25,194 research outputs found

    Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

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    In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNPR_{{\mathbb R}} under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201

    2-Server PIR with sub-polynomial communication

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    A 2-server Private Information Retrieval (PIR) scheme allows a user to retrieve the iith bit of an nn-bit database replicated among two servers (which do not communicate) while not revealing any information about ii to either server. In this work we construct a 1-round 2-server PIR with total communication cost nO(loglogn/logn)n^{O({\sqrt{\log\log n/\log n}})}. This improves over the currently known 2-server protocols which require O(n1/3)O(n^{1/3}) communication and matches the communication cost of known 3-server PIR schemes. Our improvement comes from reducing the number of servers in existing protocols, based on Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing these protocols in an algebraic way (using polynomial interpolation) and extending them using partial derivatives

    Which groups are amenable to proving exponent two for matrix multiplication?

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    The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding ω\omega in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on ω\omega and is conjectured to be powerful enough to prove ω=2\omega = 2, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove ω=2\omega = 2 in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove ω=2\omega = 2 in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over (Z/pZ)n(\mathbb{Z}/p\mathbb{Z})^n that are degree dd polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups SnS_n cannot prove nontrivial bounds on ω\omega when the embedding is via three Young subgroups---subgroups of the form Sk1×Sk2××SkS_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}---which is a natural strategy that includes all known constructions in SnS_n. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on ω\omega and methods for ruling them out.Comment: 23 pages, 1 figur

    Approximating the Largest Root and Applications to Interlacing Families

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    We study the problem of approximating the largest root of a real-rooted polynomial of degree nn using its top kk coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in kk that use the top kk coefficients to approximate the maximum root within a factor of n1/kn^{1/k} and 1+O(lognk)21+O(\tfrac{\log n}{k})^2 when klognk\leq \log n and k>lognk>\log n respectively. We also prove corresponding information-theoretic lower bounds of nΩ(1/k)n^{\Omega(1/k)} and 1+Ω(log2nkk)21+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of 2O~(n3)2^{\tilde O(\sqrt[3]n)} for all of them

    Superselectors: Efficient Constructions and Applications

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    We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel conflict resolution and data security. We prove close upper and lower bounds on the size of superselectors and we provide efficient algorithms for their constructions. Albeit our bounds are very general, when they are instantiated on the combinatorial structures that are particular cases of superselectors (e.g., (p,k,n)-selectors, (d,\ell)-list-disjunct matrices, MUT_k(r)-families, FUT(k, a)-families, etc.) they match the best known bounds in terms of size of the structures (the relevant parameter in the applications). For appropriate values of parameters, our results also provide the first efficient deterministic algorithms for the construction of such structures
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