Population protocol, Diffuse computation, Sensor networks, Passive mobility, Computationalcomplexity, Space hierarchy, Symmetric computation, Complexity class

Abstract: Implementation of a commercial application to a
grid infrastructure introduces new challenges in managing the
quality-of-service (QoS) requirements, most stem from the fact
that negotiation on QoS between the user and the service provider
should strictly be satisfied. An interesting commercial application
with a wide impact on a variety of fields, which can benefit from
the computational grid technologies, is three–dimensional (3-D)
rendering. In order to implement, however, 3-D rendering to a
grid infrastructure, we should develop appropriate scheduling
and resource allocation mechanisms so that the negotiated (QoS)
requirements are met. Efficient scheduling schemes require
modeling and prediction of rendering workload. In this paper
workload prediction is addressed based on a combined fuzzy
classification and neural network model. Initially, appropriate
descriptors are extracted to represent the synthetic world. The
descriptors are obtained by parsing RIB formatted files, which
provides a general structure for describing computer-generated
images. Fuzzy classification is used for organizing rendering
descriptor so that a reliable representation is accomplished which
increases the prediction accuracy. Neural network performs
workload prediction by modeling the nonlinear input-output
relationship between rendering descriptors and the respective
computationalcomplexity. To increase prediction accuracy, a
constructive algorithm is adopted in this paper to train the neural
network so that network weights and size are simultaneously
estimated. Then, a grid scheduler scheme is proposed to estimate
the queuing order that the tasks should be executed and the
most appopriate processor assignment so that the demanded
QoS are satisfied as much as possible. A fair scheduling policy is
considered as the most appropriate. Experimental results on a real
grid infrastructure are presented to illustrate the efficiency of the
proposed workload prediction — scheduling algorithm compared
to other approaches presented in the literature.

Abstract: The inference problem for propositional circumscription is known to
be highly intractable and, in fact, harder than the inference problem for classi-
cal propositional logic. More precisely, in its full generality this problem is P - 2
complete, which means that it has the same inherent computationalcomplexity
as the satisfiability problem for quantified Boolean formulas with two alternations
(universal-existential) of quantifiers. We use Schaefer?s framework of generalized
satisfiability problems to study the family of all restricted cases of the inference
problem for propositional circumscription. Our main result yields a complete clas-
sification of the ?truly hard? ( P -complete) and the ?easier? cases of this problem
2
(reducible to the inference problem for classical propositional logic). Specifically,
we establish a dichotomy theorem which asserts that each such restricted case either
is P -complete or is in coNP. Moreover, we provide efficiently checkable criteria
2
that tell apart the ?truly hard? cases from the ?easier? ones. We show our results both
when the formulas involved are and are not allowed to contain constants. The present
work complements a recent paper by the same authors, where a complete classifi-
cation into hard and easy cases of the model-checking problem in circumscription
was established.

Abstract: This chapter is an introduction to the basic concepts and advances of a new field, that of Computational (or Algorithmic) Game Theory. We study the computationalcomplexity of Nash equilibria and review the related algorithms proposed in the literature. Then, given the apparent difficulty of computing exact Nash equilibria, we study the efficient computation of approximate notions of Nash equilibria. Next we deal with several computational issues related to the class of congestion games, which model the selfish behavior of individuals when competing on the usage of a common set of resources. Finally, we study the price of anarchy (in the context of congestion games), which is defined as a measure of the performance degradation due to the the lack of coordination among the involved players.

Abstract: Wireless Sensor Networks consist of a large number of small, autonomous devices, that are able to interact with their inveronment by sensing and collaborate to fulfill their tasks, as, usually, a single node is incapable of doing so; and they use wireless communication to enable this collaboration. Each device has limited computational and energy resources, thus a basic issue in the applicastions of wireless sensor networks is the low energy consumption and hence, the maximization of the network lifetime.
The collected data is disseminated to a static control point – data sink in the network, using node to node - multi-hop data propagation. However, sensor devices consume significant amounts of energy in addition to increased implementation complexity, since a routing protocol is executed. Also, a point of failure emerges in the area near the control center where nodes relay the data from nodes that are farther away. Recently, a new approach has been developed that shifts the burden from the sensor nodes to the sink. The main idea is that the sink has significant and easily replenishable energy reserves and can move inside the area the sensor network is deployed, in order to acquire the data collected by the sensor nodes at very low energy cost. However, the need to visit all the regions of the network may result in large delivery delays.
In this work we have developed protocols that control the movement of the sink in wireless sensor networks with non-uniform deployment of the sensor nodes, in order to succeed an efficient (with respect to both energy and latency) data collection. More specifically, a graph formation phase is executed by the sink during the initialization: the network area is partitioned in equal square regions, where the sink, pauses for a certain amount of time, during the network traversal, in order to collect data.
We propose two network traversal methods, a deterministic and a random one. When the sink moves in a random manner, the selection of the next area to visit is done in a biased random manner depending on the frequency of visits of its neighbor areas. Thus, less frequently visited areas are favored. Moreover, our method locally determines the stop time needed to serve each region with respect to some global network resources, such as the initial energy reserves of the nodes and the density of the region, stopping for a greater time interval at regions with higher density, and hence more traffic load. In this way, we achieve accelerated coverage of the network as well as fairness in the service time of each region.Besides randomized mobility, we also propose an optimized deterministic trajectory without visit overlaps, including direct (one-hop) sensor-to-sink data transmissions only.
We evaluate our methods via simulation, in diverse network settings and comparatively to related state of the art solutions. Our findings demonstrate significant latency and energy consumption improvements, compared to previous research.

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 computationalcomplexity
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 computationalcomplexity 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: In this work, we propose an energy-efficient multicasting algorithm
for wireless networks for the case where the transmission
powers of the nodes are fixed. Our algorithm is
based on the multicost approach and selects an optimal
energy-efficient set of nodes for multicasting, taking into account:
i) the node residual energies, ii) the transmission
powers used by the nodes, and iii) the set of nodes covered.
Our algorithm is optimal, in the sense that it can
optimize any desired function of the total power consumed
by the multicasting task and the minimum of the current
residual energies of the nodes, provided that the optimization
function is monotonic in each of these parameters. Our
optimal algorithm has non-polynomial complexity, thus, we
propose a relaxation producing a near-optimal solution in
polynomial time. The performance results obtained show
that the proposed algorithms outperform established solutions
for energy-aware multicasting, with respect to both
energy consumption and network lifetime. Moreover, it is
shown that the near-optimal multicost algorithm obtains
most of the performance benefits of the optimal multicost
algorithm at a smaller computational overhead.

Abstract: We study the combinatorial structure and computationalcomplexity 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: The Internet and the Web have arguably surpassed the von Neumann Computer as the most complex computational artifacts of our times. Out of all the formidable characteristics of the Internet/Web, it seems that the most novel is its socio-economic complexity. The Internet and the Web are built, operated and used by a multitude of very diverse economic interests, which compete or collaborate in various degrees. In fact, this suggests that some very important insights about the Web may come from a fusion of ideas from Algorithms with concepts and techniques from Mathematical Economics and Game Theory. Since 2000, a new field has emerged (Algorithmic Game Theory and Computational Internet Economics), which examines exactly such ideas. We feel that this field belongs to Web Science. Some of the main topics of that field examined so far are Internet equilibria, the so called “Price of Anarchy” (a measure of loss of optimality due to selfishness [Koutsoupias,Papadimitriou (1999)], electronic markets and their equilibria, auctions, and algorithmic mechanisms design (inverse game theory or how to design a game in the net in such a clever way that individual players, motivated solely by their selfish interests, actually end up meeting the goals of the game designer!). Our paper here surveys the main concepts and results of this very promising subfield.

Abstract: We propose efficient schemes for information-theoretically secure
key exchange in the Bounded Storage Model (BSM), where the adversary
is assumed to have limited storage. Our schemes generate a
secret One Time Pad (OTP) shared by the sender and the receiver,
from a large number of public random bits produced by the sender
or by an external source. Our schemes initially generate a small
number of shared secret bits, using known techniques. We introduce
a new method to expand a small number of shared bits to a
much longer, shared key.
Our schemes are tailored to the requirements of sensor nodes
and wireless networks. They are simple, efficient to implement and
take advantage of the fact that practical wireless protocols transmit
data in frames, unlike previous protocols, which assume access to
specific bits in a stream of data. Indeed, our main contribution is
twofold.
On the one hand, we construct schemes that are attractive in
terms of simplicity, computationalcomplexity, number of bits read
from the shared random source and expansion factor of the initial
key to the final shared key.
On the other hand, we show how to transformany existing scheme
for key exchange in BSM into a more efficient scheme in the number
of bits it reads from the shared source, given that the source is
transmitted in frames.

Abstract: In this paper we propose an energy-aware broadcast algorithm for wireless networks. Our algorithm is based on the multicost approach and selects the set of nodes that by transmitting implement broadcasting in an optimally energy-efficient way. The energy-related parameters taken into account are the node transmission power and the node residual energy. The algorithm{\^a}€™s complexity however is non-polynomial, and therefore, we propose a relaxation producing a near-optimal solution in polynomial time. We also consider a distributed information exchange scheme that can be coupled with the proposed algorithms and examine the overhead introduced by this integration. Using simulations we show that the proposed algorithms outperform other solutions in the literature in terms of energy efficiency. Moreover, it is shown that the near-optimal algorithm obtains most of the performance benefits of the optimal algorithm at a smaller computational overhead.

Abstract: A key problem in Grid networks is how to efficiently manage the available infrastructure, in order to
satisfy user requirements and maximize resource utilization. This is in large part influenced by the
algorithms responsible for the routing of data and the scheduling of tasks. In this paper,wepresent several
multi-cost algorithms for the joint scheduling of the communication and computation resources that
will be used by a Grid task. We propose a multi-cost scheme of polynomial complexity that performs
immediate reservations and selects the computation resource to execute the task and determines the
path to route the input data. Furthermore, we introduce multi-cost algorithms that perform advance
reservations and thus also find the starting times for the data transmission and the task execution. We
initially present an optimal scheme of non-polynomial complexity and by appropriately pruning the set
of candidate paths we also give a heuristic algorithm of polynomial complexity. Our performance results
indicate that in a Grid network in which tasks are either CPU- or data-intensive (or both), it is beneficial
for the scheduling algorithm to jointly consider the computational and communication problems. A
comparison between immediate and advance reservation schemes shows the trade-offs with respect to
task blocking probability, end-to-end delay and the complexity of the algorithms.

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 routing games, the network performance at equilibrium can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braess's paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve the equilibrium performance by edge removal. In this work, we investigate the computationalcomplexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network. We first show that bottleneck routing games do not suffer from Braess's paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA is as large as \Omega(n^{0.121}) and cannot improve by edge removal. Thus, the network design problem for such games is NP-hard to approximate within a factor of O(n^{0.121-\eps}), for any constant \eps > 0. On the positive side, we show how to compute an almost optimal subnetwork w.r.t. the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. The running time is determined by the total number of paths, and is quasipolynomial when the number of paths is quasipolynomial.

Abstract: In this paper we propose an energy-efficient broadcast algorithm for wireless networks for the case where the transmission powers of the nodes are fixed. Our algorithm is based on the multicost approach and selects an optimal energy-efficient set of nodes for broadcasting, taking into account: i) the node residual energies, ii) the transmission powers used by the nodes, and iii) the set of nodes that are covered by a specific schedule. Our algorithm is optimal, in the sense that it can optimize any desired function of the total power consumed by the broadcasting task and the minimum of the current residual energies of the nodes, provided that the optimization function is monotonic in each of these parameters. Our algorithm has non-polynomial complexity, thus, we propose a relaxation producing a near-optimal solution in polynomial time. Using simulations we show that the proposed algorithms outperform other established solutions for energy-aware broadcasting with respect to both energy consumption and network lifetime. Moreover, it is shown that the near-optimal multicost algorithm obtains most of the performance benefits of the optimal multicost algorithm at a smaller computational overhead.

Abstract: In recent years there has been signi1cant interest in the study of random k-SAT formulae. For
a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct,
non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk (n;m) is
formed by selectinguniformly and independently m clauses from Bk and takingtheir conjunction.
Motivated by insights from statistical mechanics that suggest a possible relationship between the
?order? of phase transitions and computationalcomplexity, Monasson and Zecchina (Phys. Rev.
E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p ¸ [0; 1], a random
(2 + p)-SAT formula, F2+p(n;m), has m randomly chosen clauses over n variables, where pm
clauses are chosen from B3 and (1 − p)m from B2. Usingthe heuristic ?replica method? of
statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the
behavior of random (2 + p)-SAT formulae. In this paper we give the 1rst rigorous results for
random (2 + p)-SAT, includingthe followingsurprisingfact: for p 6 2=5, with probability
1 − o(1), a random (2 + p)-SAT formula is satis1able i@ its 2-SAT subformula is satis1able.
That is, for p 6 2=5, random (2 + p)-SAT behaves like random 2-SAT.

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: In emerging pervasive scenarios, data is collected by sensing devices in streams that occur at several distributed points of observation. The size of the data typically far exceeds the storage and computational capabilities of the tiny devices that have to collect and process them. A general and challenging task is to allow (some of) the nodes of a pervasive network to collectively perform monitoring of a neighbourhood of interest by issuing continuous aggregate queries on the streams observed in its vicinity. This class of algorithms is fully decentralized and diffusive in nature: collecting all the data at a few central nodes of the network is unfeasible in networks of low capability devices or in the presence of massive data sets. Two main problems arise in this scenario: (i) the intrinsic complexity of maintaining statistics over a data stream whose size greatly exceeds the capabilities of the device that performs the computation; (ii) composing the partial outcomes computed at different points of observation into an accurate, global statistic over a neighbourhood of interest, which entails coping with several problems, last but not least the receipt of duplicate information along multiple paths of diffusion.
Streaming techniques have emerged as powerful tools to achieve the general goals described above, in the first place because they assume a computational model in which computational and storage resources are assumed to be far exceeded by the amount of data on which computation occurs. In this contribution, we review the main streaming techniques and provide a classification of the computational problems and the applications they effectively address, with an emphasis on decentralized scenarios, which are of particular interest in pervasive networks

Abstract: A dichotomy theorem for a class of decision problems is a result asserting that certain problems in the
class are solvable in polynomial time, while the rest are NP-complete. The first remarkable such dichotomy
theorem was proved by Schaefer in 1978. It concerns the class of generalized satisfiability problems Sat?S{\L},
whose input is a CNF?S{\L}-formula, i.e., a formula constructed from elements of a fixed set S of generalized
connectives using conjunctions and substitutions by variables. Here, we investigate the complexity of
minimal satisfiability problems Min Sat?S{\L}, where S is a fixed set of generalized connectives. The input to
such a problem is a CNF?S{\L}-formula and a satisfying truth assignment; the question is to decide whether
there is another satisfying truth assignment that is strictly smaller than the given truth assignment with
respect to the coordinate-wise partial order on truth assignments. Minimal satisfiability problems were first
studied by researchers in artificial intelligence while investigating the computationalcomplexity of prop-
ositional circumscription. The question of whether dichotomy theorems can be proved for these problems
was raised at that time, but was left open. We settle this question affirmatively by establishing a dichotomy
theorem for the class of all Min Sat?S{\L}-problems, where S is a finite set of generalized connectives. We also
prove a dichotomy theorem for a variant of Min Sat?S{\L} in which the minimization is restricted to a subset of
the variables, whereas the remaining variables may vary arbitrarily (this variant is related to extensions of
propositional circumscription and was first studied by Cadoli). Moreover, we show that similar dichotomy
theorems hold also when some of the variables are assigned constant values. Finally, we give simple criteria that tell apart the polynomial-time solvable cases of these minimal satisfiability problems from the NP-
complete ones.

Abstract: Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfish routing? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93-102]. A Leader can decrease the coordination ratio by assigning flow {\'a}r on M, and then all Followers assign selfishly the (1-{\'a})r remaining flow. This is a Stackelberg Scheduling Instance(M,r,{\'a}),0≤{\'a}≤1. It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104-113] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion @b"M of flow r>0 needed by a Leader to induce M's optimum cost, as well as her optimal strategy. This shows that the optimal Leader's strategy on instances (M,r,@a>=@b"M) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling {\'a}r flow can induce ≤1/{\'a} times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19-28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on ComputationalComplexity, ECCC, (056), 2005].

Abstract: In this work, we study the combinatorial structure and the
computationalcomplexity of Nash equilibria for a certain game that
models selfish routing over a network consisting of m parallel links. We
assume a collection of n users, each employing a mixed strategy, which
is a probability distribution over links, to control the routing of its own
assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic
on those links that minimize its expected latency cost, given the network
congestion caused by the other users. The social cost of a Nash equilibrium
is the expectation, over all random choices of the users, of the
maximum, over all links, latency through a link.
We embark on a systematic study of several algorithmic problems related
to the computation of Nash equilibria for the selfish routing game we consider.
In a nutshell, these problems relate to deciding the existence of a
Nash equilibrium, constructing a Nash equilibrium with given support
characteristics, constructing the worst Nash equilibrium (the one with
maximum social cost), constructing the best Nash equilibrium (the one
with minimum social cost), or computing the social cost of a (given) Nash
equilibrium. Our work provides a comprehensive collection of efficient algorithms,
hardness results (both as NP-hardness and #P-completeness
results), and structural results for these algorithmic problems. Our results
span and contrast a wide range of assumptions on the syntax of the
Nash equilibria and on the parameters of the system.