and results.

Abstract: Braess’s paradox states that removing a part of a network may im-
prove the players’ latency at equilibrium. In this work, we study the approxima-
bility of the best subnetwork problem for the class of random
G
n;p
instances
proven prone to Braess’s paradox by (Roughgarden and Valiant, RSA 2010) and
(Chung and Young, WINE 2010). Our main contribution is a polynomial-time
approximation-preserving reduction of the best subnetwork problem for such in-
stances to the corresponding problem in a simplified network where all neighbors
of
s
and
t
are directly connected by
0
latency edges. Building on this, we obtain
an approximation scheme that for any constant
" >
0
and with high probabil-
ity, computes a subnetwork and an
"
-Nash flow with maximum latency at most
(1+
"
)
L
+
"
, where
L
is the equilibrium latency of the best subnetwork. Our ap-
proximation scheme runs in polynomial time if the random network has average
degree
O
(poly(ln
n
))
and the traffic rate is
O
(poly(lnln
n
))
, and in quasipoly-
nomial time for average degrees up to
o
(
n
)
and traffic rates of
O
(poly(ln
n
))
.

Abstract: We study a problem of scheduling client requests to servers. Each client has a particular latency requirement at each server and may choose either to be assigned to some server in order to get serviced provided that her latency requirement is met, or not to participate in the assignment at all. From a global perspective, in order to optimize the performance of such a system, one would aim to maximize the number of clients that participate in the assignment. However, clients may behave selfishly in the sense that, each of them simply aims to participate in an assignment and get serviced by some server where her latency requirement is met with no regard to overall system performance. We model this selfish behavior as a strategic game, show how to compute pure Nash equilibria efficiently, and assess the impact of selfishness on system performance. We also show that the problem of optimizing performance is computationally hard to solve, even in a coordinated way, and present efficient approximation and online algorithms.

Abstract: We study a problem of scheduling client requests to servers.
Each client has a particular latency requirement at each server and may
choose either to be assigned to some server in order to get serviced provided
that her latency requirement is met or not to participate in the
assignment at all. From a global perspective, in order to optimize the
performance of such a system, one would aim to maximize the number
of clients that participate in the assignment. However, clients may behave
selfishly in the sense that each of them simply aims to participate
in an assignment and get serviced by some server where her latency requirement
is met with no regard to the overall system performance. We
model this selfish behavior as a strategic game, show how to compute
equilibria efficiently, and assess the impact of selfishness on system performance.
We also show that the problem of optimizing performance is
computationally hard to solve, even in a coordinated way, and present
efficient approximation and online algorithms.

Abstract: In 1876 Charles Lutwidge Dodgson suggested the intriguing voting rule that today bears his name. Although Dodgson's rule is one of the most well-studied voting rules, it suffers from serious deciencies, both from the computational point of view|it is NP-hard even to approximate the Dodgson score within sublogarithmic factors|and from the social choice point of view|it fails basic social choice desiderata such as monotonicity and homogeneity.
In a previous paper [Caragiannis et al., SODA 2009] we have asked whether there are approximation algorithms for Dodgson's rule that are monotonic or homogeneous. In this paper we give denitive answers to these questions. We design a monotonic exponential-time algorithm that yields a 2-approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomial-time O(logm)-approximation algorithm (where m is the number of alternatives); this result is tight as well due to a complexity-theoretic lower bound. Furthermore, we show that a slight variation of a known voting rule yields a monotonic, homogeneous, polynomial-time O(mlogm)-approximation algorithm, and establish that it is impossible to achieve a better approximation ratio even if one just asks for homogeneity. We complete the picture by studying several additional social choice properties; for these properties, we prove that algorithms with an approximation ratio that depends only on m do not exist.

Abstract: In 1876 Charles Lutwidge Dodgson suggested the intriguing
voting rule that today bears his name. Although Dodg-
son's rule is one of the most well-studied voting rules, it suf-
fers from serious deciencies, both from the computational
point of view|it is NP-hard even to approximate the Dodg-
son score within sublogarithmic factors|and from the social
choice point of view|it fails basic social choice desiderata
such as monotonicity and homogeneity.
In a previous paper [Caragiannis et al., SODA 2009] we
have asked whether there are approximation algorithms for
Dodgson's rule that are monotonic or homogeneous. In this
paper we give denitive answers to these questions. We de-
sign a monotonic exponential-time algorithm that yields a
2-approximation to the Dodgson score, while matching this
result with a tight lower bound. We also present a monotonic
polynomial-time O(logm)-approximation algorithm (where
m is the number of alternatives); this result is tight as well
due to a complexity-theoretic lower bound. Furthermore,
we show that a slight variation of a known voting rule yields
a monotonic, homogeneous, polynomial-time O(mlogm)-
approximation algorithm, and establish that it is impossible
to achieve a better approximation ratio even if one just asks
for homogeneity. We complete the picture by studying sev-
eral additional social choice properties; for these properties,
we prove that algorithms with an approximation ratio that
depends only on m do not exist.

Abstract: We consider the frugal coverage problem, an interesting vari-
ation of set cover de¯ned as follows. Instances of the problem consist of
a universe of elements and a collection of sets over these elements; the
objective is to compute a subcollection of sets so that the number of
elements it covers plus the number of sets not chosen is maximized. The
problem was introduced and studied by Huang and Svitkina [7] due to
its connections to the donation center location problem. We prove that
the greedy algorithm has approximation ratio at least 0:782, improving
a previous bound of 0:731 in [7]. We also present a further improvement
that is obtained by adding a simple corrective phase at the end of the
execution of the greedy algorithm. The approximation ratio achieved in
this way is at least 0:806. Our analysis is based on the use of linear
programs which capture the behavior of the algorithms in worst-case
examples. The obtained bounds are proved to be tight.

Abstract: In this work, we introduce the notion of time to some well-known combinatorial optimization problems. In particular, we study problems defined on temporal graphs. A temporal graph D=(V,A) may be viewed as a time-sequence G_1,G_2,...,G_l of static graphs over the same (static) set of nodes V. Each G_t = D(t) = (V,A(t)) is called the instance of D at time t and l is called the lifetime of D. Our main focus is on analogues of traveling salesman problems in temporal graphs. A sequence of time-labeled edges (e.g. a tour) is called temporal if its labels are strictly increasing. We begin by considering the problem of exploring the nodes of a temporal graph as soon as possible. In contrast to the positive results known for the static case, we prove that, it cannot be approximated within cn, for some constant c > 0, in general temporal graphs and within (2 − \varepsilon), for every constant \varepsilon > 0, in the special case in which D(t) is connected for all 1 <= t <= l, both unless P = NP. We then study the temporal analogue of TSP(1,2), abbreviated TTSP(1,2), where, for all 1 <= t <= l, D(t) is a complete weighted graph with edge-costs from {1,2} and the cost of an edge may vary from instance to instance. The goal is to find a minimum cost temporal TSP tour. We give several polynomial-time approximation algorithms for TTSP(1,2). Our best approximation is (1.7 + \varepsilon) for the generic TTSP(1,2) and (13/8 + \varepsilon) for its interesting special case in which the lifetime of the temporal graph is restricted to n. In the way, we also introduce temporal versions of Maximum Matching, Path Packing, Max-TSP, and Minimum Cycle Cover, for which we obtain polynomial-time approximation algorithms and hardness results.

Abstract: In view of the apparent intractability of constructing Nash Equilibria (NE
in short) in polynomial time, even for bimatrix games, understanding the limitations
of the approximability of the problem is an important challenge.
In this work we study the tractability of a notion of approximate equilibria in bimatrix
games, called well supported approximate Nash Equilibria (SuppNE in short).
Roughly speaking, while the typical notion of approximate NE demands that each
player gets a payoff at least an additive term less than the best possible payoff, in a
SuppNE each player is assumed to adopt with positive probability only approximate
pure best responses to the opponent¢s strategy.
As a first step, we demonstrate the existence of SuppNE with small supports and
at the same time good quality. This is a simple corollary of Alth{\"o}fer¢s Approximation
Lemma, and implies a subexponential time algorithm for constructing SuppNE of
arbitrary (constant) precision.
We then propose algorithms for constructing SuppNE in win lose and normalized
bimatrix games (i.e., whose payoff matrices take values from {0, 1} and [0, 1] respectively).
Our methodology for attacking the problem is based on the solvability of zero sum bimatrix games (via its connection to linear programming) and provides a
0.5-SuppNE for win lose games and a 0.667-SuppNE for normalized games.
To our knowledge, this paper provides the first polynomial time algorithms constructing
{\aa}-SuppNE for normalized or win lose bimatrix games, for any nontrivial
constant 0 ≤{\aa} <1, bounded away from 1.