Abstract: The 3-coloring problem is well known to be NP-complete. It is also well known that it
remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming
the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2
o
(
n
)
on graphs with
n
vertices and diameter at most 4. In spite of the extensive studies of the 3-coloring problem with
respect to several basic parameters, the complexity status of this problem on graphs with small
diameter, i.e. with diameter at most 2, or at most 3, has been a longstanding and challenging open
question. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2,
we provide the rst subexponential algorithm for 3-coloring, with complexity 2
O
(
p
n
log
n
)
, which is
asymptotically the same as the currently best known time complexity for the graph isomorphism
problem. Furthermore we extend the notion of an articulation vertex to that of an
articulation
neighborhood
, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2
that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish
the complexity of 3-coloring, even for the case of triangle-free graphs. Namely we prove that for
every
"
2
[0
;
1), 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with
n
vertices and minimum degree
=
(
n
"
). Moreover, assuming ETH, we use three di erent ampli cation
techniques of our hardness results, in order to obtain for every
"
2
[0
;
1) subexponential asymptotic
lowerbounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum
degree
=
(
n
"
). Finally, we provide a 3-coloring algorithm with running time 2
O
(min
f
;
n
log
g
)
for
arbitrary graphs with diameter 3, where
n
is the number of vertices and
(resp.
) is the minimum
(resp. maximum) degree of the input graph. To the best of our knowledge, this algorithm is the rst
subexponential algorithm for graphs with
=
!
(1) and for graphs with
=
O
(1) and
=
o
(
n
).
Due to the above lowerbounds of the complexity of 3-coloring, the running time of this algorithm is
asymptotically almost tight when the minimum degree of the input graph is
=
(
n
"
), where
"
2
[
1
2
;
1).
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 lowerbounds 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 lowerbounds 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 lowerbounds on the performance of on-line algorithms.
Abstract: We address an important communication issue arising in
wireless cellular networks that utilize frequency division
multiplexing (FDM) technology. In such networks, many
users within the same geographical region (cell) can communicate
simultaneously with other users of the network
using distinct frequencies. The spectrum of the available
frequencies is limited; thus, efficient solutions to the call
controlproblemareessential.Theobjectiveofthecallcontrol
problem is, given a spectrum of available frequencies
and users that wish tocommunicate, to maximize the benefit,
i.e., the number of users that communicate without
signalinterference.Weconsidercellularnetworksofreuse
distance k ≥ 2 and we study the online version of the
problem using competitive analysis. In cellular networks
of reuse distance 2, the previously best known algorithm
that beats the lower bound of 3 on the competitiveness
of deterministic algorithms, works on networks with one
frequency, achieves a competitive ratio against oblivious
adversaries, which is between 2.469 and 2.651, and uses
a number of random bits at least proportional to the size
of the network.We significantly improve this result by presentingaseriesofsimplerandomizedalgorithmsthathave
competitiveratiossignificantlysmallerthan3,workonnetworks
with arbitrarily many frequencies, and use only a
constant number of random bits or a comparable weak
random source. The best competitiveness upper bound
we obtain is 16/7 using only four random bits. In cellular
networks of reuse distance k > 2, we present simple
randomized online call control algorithms with competitive
ratios, which significantly beat the lowerbounds on
the competitiveness of deterministic ones and use only
O(log k )randombits. Also,weshownewlowerboundson
thecompetitivenessofonlinecallcontrolalgorithmsincellularnetworksofanyreusedistance.
Inparticular,weshow
thatnoonline algorithm can achieve competitive ratio better
than 2, 25/12, and 2.5, in cellular networks with reuse
distancek ∈ {2, 3, 4},k = 5,andk ≥ 6, respectively.
Abstract: Direct routing is the special case of bufferless routing where N packets, once injected into the network, must be routed along specific paths to their destinations without conflicts. We give a general treatment of three facets of direct routing: (i) Algorithms. We present a polynomial time greedy algorithm for arbitrary direct routing problems whch is worst-case optimal, i.e., there exist instances for which no direct routing algorithm is better than the greedy. We apply variants of this algorithm to commonly used network topologies. In particular, we obtain near-optimal routing time using for the tree and d-dimensional mesh, given arbitrary sources and destinations; for the butterfly and the hypercube, the same result holds for random destinations. (ii) Complexity. By a reduction from Vertex Coloring, we show that Direct Routing is inapproximable, unless P=NP. (iii) LowerBounds for Buffering. We show the existence of routing problems which cannot be solved efficiently with direct routing. To solve these problems, any routing algorithm needs buffers. We give nontrivial lowerbounds on such buffering requirements for general routing algorithms.
Abstract: Two important performance parameters of distributed, rate-based flow control algorithms are their locality and convergence complexity. The former is characterized by the amount of global knowledge that is available to their scheduling mechanisms, while the latter is defined as the number of update operations performed on rates of individual sessions until max-min fairness is reached. Optimistic algorithms allow any session to intermediately receive a rate larger than its max-min fair rate; bottleneck algorithms finalize the rate of a session only if it is restricted by a certain, highly congested link of the network. In this work, we present a comprehensive collection of lower and upper bounds on convergence complexity, under varying degrees of locality, for optimistic, bottleneck, rate-based flow control algorithms. Say that an algorithm is oblivious if its scheduling mechanism uses no information of either the session rates or the network topology. We present a novel, combinatorial construction of a capacitated network, which we use to establish a fundamental lower bound of dn 4 + n 2 on the convergence complexity of any oblivious algorithm, where n is the number of sessions laid out on a network, and d, the session dependency, is a measure of topological dependencies among sessions. Moreover, we devise a novel simulation proof to establish that, perhaps surprisingly, the lower bound of dn 4 + n 2 on convergence complexity still holds for any partially oblivious algorithm, in which the scheduling mechanism is allowed to use information about session rates, but is otherwise unaware of network topology. On the positive side, we prove that the lowerbounds for oblivious and partially oblivious algorithms are both tight. We do so by presenting optimal oblivious algorithms, which converge after dn 2 + n 2 update operations are performed in the worst case. To complete the picture, we show that linear convergence complexity can indeed be achieved if information about both session rates and network topology is available to schedulers. We present a counterexample, nonoblivious algorithm, which converges within an optimal number of n update operations. Our results imply a surprising convergence complexity collapse of oblivious and partially oblivious algorithms, and a convergence complexity separation between (partially) oblivious and nonoblivious algorithms for optimistic, bottleneck rate-based flow control.
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 lowerbounds 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: In this paper we consider communication issues arising in mobile networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of available frequencies is limited; thus, efficient solutions to the frequency allocation and the call control problem are essential. In the frequency allocation problem, given users that wish to communicate, the objective is to minimize the required spectrum of frequencies so that communication can be established without signal interference. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users served. We consider cellular, planar, and arbitrary network topologies. In particular, we study the on-line version of both problems using competitive analysis. For frequency allocation in cellular networks, we improve the best known competitive ratio upper bound of 3 achieved by the folklore Fixed Allocation algorithm, by presenting an almost tight competitive analysis for the greedy algorithm; we prove that its competitive ratio is between 2.429 and 2.5. For the call control problem, we present the first randomized algorithm that beats the deterministic lower bound of 3 achieving a competitive ratio of 2.934 in cellular networks. Our analysis has interesting extensions to arbitrary networks. Also, using Yao's Minimax Principle, we prove two lowerbounds of 1.857 and 2.086 on the competitive ratio of randomized call control algorithms for cellular and arbitrary planar networks, respectively.
Abstract: In this paper we consider communication issues arising in cellular (mobile) networks that utilize frequency division multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of available frequencies is limited; thus, efficient solutions to the frequency-allocation and the call-control problems are essential. In the frequency-allocation problem, given users that wish to communicate, the objective is to minimize the required spectrum of frequencies so that communication can be established without signal interference. The objective of the call-control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users served. We consider cellular, planar, and arbitrary network topologies.
In particular, we study the on-line version of both problems using competitive analysis. For frequency allocation in cellular networks, we improve the best known competitive ratio upper bound of 3 achieved by the folklore Fixed Allocation algorithm, by presenting an almost tight competitive analysis for the greedy algorithm; we prove that its competitive ratio is between 2.429 and 2.5 . For the call-control problem, we present the first randomized algorithm that beats the deterministic lower bound of 3 achieving a competitive ratio between 2.469 and 2.651 for cellular networks. Our analysis has interesting extensions to arbitrary networks. Also, using Yao's Minimax Principle, we prove two lowerbounds of 1.857 and 2.086 on the competitive ratio of randomized call-control algorithms for cellular and arbitrary planar networks, respectively.
Abstract: We consider the performance of a number of DPLL algorithms on random 3-CNF formulas with n variables and m = rn clauses. A long series of papers analyzing so-called “myopic” DPLL algorithms has provided a sequence of lowerbounds for their satisfiability threshold. Indeed, for each myopic algorithm A it is known that there exists an algorithm-specific clause-density, rA , such that if r 2.78 and the same is true for generalized unit clause for all r > 3.1. Our results imply exponential lowerbounds for many other myopic algorithms for densities similarly close to the corresponding rA .
Abstract: We consider the problem of searching for a piece of
information in a fully interconnected computer network
(also called a complete network or clique) by exploiting
advice about its location from the network nodes. Each
node contains a database that ?knows? what kind of
documents or information are stored in other nodes
(e.g., a node could be a Web server that answers queries
about documents stored on the Web). The databases in
each node, when queried, provide a pointer that leads to
the node that contains the information. However, this
information is up-to-date (or correct) with some
bounded probability. While, in principle, one may always
locate the information by simply visiting the network
nodes in some prescribed ordering, this requires a time
complexity in the order of the number of nodes of the
network. In this paper, we provide algorithms for locating
an information node in the complete communication
network, which take advantage of advice given from
network nodes. The nodes may either give correct advice,
by pointing directly to the information node, or give
wrong advice, by pointing elsewhere. On the lowerbounds?
side, we show that no fixed-memory (i.e., with
memory independent of the network size) deterministic
algorithm may locate the information node in a constant
(independent of the network size) expected number of
steps. Moreover, if p (1/n) is the probability that a
node of an n-node clique gives correct advice, we show
that no algorithm may locate the information node in an
expected number of steps less than 1/p o(1). To study
how the expected number of steps is affected by the
amount of memory allowed to the algorithms, we give a
memoryless randomized algorithm with expected number
of steps 4/p o(1/p) o(1) and a 1-bit randomized
algorithm requiring on the average at most 2/p o(1)
steps. In addition, in the memoryless case, we also
prove a 4/p lower bound for the expected number of
steps in the case where the nodes giving faulty advice
may decide on the content of this advice in any possible
way and not merely at random (adversarial fault model).
Finally, for the case where faulty nodes behave randomly,
we give an optimal, unlimited memory deterministic
algorithm with expected number of steps bounded
from above by 1/p o(1/p) 1.
Abstract: We consider the problem of searching for a piece of information in a fully interconnected computer network or clique by exploiting
advice about its location from the network nodes Each node contains a
database that knows what kind of documents or information are stored
in other nodes e.g. a node could be a Web server that answers queries
about documents stored on the Web. The databases in each node when
queried provide a pointer that leads to the node that contains the information. However this information is up to date or correct with some
bounded probability. While in principle one may always locate the information by simply visiting the network nodes in some prescribed ordering
this requires a time complexity in the order of the number of nodes of the
network. In this paper we provide algorithms for locating an information node in the complete communication network that take advantage
of advice given from network nodes The nodes may either give correct
advice by pointing directly to the information node or give wrong advice
by pointing elsewhere We show that on the averageif the probability that a node provides correct advice is asymptotically larger than
where is the number of the computer nodes then the average time complexity for locating the information node is asymptotically or depending on the available memory.The probability may in general be a function of the number of network nodes . On the lowerbounds
side we prove that noxed memory deterministic algorithm can locate
the information node in nite expected number of steps. We also prove
a lower bound of
for the expected number of steps of any algorithm
that locates the information node in the complete network.
Abstract: In this paper we derive lowerbounds for the degree of polynomials that approximate the square root of the discrete logarithm for Elliptic Curves with orders of various specific types. These bounds can serve as evidence for the difficulty in the computation of the square root of discrete logarithms for such elliptic curves, with properly chosen parameters that result in the curve having order of any of types studied in this paper. The techniques are potentially applicable to elliptic curves of order of any specific, allowable (by Hasse’s bounds), order type that is of interest for the application in hand.
Abstract: Flow control is the main technique currently used to prevent some of the ordered traffic from entering a communication network, and to avoid congestion. A challenging aspect of flow control is how to treat all sessions "fairly " when it is necessary to turn traffic away from the network. In this work, we show how to extend the theory of max-min fair flow control to the case where priorities are assigned to different varieties of traffic, which are sensitive to traffic levels. We examine priorities expressible in the general form of increasing functions of rates, considering yet in combination the more elaborative case with unescapable upper and lowerbounds on rates of traffic sessions. We offer optimal, priority bottleneck algorithms, which iteratively adjust the session rates in order to meet a new condition of max-min fairness under priorities and rate bounds. In our setting, which is realistic for today's technology of guaranteed quality of service, traffic may be turned away not only to avoid congestion, but also to respect particular minimum requirements on bandwidth. Moreover, we establish lowerbounds on the competitiveness of network-oblivious schemes compared to optimal schemes with complete knowledge of network structure. Our theory extends significantly the classical theory of max-min fair flow control [2]. Moreover, our results on rejected traffic are fundamentally different from those related to call control and bandwidth allocation, since not only do we wish to optimize the number and rates of accepted sessions, but we also require priority fairness.
Abstract: Evolutionary dynamics have been traditionally studied in the context of homogeneous populations, mainly described by the Moran process [15]. Recently, this approach has been generalized in [13] by arranging individuals on the nodes of a network (in general, directed). In this setting, the existence of directed arcs enables the simulation of extreme phenomena, where the fixation probability of a randomly placed mutant (i.e. the probability that the offsprings of the mutant eventually spread over the whole population) is arbitrarily small or large. On the other hand, undirected networks (i.e. undirected graphs) seem to have a smoother behavior, and thus it is more challenging to find suppressors/amplifiers of selection, that is, graphs with smaller/greater fixation probability than the complete graph (i.e. the homogeneous population). In this paper we focus on undirected graphs. We present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lowerbounds for the fixation
probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in [13]. In our new evolutionary model, all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. That is, the behavior of the individuals in our new model and in the model of [13] can be interpreted as an “aggregation” vs. an “all-or-nothing” strategy, respectively. We prove that our new model of mutual influences admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the “healthy” state of the system.
Abstract: We address the call control problem in wireless cellular networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region (cell) can communicate simultaneously with other users of the network using distinct frequencies. The available frequency spectrum is limited; hence, its management should be done efficiently. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate in a cellular network, to maximize the number of users that communicate without signal interference. We study the online version of the problem in cellular networks using competitive analysis and present new upper and lowerbounds.
Abstract: In sponsored search auctions, advertisers compete for a number
of available advertisement slots of different quality. The
auctioneer decides the allocation of advertisers to slots using
bids provided by them. Since the advertisers may act
strategically and submit their bids in order to maximize their
individual objectives, such an auction naturally defines a
strategic game among the advertisers. In order to quantify
the efficiency of outcomes in generalized second price auctions,
we study the corresponding games and present new
bounds on their price of anarchy, improving the recent results
of Paes Leme and Tardos [16] and Lucier and Paes
Leme [13]. For the full information setting, we prove a surprisingly
low upper bound of 1.282 on the price of anarchy
over pure Nash equilibria. Given the existing lowerbounds,
this bound denotes that the number of advertisers has almost
no impact on the price of anarchy. The proof exploits
the equilibrium conditions developed in [16] and follows by
a detailed reasoning about the structure of equilibria and a
novel relation of the price of anarchy to the objective value
of a compact mathematical program. For more general equilibrium
classes (i.e., mixed Nash, correlated, and coarse correlated
equilibria), we present an upper bound of 2.310 on
the price of anarchy. We also consider the setting where
advertisers have incomplete information about their competitors
and prove a price of anarchy upper bound of 3.037
over Bayes-Nash equilibria. In order to obtain the last two
bounds, we adapt techniques of Lucier and Paes Leme [13]
and significantly extend them with new arguments
Abstract: Random Intersection Graphs is a new class of random graphs introduced in [5], in which each of n vertices randomly and independently chooses some elements from a universal set, of cardinality m. Each element is chosen with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=n{\'a}, for any real {\'a} different than one, we establish here, for the first time, tight lowerbounds p0(n,m), on the value of p, as a function of n and m, above which the graph Gn,m,p is almost certainly Hamiltonian, i.e. it contains a Hamilton Cycle almost certainly. Our bounds are tight in the sense that when p is asymptotically smaller than p0(n,m) then Gn,m,p almost surely has a vertex of degree less than 2. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection model. Interestingly, Hamiltonicity appears well below the general thresholds, of [4], at which Gn,m,p looks like a usual random graph. Thus our bounds are much stronger than the trivial bounds implied by those thresholds.
Our results strongly support the existence of a threshold for Hamiltonicity in Gn,m,p.
Abstract: We study the on-line versions of two fundamental graph problems, maximum independent set and minimum coloring, for the case of disk graphs which are graphs resulting from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lowerbounds for deterministic on-line independent set algorithms and present new upper and lowerbounds; we also present an improved upper bound for on-line coloring.
Abstract: In this work, we study the impact of dynamically changing link capacities on the delay bounds of LIS (Longest-In-System) and SIS (Shortest-In-System) protocols on specific networks (that can be modelled as Directed Acyclic Graphs (DAGs)) and stability bounds of greedy contention–resolution protocols running on arbitrary networks under the Adversarial Queueing Theory. Especially, we consider the model of dynamic capacities, where each link capacity may take on integer values from [1,C] with C>1, under a (w,\~{n})-adversary. We show that the packet delay on DAGs for LIS is upper bounded by O(iw\~{n}C) and lower bounded by {\`U}(iw\~{n}C) where i is the level of a node in a DAG (the length of the longest path leading to node v when nodes are ordered by the topological order induced by the graph). In a similar way, we show that the performance of SIS on DAGs is lower bounded by {\`U}(iw\~{n}C), while the existence of a polynomial upper bound for packet delay on DAGs when SIS is used for contention–resolution remains an open problem. We prove that every queueing network running a greedy contention–resolution protocol is stable for a rate not exceeding a particular stability threshold, depending on C and the length of the longest path in the network.
Abstract: We study the on-line version of the maximum independent set problem, for the case of disk graphs which are graphs resulting
from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lowerbounds for deterministic on-line independent set algorithms and present new upper and lowerbounds.
Abstract: In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction, we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lowerbounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions.
Abstract: In this work, we study protocols (i.e. distributed algorithms) so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair, intuitively meaning that it must assign to every reachable configuration of the system a non-zero probability to occur. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. In particular, in every interaction, the protocol may activate an inactive connection, deactivate an active one, or leave the state of a connection unchanged. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network (i.e. one that does not change any more). We give protocols (optimal in some cases) and lowerbounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. Our universality protocols use a subset of the population (waste) in order to distributedly construct there a TM able to decide a graph class in some given space. Then, the protocols repeatedly construct in the rest of the population (useful space) a graph equiprobably drawn from all possible graphs. The TM works on this and accepts if the presented graph is in the class. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions. Delicate composition and reinitialization issues have to be solved for these general constructions to work.
Abstract: We address an important communication issue in wireless cellular networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region (cell) can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of the available frequencies is limited; thus, efficient solutions to the call control problem are essential. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users that communicate without signal interference. We consider cellular networks of reuse distance kge 2 and we study the on-line version of the problem using competitive analysis.
In cellular networks of reuse distance 2, the previously best known algorithm that beats the lower bound of 3 on the competitiveness of deterministic algorithms works on networks with one frequency, achieves a competitive ratio against oblivious adversaries which is between 2.469 and 2.651, and uses a number of random bits at least proportional to the size of the network. We significantly improve this result by presenting a series of simple randomized algorithms that have competitive ratios smaller than 3, work on networks with arbitrarily many frequencies, and use only a constant number of random bits or a comparable weak random source. The best competitiveness upper bound we obtain is 7/3.
In cellular networks of reuse distance k>2, we present simple randomized on-line call control algorithms with competitive ratios which significantly beat the lowerbounds on the competitiveness of deterministic ones and use only random bits. Furthermore, we show a new lower bound on the competitiveness of on-line call control algorithms in cellular networks of reuse distance kge 5.
Abstract: This work extends what is known so far for a basic model of
evolutionary antagonis
m in undirected ne
tworks (graphs).
More specif-
ically, this work studies the generalized Moran process, as introduced
by Lieberman, Hauert, and Nowak [Nature, 433:312-316, 2005], where
the individuals of a population reside on the vertices of an undirected
connected graph. The initial population has a single
mutant
of a
fitness
value
r
(typically
r>
1), residing at some vertex
v
of the graph, while
every other vertex is initially occupied by an individual of fitness 1. At
every step of this process, an individual (i.e. vertex) is randomly chosen
for reproduction with probability proportional to its fitness, and then it
places a copy of itself on a random neighbor, thus replacing the individ-
ual that was residing there. The main quantity of interest is the
fixation
probability
, i.e. the probability that eventually the whole graph is occu-
pied by descendants of the mutant. In this work we concentrate on the
fixation probability when the mutant is initially on a specific vertex
v
,
thus refining the older notion of Lieberman et al. which studied the fix-
ation probability when the initial mutant is placed at a random vertex.
We then aim at finding graphs that have many “strong starts” (or many
“weak starts”) for the mutant. Thus we introduce a parameterized no-
tion of
selective amplifiers
(resp.
selective suppressors
)ofevolution.We
prove the existence of
strong
selective amplifiers (i.e. for
h
(
n
)=
Θ
(
n
)
vertices
v
the fixation probability of
v
is at least 1
−
c
(
r
)
n
for a func-
tion
c
(
r
) that depends only on
r
), and the existence of quite strong
selective suppressors. Regarding the traditional notion of fixation prob-
ability from a random start, we provi
de strong upper and lowerbounds:
first we demonstrate the non-existence of “strong universal” amplifiers,
and second we prove the
Thermal Theorem
which states that for any
undirected graph, when the mutant starts at vertex
v
, the fixation prob-
ability at least (
r
−
1)
/
(
r
+
deg
v
deg
min
). This theorem (which extends the
“Isothermal Theorem” of Lieberman et al. for regular graphs) implies
an almost tight lower bound for the usual notion of fixation probability.
Our proof techniques are original and are based on new domination ar-
guments which may be of general interest in Markov Processes that are
of the general birth-death type.
Abstract: In this work we consider temporal networks, i.e. networks defined by a labeling $\lambda$ assigning to each edge of an underlying graph G a set of discrete time-labels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on path problems of temporal networks. In particular, we consider time-respecting paths, i.e. paths whose edges are assigned by $\lambda$ a strictly increasing sequence of labels. We begin by giving two efficient algorithms for computing shortest time-respecting paths on a temporal network. We then prove that there is a natural analogue of Menger’s theorem holding for arbitrary temporal networks. Finally, we propose two cost minimization parameters for temporal network design. One is the temporality of G, in which the goal is to minimize the maximum number of labels of an edge, and the other is the temporal cost of G, in which the goal is to minimize the total number of labels used. Optimization of these parameters is performed subject to some connectivity constraint. We prove several lower and upper bounds for the temporality and the temporal cost of some very basic graph families such as rings, directed acyclic graphs, and trees.
Abstract: We consider non-cooperative unsplittable congestion games where players share resources, and each player's strategy is pure and consists of a subset of the resources on which it applies a fixed weight. Such games represent unsplittable routing flow games and also job allocation games. The congestion of a resource is the sum of the weights of the players that use it and the player's cost function is the sum of the utilities of the resources on its strategy. The social cost is the total weighted sum of the player's costs. The quality of Nash equilibria is determined by the price of anarchy (PoA) which expresses how much worse is the social outcome in the worst equilibrium versus the optimal coordinated solution. In the literature the predominant work has only been on games with polynomial utility costs, where it has been proven that the price of anarchy is bounded by the degree of the polynomial. However, no results exist on general bounds for non-polynomial utility functions.
Here, we consider general versions of these games in which the utility of each resource is an arbitrary non-decreasing function of the congestion. In particular, we consider a large family of superpolynomial utility functions which are asymptotically larger than any polynomial. We demonstrate that for every such function there exist games for which the price of anarchy is unbounded and increasing with the number of players (even if they have infinitesimal weights) while network resources remain fixed. We give tight lower and upper bounds which show this dependence on the number of players. Furthermore we provide an exact characterization of the PoA of all congestion games whose utility costs are bounded above by a polynomial function. Heretofore such results existed only for games with polynomial cost functions.
Abstract: One of the most challenging problems in probability and complexity theory is
to establish and determine the satisfiability threshold, or phase transition, for
random k-SAT instances: Boolean formulas consisting of clauses with exactly k
literals. As the previous part of the volume has explored, empirical observations
suggest that there exists a critical ratio of the number of clauses to the number
of variables, such that almost all randomly generated formulas with a higher
ratio are unsatisfiable while almost all randomly generated formulas with a lower
ratio are satisfiable. The statement that such a crossover point really exists is
called the satisfiability threshold conjecture. Experiments hint at such a direction,
but as far as theoretical work is concerned, progress has been difficult. In an
important advance, Friedgut [23] showed that the phase transition is a sharp one,
though without proving that it takes place at a “fixed” ratio for large formulas.
Otherwise, rigorous proofs have focused on providing successively better upper
and lowerbounds for the value of the (conjectured) threshold. In this chapter, our
Abstract: The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable
ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up
to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper,
we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own
right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for
the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability
threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the
binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally,
to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic
models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for
unconditional probabilities, which were previously known.
Abstract: We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a WDM optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of w wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring problem (maxRPC).
Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain bounds on the approximability of several algorithms. Our results also apply to the problem maxPC where paths instead of connections are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit.
