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: 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 approximationalgorithms 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 approximationalgorithms 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-approximationalgorithms 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 approximationalgorithms 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 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 approximationalgorithms, 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 approximationalgorithms 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 approximationalgorithms that find path colorings by exploiting fractional path colorings. Also, we discuss upper and lower bounds on the performance of on-line algorithms.

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: 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 approximationalgorithms and inapproximability results.

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 approximationalgorithms 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: 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: 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 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: 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 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: 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 approximationalgorithms 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 approximationalgorithms 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 approximationalgorithms 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 approximationalgorithms 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.