 July 9–13, 2007
Wrocław — Poland
ICALP 2007
34th International Colloquium on
Automata, Languages and Programming
Colocated with LICS 2007, LC 2007, and PPDP 2007   ## List of accepted papers

(In order of appearance)

### Track A: Algorithms, Automata, Complexity and Games

Competitive Algorithms for Due Date Scheduling
Nikhil Bansal, Ho-Leung Chan and Kirk Pruhs

We consider several online scheduling problems that arise when customers request make-to-order products from a company. At the time of the order, the company must quote a due date to the customer. To satisfy the customer, the company must produce the good by the due date. The company must have an online algorithm with two components: The first component sets the due dates, and the second component schedules the resulting jobs with the goal of meeting the due dates.

The most basic quality of service measure for a job is the quoted lead time, which is the difference between the due date and the release time. We first consider the basic problem of minimizing the average quoted lead time. We show that there is a 1+ε-speed O(log k / ε)-competitive algorithm for this problem (here k is the ratio of the maximum work of a job to the minimum work of a job), and that this algorithm is essentially optimally competitive. This result extends to the case that each job has a weight and the objective is weighted quoted lead time.

We then introduce the following general setting: there is a non-increasing profit function pi(t) associated with each job Ji. If the customer for job Ji is quoted a due date of di, then the profit obtained from completing this job by its due date is pi(di). We consider the objective of maximizing profits. We show that if the company must finish each job by its due date, then there is no O(1)-speed poly-log-competitive algorithm. However, if the company can miss the due date of a job, at the cost of forgoing the profits from that job, then we show that there is a (1+ε)-speed O(1+1/ε)-competitive algorithm, and that this algorithm is essentially optimally competitive.

Mechanism Design for Fractional Scheduling on Unrelated Machines
George Christodoulou, Elias Koutsoupias and Annamária Kovács

In this paper, we consider the mechanism design version of the fractional variant of the scheduling problem on unrelated machines. We give a lower bound of 2–1/n for any fractional truthful mechanism, while we propose a truthful mechanism that achieves approximation of 1+(n–1)/2, for n machines. We also focus on an interesting family of allocation algorithms, the task-independent algorithms. We give a lower bound of 1+(n–1)/2, that holds for every (not only monotone) allocation algorithm of this class. Under this consideration, our truthful independent mechanism is the best that we can hope from this family of algorithms.

Estimating Sum by Weighted Sampling
Rajeev Motwani, Rina Panigrahy and Ying Xu

We study the classic problem of estimating the sum of n variables. The traditional uniform sampling approach requires a linear number of samples to provide any non-trivial guarantees on the estimated sum. In this paper we consider various sampling methods besides uniform sampling, in particular sampling a variable with probability proportional to its value, referred to as linear weighted sampling. If only linear weighted sampling is allowed, we show an algorithm for estimating sum with Õ(√n) samples, and it is almost optimal in the sense that Ω(√n) samples are necessary for any reasonable sum estimator. If both uniform sampling and linear weighted sampling are allowed, we show a sum estimator with Õ(∛n) samples. More generally, we may allow general weighted sampling where the probability of sampling a variable is proportional to any function of its value. We prove a lower bound of Ω(∛n) samples for any reasonable sum estimator using general weighted sampling, which implies that our algorithm combining uniform and linear weighted sampling is an almost optimal sum estimator.

Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima
Johannes Blömer and Stefanie Naewe

In this paper we introduce a new lattice problem, the subspace avoiding problem (SAP). We describe a probabilistic single exponential time algorithm for SAP for arbitrary ℓp norms. We also describe polynomial time reductions for four classical problems from the geometry of numbers, the shortest vector problem SVP, the closest vector problem CVP, the successive minima problem SMP, and the shortest independent vectors problem SIVP to SAP, establishing probabilistic single exponential time algorithms for them. The result generalize and extend previous results of Ajtai, Kumar and Sivakumar. The results on SMP and SIVP are new for all norms. The results on SVP and CVP generalize previous results of Ajtai et al. for the ℓ2 norm to arbitrary ℓp norms.

Low Distortion Spanners
Seth Pettie

A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say HG is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood.

In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1+ε,β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.

Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs
André Berger and Michelangelo Grigni

We present a linear time algorithm exactly solving the 2-edge connected spanning subgraph (2-ECSS) problem in a graph of bounded treewidth. Using this with Klein's diameter reduction technique, we find a linear time PTAS for the problem in unweighted planar graphs, and the first PTAS for the problem in weighted planar graphs.

Labeling Schemes for Vertex Connectivity
Amos Korman

This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any n-node graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are k-vertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k2log n on the number of bits used in a label. The best previous upper bound for the label size of such labeling scheme is 2klog n.

Key words: Graph algorithms, vertex-connectivity, labeling schemes.

Unbounded-Error One-Way Classical and Quantum Communication Complexity
Kazuo Iwama, Harumichi Nishimura, Rudy Raymond and Shigeru Yamashita

This paper studies the gap between quantum one-way communication complexity Q(f) and its classical counterpart C(f), under the unbounded-error setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for any (total or partial) Boolean function f, Q(f)=⎡C(f)/2⎤, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining an exact bound for the existence of (m,n,p)-QRAC which is the n-qubit random access coding that can recover any one of m original bits with success probability ≥p. We can prove that (m,n,>1/2)-QRAC exists if and only if m≤22n–1. Previously, only the non-existence of (22n,n,>1/2)-QRAC was known.

A Lower Bound on Entanglement-Assisted Quantum Communication Complexity
Ashley Montanaro and Andreas Winter

We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed inputs can be turned into a quantum communication protocol. As an application of this bound, we give a very simple proof of the statement that almost all Boolean functions on n+n bits have communication complexity linear in n, even in the presence of unlimited entanglement.

Separating Deterministic from Nondeterministic NOF Multiparty Communication Complexity
Paul Beame, Matei David, Toniann Pitassi and Philipp Woelfel

We solve some fundamental problems in the number-on-forehead (NOF) k-party communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a one-sided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k=nO(1) players, where n is the number of bits on each players' forehead. This separates the analogues of RP and P in the NOF communication model. We also show that there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of any explicit function which allows such separations. However, for the 3-player case we exhibit an explicit function which has Ω(log log n) randomized complexity for private coins but only constant complexity for public coins.

It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic k-party communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the number-in-hand model, is not complete in the NOF model under cylindrical reductions.

An Optimal Decomposition Algorithm for Tree Edit Distance
Erik D. Demaine, Shay Mozes, Benjamin Rossman and Oren Weimann

The edit distance between two ordered rooted trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case O(n3)-time algorithm for this problem, improving the previous best O(n3log n)-time algorithm. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms — which also includes the previous fastest algorithms — by tightening the known lower bound of Ω(n2log2n) to Ω(n3), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm2(1+log (n/m))) when the two trees have sizes m and n where m<n.

On Commutativity Based Edge Lean Search
Dragan Bošnački, Edith Elkind, Blaise Genest and Doron Peled

Exploring a graph through search is one of the most basic building blocks of various applications. In a setting with a huge state space, such as in testing and verification, optimizing the search may be crucial. We consider the problem of visiting all states in a graph where edges are generated by actions and the (reachable) states are not known in advance. Some of the actions may commute, i.e., they result in the same state for every order in which they are taken (this is the case when the actions are performed independently by different processes). We show how to use commutativity to achieve full coverage of the states while traversing considerably fewer edges.

Commitment Under Uncertainty: Two-Stage Stochastic Matching Problems
Irit Katriel, Claire Kenyon-Mathieu and Eli Upfal

We define and study two versions of the bipartite matching problem in the framework of two-stage stochastic optimization with recourse. In one version the uncertainty is in the second stage costs of the edges, in the other version the uncertainty is in the set of vertices that needs to be matched. We prove lower bounds, and analyze efficient strategies for both cases. These problems model real-life stochastic integral planning problems such as commodity trading, reservation systems and scheduling under uncertainty.

On the Complexity of Hard-Core Set Constructions
Chi-Jen Lu, Shi-Chun Tsai and Hsin-Lung Wu

We study a fundamental result of Impagliazzo (FOCS'95) known as the hard-core set lemma. Consider any function f:{0,1}n→{0,1} which is "mildly-hard", in the sense that any circuit of size s must disagree with f on at least δ fraction of inputs. Then the hard-core set lemma says that f must have a hard-core set H of density δ on which it is "extremely hard", in the sense that any circuit of size s'=O(s/(1/ε2 log(1/(εδ)))) must disagree with f on at least (1-ε)/2 fraction of inputs from H.

There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low-level complexity class. We introduce two models of hard-core set constructions, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models.

First, we show that in any strongly black-box construction, one can only prove the hardness of a hard-core set for smaller circuits of size at most s'=O(s/(1/ε2 log(1/δ))). Next, we show that any weakly black-box construction must be inherently non-uniform — to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with Ω(1/ε log|G|) bits of advice. Finally, we show that weakly black-box constructions in general cannot be realized in a low-level complexity class such as AC0[p] — the assumption that f is hard for AC0[p] is not sufficient to guarantee the existence of a hard-core set.

Approximation by DNF: Examples and Counterexamples
Ryan O'Donnell and Karl Wimmer

Say that f:{0,1}n→{0,1} ε-approximates g:{0,1}n→{0,1} if the functions disagree on at most an ε fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits:

1. For every constant 0<ε<1/2 there is a DNF of size 2O(√n) that ε-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent.
2. There is a monotone function :{0,1}n→{0,1} with total influence (AKA average sensitivity) B(ℱ)≤O(log n) such that any DNF or CNF that .01-approximates ℱ requires size 2Ω(n/log n) and such that any unbounded fan-in AND-OR-NOT circuit that .01-approximates requires size Ω(n/log n). This disproves a conjecture of Benjamini, Kalai, and Schramm.

Exotic Quantifiers, Complexity Classes, and Complete Problems
Peter Bürgisser and Felipe Cucker

We define new complexity classes in the Blum-Shub-Smale theory of computation over the reals, in the spirit of the polynomial hierarchy, with the help of infinitesimal and generic quantifiers. Basic topological properties of semialgebraic sets like boundedness, closedness, compactness, as well as the continuity of semialgebraic functions are shown to be complete in these new classes. All attempts to classify the complexity of these problems in terms of the previously studied complexity classes have failed. We also obtain completeness results in the Turing model for the corresponding discrete problems. In this setting, it turns out that infinitesimal and generic quantifiers can be eliminated, so that the relevant complexity classes can be described in terms of usual quantifiers only.

Online Conflict-Free Colorings for Hypergraphs
Amotz Bar-Noy, Panagiotis Cheilaris, Svetlana Olonetsky and Shakhar Smorodinsky

We provide a framework for online conflict-free coloring (CF-coloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CF-coloring any k-degenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CF-coloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results.

We introduce deterministic online CF-coloring algorithms for points on the line with respect to intervals and for points on the plane with respect to halfplanes (or unit discs) that use Θ(log n) colors and recolor O(n) points in total.

Distributed Computing with Advice: Information Sensitivity of Graph Coloring
Pierre Fraigniaud, Cyril Gavoille, David Ilcinkas and Andrzej Pelc

We study the problem of the amount of information (advice) about a graph that must be given to its nodes in order to achieve fast distributed computations. The required size of the advice enables to measure the information sensitivity of a network problem. A problem is information sensitive if little advice is enough to solve the problem rapidly (i.e., much faster than in the absence of any advice), whereas it is information insensitive if it requires giving a lot of information to the nodes in order to ensure fast computation of the solution. In this paper, we study the information sensitivity of distributed graph coloring.

Universal Algebra and Hardness Results for Constraint Satisfaction Problems
Benoît Larose and Pascal Tesson

We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain analysis of the complexity of reductions used in the algebraic study of CSPs. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NP-complete and they match the recent classification for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the idempotent reduct of a preprimal algebra.

On the power of k-consistency
Albert Atserias, Andrei Bulatov and Victor Dalmau

The k-consistency algorithm for constraint-satisfaction problems proceeds, roughly, by finding all partial solutions on at most k variables and iteratively deleting those that cannot be extended to a partial solution by one more variable. It is known that if the core of the structure encoding the scopes of the constraints has treewidth at most k, then the k-consistency algorithm is always correct. We prove the exact converse to this: if the core of the structure encoding the scopes of the constraints does not have treewidth at most k, then the k-consistency algorithm is not always correct. This characterizes the exact power of the k-consistency algorithm in structural terms.

Complexity of Propositional Proofs Under a Promise
Nachum Dershowitz and Iddo Tzameret

We study — within the framework of propositional proof complexity — the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where "many" stands for an explicitly specified function λ in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises:

1. Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε∙2n, for any constant 0<ε<1.
2. There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2δn (and the number of clauses is o(n3/2)), for any constant 0<δ<1.

Sharp Tractability Borderlines for Finding Connected Motifs in Vertex-Colored Graphs
Michael R. Fellows, Guillaume Fertin, Danny Hermelin and Stéphane Vialette

We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem has applications in metabolic network analysis, an important area in bioinformatics. We give two positive results and three negative results that together draw sharp borderlines between tractable and intractable instances of the problem.

Parameterized Algorithms for Directed Maximum Leaf Problems
Noga Alon, Fedor V. Fomin, Gregory Gutin, Michael Krivelevich and Saket Saurabh

We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family ℒ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in ℒ. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph D∊ℒ of order n with minimum in-degree at least 3 contains a rooted spanning tree, then D contains one with at least (n/2)1/5-1 leaves.

Parameterized Approximability of the Disjoint Cycle Problem
Martin Grohe and Magdalena Grüber

We give an fpt approximation algorithm for the directed vertex disjoint cycle problem. Given a directed graph G with n vertices and a positive integer k, the algorithm constructs a family of at least k/ρ(k) disjoint cycles of G if the graph G has a family of at least k disjoint cycles (and otherwise may still produce a solution, or just report failure). Here ρ is a computable function such that k/ρ(k) is nondecreasing and unbounded. The running time of our algorithm is polynomial.

The directed vertex disjoint cycle problem is hard for the parameterized complexity class W, and to the best of our knowledge our algorithm is the first fpt approximation algorithm for a natural W-hard problem.

Key words: approximation algorithms, fixed-parameter tractability, parameterized complexity theory.

Linear Problem Kernels for NP-Hard Problems on Planar Graphs
Jiong Guo and Rolf Niedermeier

We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for CONNECTED VERTEX COVER, MINIMUM EDGE DOMINATING SET, MAXIMUM TRIANGLE PACKING, and EFFICIENT DOMINATING SET on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.

Balanced Families of Perfect Hash Functions and Their Applications
Noga Alon and Shai Gutner

The construction of perfect hash functions is a well-studied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to [k] is a δ-balanced (n,k)-family of perfect hash functions if for every S⊆[n], |S|=k, the number of functions that are 1-1 on S is between T/δ and δT for some constant T>0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on S, for each S of size k. In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking δ to be close to 1) for every such S. Our main result is that for any constant δ>1, a δ-balanced (n,k)-family of perfect hash functions of size 2O(k log log k) log n can be constructed in time 2O(k log log k) n log n. Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length k and the number of simple cycles of size k for any kO(log n / log log log n) in a graph with n vertices. The approximation is up to any fixed desirable relative error.

Key words: approximate counting of subgraphs, color-coding, perfect hashing.

An Exponential Improvement on the MST Heuristic for Minimum Energy Broadcasting in Ad Hoc Wireless Networks
Ioannis Caragiannis, Michele Flammini and Luca Moscardelli

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

Succinct Ordinal Trees Based on Tree Covering
Meng He, J. Ian Munro and S. Srinivasa Rao

Various methods have been used to represent a tree of n nodes in essentially the information-theoretic minimum space while supporting various navigational operations in constant time, but different representations usually support different operations. Our main contribution is a succinct representation of ordinal trees, based on that of Geary et al., that supports all the navigational operations supported by various succinct tree representations while requiring only 2n+o(n) bits. It also supports efficient level-order traversal, a useful ordering previously supported only with a very limited set of operations.

Our second contribution expands on the notion of a single succinct representation supporting more than one traversal ordering, by showing that our method supports two other encoding schemes as abstract data types. In particular, it supports extracting a word (O(lg n) bits) of the balanced parenthesis sequence or depth first unary degree sequence in O(f(n)) time, using at most n/f(n) + o(n) additional bits, for any f(n) in O(lg n) and Ω(1).

A Framework for Dynamizing Succinct Data Structures
Ankur Gupta, Wing-Kai Hon, Rahul Shah and Jeffrey Scott Vitter

We present a framework to dynamize succinct data structures, to encourage their use over non-succinct versions in a wide variety of important application areas. Our framework can dynamize most state-of-the-art succinct data structures for dictionaries, ordinal trees, labeled trees, and text collections. Of particular note is its direct application to XML indexing structures that answer subpath queries. Our framework focuses on achieving information-theoretically optimal space along with near-optimal update/query bounds.

As the main part of our work, we consider the following problem central to text indexing: Given a text T over an alphabet Σ, construct a compressed data structure answering the queries char(i), ranks(i), and selects(i) for a symbol s∊Σ. Many data structures consider these queries for static text T. We build on these results and give the best known query bounds for the dynamic version of this problem, supporting arbitrary insertions and deletions of symbols in T.

Specifically, with an amortized update time of O(nε), any static succinct data structure D for T, taking t(n) time for queries, can be converted by our framework into a dynamic succinct data structure that supports ranks(i), selects(i), and chars(i) queries in O(t(n) + log log n) time, for any constant ε>0. When |Σ|=polylog(n), we achieve O(1) query times. Our update/query bounds are near-optimal with respect to the lower bounds.

In-Place Suffix Sorting
Gianni Franceschini and S. Muthukrishnan

Given string T=T[1,...,n], the suffix sorting problem is to lexicographically sort the suffixes T[i,...,n] for all i. This problem is central to the construction of suffix arrays and trees with many applications in string processing, computational biology and compression. A bottleneck in these applications is the amount of workspace needed to perform suffix sorting beyond the space needed to store the input as well as the output. In particular, emphasis is even on the constant c in the O(n)=cn space algorithms known for this problem.

Currently the best previous result takes O(nv + n log n) time and O(n/√v) extra space, for any v∊[1,√n] for strings from a general alphabet. We improve this and present the first known in-place suffix sorting algorithm. Our algorithm takes O(n log n) time using O(1) workspace and is optimal in the worst case for the general alphabet.

Strong Price of Anarchy for Machine Load Balancing
Amos Fiat, Haim Kaplan, Meital Levy and Svetlana Olonetsky

As defined by Aumann in 1959, a strong equilibrium is a Nash equilibrium that is resilient to deviations by coalitions. We give tight bounds on the strong price of anarchy for load balancing on related machines. We also give tight bounds for k-strong equilibria, where the size of a deviating coalition is at most k.

Keywords: Game theory, Strong Nash equilibria, Load balancing, Price of Anarchy

Efficient Algorithms for Constant Well Supported Approximate Equilibria in Bimatrix Games
Spyros C. Kontogiannis and Paul G. Spirakis

In this work we study the tractability of well supported approximate Nash Equilibria (SuppNE in short) in bimatrix games. 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 of great importance.

We initially prove that SuppNE are immune to the addition of arbitrary real vectors to the rows (columns) of the row (column) player's payoff matrix. Consequently we propose a polynomial time algorithm (based on linear programming) that constructs a 0.5-SuppNE for arbitrary win lose games.

We then parameterize our technique for win lose games, in order to apply it to arbitrary (normalized) bimatrix games. Indeed, this new technique leads to a weaker φ-SuppNE for win lose games, where φ = (√5–1)/2 is the golden ratio. Nevertheless, this parameterized technique extends nicely to a technique for arbitrary [0,1]-bimatrix games, which assures a 0.658-SuppNE in polynomial time.

To our knowledge, these are the first polynomial time algorithms providing ε-SuppNE of normalized or win lose bimatrix games, for some nontrivial constant ε∊[0,1), bounded away from 1.

Keywords: Bimatrix Games, Well Supported Approximate Equilibria

Holographic Algorithms: The Power of Dimensionality Resolved
Jin-Yi Cai and Pinyan Lu

Valiant's theory of holographic algorithms is a novel methodology to achieve exponential speed-ups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2-dimensional bases are universal for holographic algorithms.

Reconciling Data Compression and Kolmogorov Complexity
Laurent Bienvenu and Wolfgang Merkle

While data compression and Kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively. This gap is bridged by introducing decidable Turing machines and a corresponding notion of Kolmogorov complexity, where compressors and suitably normalized decidable machines are essentially the same concept.

Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Löf randomness, Schnorr randomness, Kurtz randomness, and computable dimension. These results can also be reformulated in terms of time-bounded Kolmogorov complexity. Other applications of decidable machines are presented, such as a simplified proof of the Miller-Yu theorem (characterizing Martin-Löf randomness by the plain complexity of the initial segments) and a new characterization of computably traceable sequences via a natural lowness notion for decidable machines.

Size Competitive Meshing without Large Angles
Gary L. Miller, Todd Phillips and Donald Sheehy

We present a new meshing algorithm for the plane, Overlay Stitch Meshing (OSM), accepting as input an arbitrary Planar Straight Line Graph and producing a triangulation with all angles smaller than 170°. The output triangulation has competitive size with any optimal size mesh having equally bounded largest angle. The competitive ratio is O(log(L/s)) where L and s are respectively the largest and smallest features in the input. OSM runs in O(n log(L/s) + m) time/work where n is the input size and m is the output size. The algorithm first uses Sparse Voronoi Refinement to compute a quality overlay mesh of the input points alone. This triangulation is then combined with the input edges to give the final mesh.

Lower Bounds for Quantile Estimation in Random-Order and Multi-Pass Streaming
Sudipto Guha and Andrew McGregor

We present lower bounds on the space required to estimate the quantiles of a stream of numerical values. Quantile estimation is perhaps the most studied problem in the data stream model and it is relatively well understood in the basic single-pass data stream model in which the values are ordered adversarially. Natural extensions of this basic model include the random-order model in which the values are ordered randomly and the multi-pass model in which an algorithm is permitted a limited number of passes over the stream. We present lower bounds that complement existing upper bounds in both models. One consequence is an exponential separation between the random-order and adversarial-order models: using Ω(polylog n) space, exact selection requires Ω(log n) passes in the adversarial-order model while O(log log n) passes are sufficient in the random-order model.

Streaming and Fully Dynamic Centralized Algorithms for Constructing and Maintaining Sparse Spanners
Michael Elkin

We present a streaming algorithm for constructing sparse spanners and show that our algorithm out-performs significantly the state-of-the-art algorithm for this task. Specifically, the processing time-per-edge of our algorithm is drastically smaller than that of the state-of-the-art algorithm, and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters for the state-of-the-art algorithm.

We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has a very small incremental update time, and a non-trivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides non-trivial bounds on both incremental and decremental update time for a wide range of stretch parameter t.

Checking and Spot-Checking the Correctness of Priority Queues
Matthew Chu, Sampath Kannan and Andrew McGregor

We revisit the problem of memory checking considered by Blum et al. In this model, a checker monitors the behavior of a data structure residing in unreliable memory given an arbitrary sequence of user defined operations. The checker is permitted a small amount of separate reliable memory and must fail a data structure if it is not behaving as specified and pass it otherwise. How much additional reliable memory is required by the checker? First, we present a checker for an implementation of a priority queue. The checker uses O(√n log n) space where n is the number of operations performed. We then present a spot-checker using only O(ε–1 log δ–1 log n) space, that, with probability at least 1–δ, will fail the priority queue if it is ε-far (defined appropriately) from operating like a priority queue and pass the priority queue if it operates correctly. Finally, we then prove a range of lower bounds that complement our checkers.

On the Chromatic Number of Random Graphs
Amin Coja-Oghlan, Konstantinos Panagiotou and Angelika Steger

In this paper we study the chromatic number χ(Gn,p) of the binomial random graph Gn,p, where p = p(n) ≤ n–3/4 – δ, for every fixed δ>0. We prove that a.a.s. χ(Gn,p) is ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ–1)log(ℓ–1) ≤ np.

Quasi-Randomness and Algorithmic Regularity for Graphs with General Degree Distributions
Noga Alon, Amin Coja-Oghlan, Hiệp Hàn, Mihyun Kang, Vojtěch Rödl and Mathias Schacht

We deal with two very related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure how much a given graph "resembles" a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we present a novel concept of regularity that takes into account the graph's degree distribution, and show that if G=(V,E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor, we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time.

Key words: quasi-random graphs, Laplacian eigenvalues, sparse graphs, regularity lemma, Grothendieck's inequality

Complexity of the Cover Polynomial
Markus Bläser and Holger Dell

The cover polynomial introduced by Chung and Graham is a two-variate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is P-hard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomial: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial is P-hard under polynomial-time Turing reductions, while only three points are easy. Our construction uses a gadget which is easier to analyze and more general than the XOR-gadget used by Valiant in his proof that the permanent is P-complete.