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Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v_1,...,v_m in R^d and a constraint family B of subsets of [m], find a set S in B that maximizes the squared volume of the simplex spanned by the vectors in S. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is...
We consider the following basic problem: given an n-variate degree-d homogeneous polynomial f with real coefficients, compute a unit vector x in R{\string^}n that maximizes abs(f(x)). Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree-2 case is efficiently solvable as...
We show that approximate similarity (near neighbour) search can be solved in high dimensions with performance matching state of the art (data independent) Locality Sensitive Hashing, but with a guarantee of no false negatives. Specifically we give two data structures for common problems. For c-approximate near neighbour in Hamming space, for which we get query time dn^{1/c+o(1)} and space dn^{1+1/c+o(1)}...
We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters β (the interaction) and λ (the external field), except for the case |λ|=1 (the...
We present a novel input sensitive analysis of a deterministic online algorithm \cite{r_approx16} for the minimum metric bipartite matching problem. We show that, in the adversarial model, for any metric space \metric and a set of n servers S, the competitive ratio of this algorithm is O(\mu_{\metric}(S)\log^2 n); here \mu_{\metric}(S) is the maximum ratio of the traveling salesman tour and the diameter...
We present a deterministic distributed algorithm that computes a (2δ-1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree δ, in O(log^8 δ ⋅ log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g.,...
We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation Õ(n)-time algorithms [2], [3], which are matched with \tilde Ω(n)-time lower bounds [4], [5],\footnote{\tilde \Theta, Õ...
In the set disjointess problem, we have k players, each with a private input X^i ⊆ [n], and the goal is for the players to determine whether or not their sets have a global intersection. The players communicate over a shared blackboard, and we charge them for each bit that they write on the board.We study the trade-off between the number of interaction rounds we allow the players, and the...
We prove new lower bounds on the sizes of proofs in the Cutting Plane proof system, using a concept that we call unsatisfiability certificate. This approach is, essentially, equivalent to the well-known feasible interpolation method, but is applicable to CNF formulas that do not seem suitable for interpolation. Specifically, we prove exponential lower bounds for random k-CNFs, where k is the logarithm...
Presents the introductory welcome message from the conference proceedings. May include the conference officers' congratulations to all involved with the conference event and publication of the proceedings record.
We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers...
We give a deterministic \tilde{O}(\log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(\log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by...
In the minimum planarization} problem, given some n-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a \log^{O(1)} n-approximation algorithm for this problem on general graphs with running time n^{O(\log n/\log\log n)}. We also obtain a O(n^≥)-approximation...
Consider the problem of estimating the (un)reliability of an n-vertex graph when edges fail with probability p. We show that the Recursive Contraction Algorithms for minimum cuts, essentially unchanged and running in n2+o(1) time, yields an unbiased estimator of constant relative variance (and thus an FPRAS with the same time bound) whenever pc
We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in O(\log^3 n) time on n^{O(\log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm...
We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples. As a special case, this gives a new proof for the time-space lower bound for parity learning [R16]. Our result is stated in terms of the norm of the matrix...
Interactive coding, pioneered by Schulman (FOCS 92, STOC 93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversarys discretion, as they pass through the communication channel. Braverman, Gelles, Mao, and Ostrovsky (2015) proposed a far-reaching generalization...
In a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP(Γ)...
More than 25 years ago, inspired by applications in computer graphics, Chazelle \etal (FOCS 1991) studied the following question: Is it possible to cut any set of n lines or other objects in \Reals^3 into a subquadratic number of fragments such that the resulting fragments admit a depth order? They managed to prove an O(n^{9/4}) bound on the number of fragments, but only for the very special case...
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