18,345 research outputs found

    The matching polytope does not admit fully-polynomial size relaxation schemes

    Full text link
    The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed 0<ε<10 < \varepsilon < 1, every polyhedral (1+ε/n)(1 + \varepsilon / n)-approximation requires an exponential number of inequalities, where nn is the number of vertices. This is sharp given the well-known ρ\rho-approximation of size O((nρ/(ρ1)))O(\binom{n}{\rho/(\rho-1)}) provided by the odd-sets of size up to ρ/(ρ1)\rho/(\rho-1). Thus matching is the first problem in PP, whose natural linear encoding does not admit a fully polynomial-size relaxation scheme (the polyhedral equivalent of an FPTAS), which provides a sharp separation from the polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets mentioned above. Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the main lower bounding technique is different. While the original proof is based on the hyperplane separation bound (also called the rectangle corruption bound), we employ the information-theoretic notion of common information as introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/], which allows to analyze perturbations of slack matrices. It turns out that the high extension complexity for the matching polytope stem from the same source of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure

    Non-asymptotic Upper Bounds for Deletion Correcting Codes

    Full text link
    Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for qq-ary alphabet and string length nn is shown to be of size at most qnq(q1)(n1)\frac{q^n-q}{(q-1)(n-1)}. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The non-asymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known non-asymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
    corecore