Showing papers in "Theory of Computing in 2010"

Can You Beat Treewidth

Abstract: It is well-known that constraint satisfaction problems (CSP) can be solved in time nO(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidth-based algorithm, even if we restrict the problem to some special class of primal graphs. Formally, let g be an arbitrary class of graphs and assume that there is an algorithm A solving binary CSP for instances whose primal graph is in g. We prove that if the running lime of A is f(G)nO(k/logk), where k is the treewidth of the primal graph G and f is an arbitrary function, then the Exponential Time Hypothesis fails. We prove the result also in the more general framework of the homomorphism problem for bounded-arity relational structures. For this problem, the treewidth of the core of the left-hand side structure plays the same role as the. treewidth of the primal graph above.

The Submodular Welfare Problem with Demand Queries

Abstract: We consider the Submodular Welfare Problem where we have m items and n players with given utility functions wi : 2 (m) ! R+. The utility functions are assumed to be monotone and submodular. We want to find an allocation of disjoint sets S1; S2;:::; Sn of items maximizing Âi wi(Si). A (1 1=e)-approximation for this problem in the demand oracle model has been given by Dobzinski and Schapira (5). We improve this algorithm by presenting a (1 1=e+e)-approximation for some small fixed e > 0. We also show that the Submodular Welfare Problem is NP -hard to approximate within a ratio better than some r < 1. Moreover, this holds even when for each player there are only a constant number of items that have nonzero utility. The constant size restriction on utility functions makes it easy for players to efficiently answer any "reasonable" query about their utility functions. In contrast, for classes of instances that were used for previous hardness of approximation results, we present an incentive compatible (in expectation) mechanism based on fair division queries that achieves an optimal solution.

Elusive Functions and Lower Bounds for Arithmetic Circuits

Abstract: A basic fact in linear algebra is that the image of the curve f(x)=(x1,x2,x3,...,xm), say over C, is not contained in any m-1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree~1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f:C → Cm, such that, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness implies super-polynomial lower bounds for computing the permanent over~C. More generally, we say that a polynomial-mapping f:Fn → Fm is (s,r)-elusive, if for every polynomial-mapping Γ:Fs → Fm of degree r, Im(f) ⊄ Im(Γ). We show that for many settings of the parameters n,m,s,r, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every r n → Fn2, of degree O(r), that is (s,r)-elusive for s = n1+Ω(1/r). We use this to construct for any r, an explicit example for an n-variate polynomial of total-degree O(r), with coefficients in {0,1,}such that, any depth r arithmetic circuit for this polynomial (over any field) is of size ≥ n1+Ω(1/r). In particular, for any constant r, this gives a constant degree polynomial, such that, any depth r arithmetic circuit for this polynomial is of size ≥ n1+Ω(1). Previously, only lower bounds of the type Ω(n • λr (n)), where λr (n) are extremely slowly growing functions (e.g., λ5(n) = log n, and λ7(n) = log* log*n), were known for constant-depth arithmetic circuits for polynomials of constant degree.

A Separation of NP and coNP in Multiparty Communication Complexity

Abstract: We prove that coNP * MA in the number-on-forehead model of multiparty communication complexity for up to k = (1 e) log n players, where e > 0 is any con- stant. Specifically, we construct an explicit function F : (f0; 1g n ) k ! f0; 1g with co- nondeterministic complexity O(log n) and Merlin-Arthur complexity n W(1) . The problem was open for k > 3. As a corollary, we obtain an explicit separation of NP and coNP for up to k = (1 e) log n players, complementing an independent result by Beame et al. (2010) who separate these classes nonconstructively for up to k = 2 (1 e)n players.

Reordering Buffers for General Metric Spaces

Abstract: In the reordering buffer problem, we are given an input sequence of requests for service each of which corresponds to a point in a metric space. The cost of serving the requests heavily depends on the processing order. Serving a request induces cost corresponding to the distance between itself and the previously served request, measured in the underlying metric space. A reordering buffer with storage capacity k can be used to reorder the input sequence in a restricted fashion so as to construct an output sequence with lower service cost. This simple and universal framework is useful for many applications in computer science and economics, e.g., disk scheduling, rendering in computer graphics, or painting shops in car plants.In this paper, we design online algorithms for the reordering buffer problem. Our main result is a strategy with a polylogarithmic competitive ratio for general metric spaces. Previous work on the reordering buffer problem only considered very restricted metric spaces. We obtain our result by first developing a deterministic algorithm for arbitrary weighted trees with a competitive ratio of O(D · log k), where D denotes the unweighted diameter of the tree, i.e., the maximum number of edges on a path connecting two nodes. Then we show how to improve this competitive ratio to O(log2 k) for metric spaces that are derived from HSTs. Combining this result with the results on probabilistically approximating arbitrary metrics by tree metrics, we obtain a randomized strategy for general metric spaces that achieves a competitive ratio of O(log2 k · log n) in expectation against an oblivious adversary. Here n denotes the number of distinct points in the metric space. Note that the length of the input sequence can be much larger than n.

A New Quantum Lower Bound Method, with an Application to a Strong Direct Product Theorem for Quantum Search

Abstract: We present a new method for proving lower bounds on quantum query al- gorithms. The new method is an extension of the adversary method, by analyzing the eigenspace structure of the problem. Using the new method, we prove a strong direct product theorem for quantum search. This result was previously proved by Klauck, ˇ Spalek, and de Wolf (FOCS'04) using the polynomials method. No proof using the adversary method was known before.

Routing Without Regret: On Convergence to Nash Equilibria of Regret-Minimizing Algorithms in Routing Games

Abstract: There has been substantial work developing simple, efficient no-regret algorithms for a wide class of repeated decision-making problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversarially-changing environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a no-regret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games have substantially more structure.In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimal agents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that depends polynomially on the players' regret bounds and the maximum slope of any latency function. We also show that price-of-anarchy results may be applied to these approximate equilibria, and also consider the finite-size (non-infinitesimal) load-balancing model of Azar [2].

Separating Deterministic from Randomized Multiparty Communication Complexity

Abstract: We solve some fundamental problems in the number-on-forehead (NOF) k- player communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided false- positives error probability of 1/3, but which has linear communication complexity for deterministic protocols and, in fact, even for the more powerful nondeterministic protocols. The result holds for every e > 0 and every k 2 (1 e)n players, where n is the number of bits on each player's forehead. As a consequence, we obtain the NOF communication class separation coRP6 NP. This in particular implies that P6 RP and NP6 coNP. We also show that for every e > 0 and every k n 1 e , 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; no explicit function is known to satisfy nontrivial lower bounds for k log n players. However, for every e > 0 and every k (1 e) log n players, the NP6 coNP separation (and even the coNP6 MA separation) was obtained independently by Gavinsky and Sherstov (2010) using an explicit construction. In this work, for k (1=9) log n players, we exhibit an explicit function which has communication

Monotone Expanders: Constructions and Applications

Abstract: The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. constant-degree dimension expanders in finite fields, resolving a question of Barak,

Decision Trees and Influence: an Inductive Proof of the OSSS Inequality

Abstract: We give a simple proof of the OSSS inequality (O'Donnell, Saks, Schramm, Servedio, FOCS 2005) The inequality states that for any decision tree T calculating a Boolean function f : {0,1} n ! { 1,1}, we have Var(f) Âidi(T)Infi(f), where di(T) is the probability that the input variable xi is read by T and Infi(f) is the influence of the ith variable on f

Rounds vs. Queries Tradeoff in Noisy Computation

Abstract: WeshowthatanoisyparalleldecisiontreemakingO(n)queriesneedsW(log n) rounds to compute OR of n bits. This answers a question of Newman (IEEE Conference on Computational Complexity, 2004, 113-124). We prove more general tradeoffs between the number of queries and rounds. We also settle a similar question for computing MAX in the noisy comparison tree model; these results bring out interesting differences among the noise models.