In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, 2013]]>Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 39, no. 8, pp. 3553-3570, 2010]]>

We say that a cryptographic scheme is Continuous Leakage-Resilient (CLR), if it allows users to refresh their secret keys, using only fresh local randomness, such that: The scheme remains functional after any number of key refreshes, although the public key never changes. Thus, the \outside world" is neither aected by these key refreshes, nor needs to know about their ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 511-520, 2010]]>Boosting is a general method for improving the accuracy of learning algorithms. We use boosting to construct improved privacy-pre serving synopses of an input database. These are data structures that yield, for a given set Q of queries over an input database, reasonably accurate estimates of the responses to every query in Q, even when the number of queries ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 51-60, 2010]]>We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0,1]n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 575-584, 2010]]>We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no sub graph isomorphic to a subdivision of a fixed graph, and graphs that ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 133-142, 2010]]>In recent years, there has been a major effort to design cryptographic schemes that remain secure even when arbitrary information about the secret key is leaked (e.g., via side-channel attacks). We explore the possibility of achieving security under emph{continual} leakage from the emph{entire} secret key by designing schemes in which the secret key is updated over ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 501-510, 2010]]>We present a new algorithm for learning a convex set in n-dimensional space given labeled examples drawn from any Gaussian distribution. The complexity of the algorithm is bounded by a fixed polynomial in n times a function of k and ϵ where k is the dimension of the normal subspace (the span of normal vectors to supporting hyperplanes of ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 124-130, 2010]]>We give new pseudorandom generators for regular read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is either 0 or 2. For every width d and length n, our pseudorandom generator uses a seed of length O((log d + log log n + log(1/ϵ)) log ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 40-47, 2010]]>We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size O(n5/3) = O(n1.66⋯) and, when queried about vertices at distance d, returns a path of length 2d + 1. For weighted graphs with m = n2/...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 815-823, 2010]]>We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: * high-dimensional problems, where the goal is to show large space lower bounds. * constant-dimensional geometric problems, where the goal is to bound the query time for ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. abs/1010.3, 2010]]>We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifier G with n-1+m/k edges, such that the condition number of G with G is bounded above by Õ(k log2 n), with probability 1-p. The algorithm runs in time Õ((m log n ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 235-244, 2010]]>We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 775-784, 2010]]>is optimal, and succeeds when β ≥ c/ √ n for al arge enough constant c. The coin problem is open and interesting in models that cannot compute the majority function. In this paper we study the coin problem in the model of read-once width-w branching programs. We prove that in order to succeed in this model, β ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 30-39, 2010]]>We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of $q \ge 4$ alternatives and n voters will be manipulable with probability at least $10^{-4} \eps^2 n^{-3} q^{-30}$, where $\eps$ is the minimal statistical distance between f and the family ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 319-328, 2010]]>We study the design of truthful mechanisms for set systems, i.e., scenarios where a customer needs to hire a team of agents to perform a complex task. In this setting, frugality [2] provides a measure to evaluate the "cost of truthfulness", that is, the overpayment of a truthful mechanism relative to the "fair" payment. We propose a uniform scheme ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 755-764, 2010]]>Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertex-disjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m√n) time, where m and n are the number of edges ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 673-682, 2010]]>The main question in the on-line chain partitioning problem is to determine whether there exists an algorithm that partitions on-line posets of width at most w into polynomial number of chains see Trotter's chapter Partially ordered sets in the Handbook of Combinatorics. So far the best known on-line algorithm of Kierstead used at most (5ω - 1)/...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 347-354, 2010]]>Abstract-We consider the problem of testing if a given function f:F2n → F2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. Alon et al. [1] proposed and analyzed a natural 2d+1-query test for this problem. This test turned out to be intimately related to the Gowers norm. ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 488-497, 2010]]>The generalized nested dissection method, developed by Lipton, Rose, and Tarjan, is a seminal method for solving a linear system Ax=b where A is a symmetric positive definite matrix. The method runs extremely fast whenever A is a well-separable matrix (such as matrices whose underlying support is planar or avoids a fixed minor). In this work we extend ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 225-234, 2010]]>Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 143-152, 2010]]>Understanding the average-case complexity of natural problems on natural distributions is an important challenge for computational complexity. In this paper, we consider the average-case monotone complexity of the k-clique problem (for constant k) on Erd} os-R enyi random graphs

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 193-201, 2010]]>Let G be a directed edge-weighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest s-to-t path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the naive solution to this problem - removing ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 655-662, 2010]]>A fundamental question in leakage-resilient cryptography is: can leakage resilience always be amplified by parallel repetition? It is natural to expect that if we have a leakage-resilient primitive tolerating ℓ bits of leakage, we can take n copies of it to form a system tolerating nℓ bits of leakage. In this paper, we show that this is not ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 521-530, 2010]]>We study differential privacy in a distributed setting where two parties would like to perform analysis of their joint data while preserving privacy for both datasets. Our results imply almost tight lower bounds on the accuracy of such data analyses, both for specific natural functions (such as Hamming distance) and in general. Our bounds expose a sharp contrast between the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 81-90, 2010]]>We initiate the study of testing properties of images that correspond to sparse 0/1-valued matrices of size n × n. Our study is related to but different from the study initiated by Raskhodnikova (Proceedings of RANDOM, 2003), where the images correspond to dense 0/1-valued matrices. Specifically, while distance between images in the model studied by Raskhodnikova ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 468-477, 2010]]>We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 633-642, 2010]]>Given a network, a set of demands and a cost function f(.), the min-cost network design problem is to route all demands with the objective of minimizing Σe f(ℓe), where ℓe is the total traffic load under the routing. We focus on cost functions of the form f(x) = σ + xα for x > 0, with f(...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 585-592, 2010]]>It is shown that for each t, there is a separator of size O(t√n) in any n-vertex graph G with no Kt-minor. This settles a conjecture of Alon, Seymour and Thomas (J. Amer. Math. Soc, 1990 and STOC'90), and generalizes a result of Djidjev (1981), and Gilbert, Hutchinson and Tarjan (J. Algorithm, 1984), independently, who proved ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 153-162, 2010]]>We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present a (polylog(n),poly(log log n))approximation, which means that if there exists a solution that routes X demands integrally on edge-disjoint paths (i.e. with ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 277-286, 2010]]>We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search. Our first main result is a simple deterministic algorithm running in time 2n-Ω(n) for satisfiability of formulae of linear size in n, where n is the number of variables in the formula. ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 183-192, 2010]]>We construct the first general secure computation protocols that require no trusted infrastructure other than authenticated communication, and that satisfy a meaningful notion of security that is preserved under universal composition- assuming only the existence of enhanced trapdoor permutations. The notion of security fits within a generalization of the "angelbased" framework of Prabhakaran and Sahai (STOC'04) and implies super-...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 541-550, 2010]]>For every ∈ > 0, and integer q ≥ 3, we show that given an N-vertex graph that has an induced q-colorable subgraph of size (1 - ∈)N, it is NP-hard to find an independent set of size N/q2.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 212-221, 2010]]>We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0,1] is O(n2), in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 663-672, 2010]]>We study truthful mechanisms for hiring a team of agents in three classes of set systems: Vertex Cover auctions, How auctions, and cut auctions. For Vertex Cover auctions, the vertices are owned by selfish and rational agents, and the auctioneer wants to purchase a vertex cover from them. For k-flow auctions, the edges are owned by the agents, and ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 745-754, 2010]]>Coin flipping is one of the most fundamental tasks in cryptographic protocol design. Informally, a coin flipping protocol should guarantee both (1) Completeness: an honest execution of the protocol by both parties results in a fair coin toss, and (2) Security: a cheating party cannot increase the probability of its desired outcome by any significant amount. Since its introduction by ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 613-622, 2010]]>We prove that planar graphs have poly-logarithmic queue number, thus improving upon the previous polynomial upper bound. Consequently, planar graphs admit 3D straight-line crossing-free grid drawings in small volume.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 365-374, 2010]]>In this paper we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MODp gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ϵ > 0. In fact, we obtain ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 695-704, 2010]]>We prove that any monotone switching network solving directed connectivity on N vertices must have size NΩ(log N).

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 553-562, 2010]]>Finding the length of the longest increasing subsequence (LIS) is a classic algorithmic problem. Let n denote the size of the array. Simple O(n log n) time algorithms are known that determine the LIS exactly. In this paper, we develop a randomized approximation algorithm, that for any constant δ > 0, runs in time polylogarithmic in n and estimates ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 458-467, 2010]]>An approximate membership data structure is a randomized data structure for representing a set which supports membership queries. It allows for a small false positive error rate but has no false negative errors. Such data structures were first introduced by Bloom in the 1970's, and have since had numerous applications, mainly in distributed systems, database systems, and networks. The ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 797-804, 2010]]>We correct a claim from [Vem97] and provide a status update.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 123-123, 2010]]>We present round-efficient protocols for secure multi-party computation with a dishonest majority that rely on black-box access to the underlying primitives. Our main contributions are as follows: · a O(log* n)-round protocol that relies on black-box access to dense cryptosystems, homomorphic encryption schemes, or lossy encryption schemes. This improves upon the recent O(1)log* ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 531-540, 2010]]>We generalize algorithms from computational learning theory that are successful under the uniform distribution on the Boolean hypercube {0,1}n to algorithms successful on permutation invariant distributions. A permutation invariant distribution is a distribution where the probability mass remains constant upon permutations in the instances. While the tools in our generalization mimic those used for the Boolean hypercube, the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 113-122, 2010]]>We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V,E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 427-436, 2009]]>We consider the problem of estimating cas- caded aggregates over a matrix presented as a sequence of updates in a data stream. A cascaded aggregate P - Q is de- fined by evaluating aggregate Q repeatedly over each row of the matrix, and then evaluating aggregate P over the resulting vector of values. This problem was introduced by Cormode and Muthukrishnan, ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 765-774, 2009]]>We provide a full characterization of applicability of The Local Consistency Checking algo- rithm to solving the non-uniform Constraint Satisfaction Problems. This settles the conjecture of Larose and Zadori.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 595-603, 2009]]>With the work of Khot and Vishnoi (18) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp( (log log n)1=4) vectors. For a stronger ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 575-585, 2009]]>The general adversary bound is a semi-denite program (SDP) that lower-bounds the quan- tum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 544-551, 2009]]>A large number of interesting combinatorial opti- mization problems like Max Cut, Max k-Sat, and Unique Games fall under the class of constraint satisfaction problems (CSPs). Recent work (32) by one of the authors identifies a semidefinite programming (SDP) relaxation that yields the optimal approximation ratio for every CSP, under the Unique Games Conjecture (UGC). Very recently (33), the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 586-594, 2009]]>Applications in complex systems such as the Inter- net have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 755-764, 2009]]>The question of whether or not parallel repetition reduces the soundness error is a fundamental question in the theory of protocols. While parallel repetition reduces (at an exponential rate) the error in interactive proofs and (at a weak exponential rate) in special cases of interactive arguments (e.g., 3-message protocols — Bellare, Impagliazzo and Naor (FOCS '97), and public-coin ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 241-250, 2009]]>A number of recent results on optimization prob- lems involving submodular functions have made use of the "mul- tilinear relaxation" of the problem (3), (8), (24), (14), (13). We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 651-670, 2009]]>We present a protocol which allows a client to have a server carry out a quantum computation for her such that the client's inputs, outputs and computation remain perfectly private, and where she does not require any quantum computational power or memory. The client only needs to be able to prepare single qubits randomly chosen from a finite set ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 517-526, 2009]]>This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2-approximation algorithm for the submodular vertex cover problem based on the half-integrality of the continuous ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 671-680, 2009]]>We extend the "method of multiplicities" to get the following results, of interest in combi- natorics and randomness extraction. 1. We show that every Kakeya set in Fn q, the n-dimensional vector space over the finite field on q elements, must be of size at least qn/2n. This bound is tight to within a 2 + o(1) ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 181-190, 2009]]>We present a fully online randomized algorithm for the classical pattern matching problem that uses merely O(log m) space, breaking the O(m) barrier that held for this problem for a long time. Our method can be used as a tool in many practical applications, including monitoring Internet traffic and firewall applications. In our online model we first receive ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 315-323, 2009]]>We give approximation algorithms for the Generalized Steiner Network (GSN) problem. The input consists of a graph G = (V; E) with edge or node costs, a node subset S V , and connectivity requirements fr(s; t) : s; t 2 T V g. The goal is to nd a minimum cost subgraph H in which every node pair s; t 2 ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 417-426, 2009]]>In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimum-cost subset Eof edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 437-441, 2009]]>We study ( k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the rst deterministic polynomial time blackbox identity test for ( k) circuits over the

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 198-207, 2009]]>Given a set D = fd1;d2;:::;dDg of D strings of total length n, our task is to report the "most relevant" strings for a given query pattern P. This involves somewhat more advanced query functionality than the usual pattern matching, as some notion of "most relevant" is involved. In information retrieval literature, this task is best achieved by using ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 713-722, 2009]]>We prove that the number of Pareto-optimal so- lutions in any multiobjective binary optimization problem with a finite number of linear objective functions is polynomial in the model of smoothed analysis. This resolves a conjecture of Ren´ e Beier (5). Moreover, we give polynomial bounds on all finite moments of the number of Pareto-optimal solutions, which yields the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 681-690, 2009]]>Abstract The main result of this paper is a,generic composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well- understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specic PCPs that were being ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 472-481, 2009]]>Thorup and Zwick, in their seminal work, intro- duced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn1+1/k) that can answer queries in time O(k) with a distance estimate that is at ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 703-712, 2009]]>For arbitrarily small constants epsilon, delta ¿.¿ > 0, we present a long code test with one free bit, completeness 1-epsilon and soundness delta. Using the test, we prove the following two inapproximability results:1. Assuming the Unique Games Conjecture of Khot, given an n-vertex graph that has two disjoint independent sets of size (1/2-¿)...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 453-462, 2009]]>We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of bounded-degree graphs with an excluded minor, and in general, for any hyperfinite class o f bounded-degree graphs. These oracles utilize only local compu- tation to consistently answer queries about a global partition that breaks the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 22-31, 2009]]>Linial, London and Rabinovich (16) and Aumann and Rabani (3) proved that the min-cut max-flow ratio for general maximum concurrent flow problems (when there are k commodi- ties) is O(logk). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are poly-logarithmic in k for a ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 3-12, 2009]]>We prove the existence of an algorithm A for computing 2-d or 3-dconvex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A' in a certain class A, the running time of A on the worst permutation of S for A is at most ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 129-138, 2009]]>We construct integrality gap instances for SDP relaxation of the MAXIMUM CUT and the SPARSEST CUT problems. If the triangle inequality constraints are added to the SDP, then the SDP vectors naturally define an n-point negative type metric where n is the number of vertices in the problem instance. Our gap-instances satisfy a stronger constraint that every sub-...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 565-574, 2009]]>Coin flipping is a fundamental cryptographic primitive that enables two distrustful and far apart parties to create a uniformly random bit (Blu81). Quantum information allows for protocols in the information theoretic setting where no dishonest party can perfectly cheat. The previously best-known quantum protocol by Ambainis achieved a cheating probability of at most 3/4 (Amb01). On the other ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 527-533, 2009]]>In orthogonal range reporting we are to preprocess N points in d-dimensional space so that the points inside a d-dimensional axis-aligned query box can be reported efficiently. This is a fundamental problem in various fields, including spatial databases and compu- tational geometry. In this paper we provide a number of improvements for three and higher dimensional orthogonal ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 149-158, 2009]]>We give an explicit construction of an�-biased set overk bits of sizeO � k

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 191-197, 2009]]>We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1 ϵ)-fraction of the examples, it is NP-hard to find a halfspace that is correct on (1 2 + ϵ)-fraction of the examples, for arbitrary constant ϵ > 0. ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 385-394, 2009]]>Canetti, Goldreich, Goldwasser, and Micali (STOC 2000) introduced the notion of resettable zero-knowledge proofs, where the protocol must be zero-knowledge even if a cheating verifier can reset the prover and have several interactions in which the prover uses the same random tape. Soon afterwards, Barak, Goldreich, Goldwasser, and Lindell (FOCS 2001) studied the closely related notion of resettable ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 251-260, 2009]]>We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments. This inequality allows us to simplify and strengthen several known direct-product theorems and establish new threshold direct-product ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 221-229, 2009]]>We present two probability inequalities. The simpler first inequality weakens both hypotheses in Hoffding-Azumaine quality. Using it, we generalize concentration results previously known for the uniform density for the TSP, MWST and Random Projections to long-tailed inhomogeneous distributions. The second more complicated inequality further weakens the moment requirements and using it, we prove the best possible concentration for ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 211-220, 2009]]>We show how to efficiently extract truly ran- dom bits from two independent sources of linear min-entropy, under a computational assumption. The assumption we rely on is the existence of an efficiently computable permutation f 1, such that for any source X ∈{ 0,1}n with linear min- entropy, any circuit of size poly(n) cannot invert f(...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 617-626, 2009]]>We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic im- provements on combinatorial algorithms for dense BMM in many years, improving on the "Four Russians" O(n3/(wlogn)) bound for machine models with wordsize w. (For a pointer machine, we can set w = logn.) The ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 745-754, 2009]]>We provide the first sub-linear sketching algorithm for estimating the planar Earth-Mover Distance with a constant approximation. For sets living in the two-dimensional grid (¢)2, we achieve space ¢† for approximation O(1/†), for any desired 0 < † < 1. Our sketch has immediate applications to the streaming and nearest neighbor search problems.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 324-330, 2009]]>The PCP theorem asserts the existence of proofs that can be verified by a verifier that reads only a very small part of the proof. The theorem was originally proved by Arora and Safra (J. ACM 45(1)) and Arora et al. (J. ACM 45(3)) using sophisticated algebraic tools. More than a decade later, Dinur (J. ACM 54(3)) ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 463-471, 2009]]>Motivated by applications in cryptography, we con- sider a generalization of randomness extraction and the related no- tion of privacy amplification to the case of two correlated sources. We introduce the notion of correlation extractors, which extract nearly perfect independent instances of a given joint distribution from imperfect, or "leaky," instances of the same distribution. More concretely, suppose that Alice ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 261-270, 2009]]>Under CPA and CCA1 attacks, a secure bit encryp- tion scheme can be applied bit-by-bit to construct a secure many-bit encryption scheme. The same construction fails, however, under a CCA2 attack. In fact, since the notion of CCA2 security was introduced by Rackoff and Simon (21), it has been an open question to determine whether single bit ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 607-616, 2009]]>We study solution sets to systems of generalized linear equations of the form

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 43-52, 2009]]>This paper ties the line of work on algorithms that find an O(¿(log n))-approximation to the SPARSEST CUT together with the line of work on algorithms that run in subquadratic time by using only single-commodity flows. We present an algorithm that simultaneously achieves both goals, finding an O(¿(log (n)/¿))-approximation using O(n¿ ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 363-372, 2009]]>Gupta et al. (13) introduced a very general multi- commodity flow problem in which the cost of a given flow solution on a graph G = (V;E) is calculated by first computing the link loads via a load-function ', that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 32-40, 2009]]>In this paper, we analyze the following communication complexity problem. It is a variant of the set-disjointness problem, denoted PDISJlog N, where each of Alice and Bob gets as an input a subset of [N] of size at most log N, with the promise that the intersection of the two subsets is of size at most 1. We provide ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 63-72, 2009]]>We present several results about Delaunay triangu- lations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffle-operation in constant ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 139-148, 2009]]>We give a new model of learning motivated by smoothed analysis (Spielman and Teng, 2001). In this model, we analyze two new algorithms, for PAC-learning DNFs and agnostically learning decision trees, from random examples drawn from a constant-bounded product distributions. These two problems had previously been solved using membership queries (Jackson, 1995; Gopalan et al, 2005). Our analysis ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 395-404, 2009]]>We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has a vertex set X of order at most k such that G¡X is planar (we call this class of graphs k-apex), and if this is the case, computes a drawing of the graph in the plane after ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 639-648, 2009]]>We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the kth smallest eigenvalue of the Laplacian on a bounded-degree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 735-744, 2009]]>Abstract—Graphs,resulting from,human,behavior,(the web graph, friendship graphs, etc.) have hitherto been viewed,as,a,monolithic,class of graphs,with,similar characteristics; for instance, their degree distributions are markedly,heavy-tailed. In this paper,we,take our under- standing,of behavioral,graphs,a step further by showing that an intriguing empirical,property,of web,graphs,— their compressibility — ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 331-340, 2009]]>In the classical balls-and-bins paradigm, where n balls are placed independently and uniformly in n bins, typically the number of bins with at least two balls in them is £(n) and the maximum number of balls in a bin is £( log n log log n). It is well known that when each round ofiers k independent ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 230-238, 2009]]>We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, vol. 16, pp. 485-494, 2009]]>We generalize the Kahn-Kalai-Linial (KKL) Theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes. An example corollary is that for every f : (n) k ! f0; ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 725-734, 2009]]>For any fixed 1 > ¿ > 0 we present a fully dynamic algorithm for maintaining (2 + ¿)-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the expected amortized update time. Our query time O(log log ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 693-702, 2009]]>Conference: IEEE Symposium on Foundations of Computer Science - FOCS, 2009]]>

The central problem in computational mechanism de- sign is the tension between incentive compatibility and computational efficiency. We establish the first signifi- cant approximability gap between algorithms that are both truthful and computationally-efficient, and algorithms that only achieve one of these two desiderata. This is shown in the context of a novel mechanism design problem which we call the ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 250-259, 2008]]>We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector x2 Rn from its lower-dimensional sketch Ax2 Rm. Specif- ically, we focus on the sparse recovery problem in the L1 norm: for a parameter k, given the sketch Ax, compute an approximation ^ x of x such that the L1 approximation errorkx ^ ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 199-207, 2008]]>We present the first monotone randomized polynomial- time approximation scheme (PTAS) for minimizing the makespan of parallel related machines (Q||Cmax), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming P 6= NP). Our algorithmic ...

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 15-24, 2008]]>The Gibbard-Satterthwaite theorem states that every non- trivial voting method among at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipu- lation by a single random voter will succeed with non- negligible probability for every neutral voting method among 3 alternatives that is far from being a dictatorship.

Conference: IEEE Symposium on Foundations of Computer Science - FOCS, pp. 243-249, 2008]]>