The k-path problem asks whether a given graph contains a simple path of length k. Along with other prominent parameterized problems, it reduces to the problem of detecting multilinear terms of degree k (k-MLD), making...
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The k-path problem asks whether a given graph contains a simple path of length k. Along with other prominent parameterized problems, it reduces to the problem of detecting multilinear terms of degree k (k-MLD), making the latter a fundamental problem in parameterized algorithms. This has generated significant efforts directed at devising fast deterministic algorithms for k-MLD, and there are now at least two independent approaches that yield the same record bound on the running time. Namely the combinatorial representative-set based approach of Fomin et al. (JACM'16), and the algebraic techniques of Pratt (FOCS'19) and Brand and Pratt (ICALP'21). In this note, we explore the relationship between the latter results, based on partial differentials, and a previous algebraic approach based on the exterior algebra (Brand, ESA'19;Brand, Dell and Husfeldt, STOC'18). We do so by studying the relevant algebraic objects. These are on the one hand (1) the subalgebras of the tensor square of the exterior algebra generated in degree one. On the other hand, we consider (2) the space of partial derivatives of generic determinants, and closely related, (3) the space of minors of generic matrices. We prove that (2) arises as a quotient of (1), and that there is an isomorphism between the objects (1) and (3). Hence, the techniques are essentially equivalent, and the quotient relation between (2) and both of (1) and (3) hints at a possible refinement of the techniques. (C) 2021 Published by Elsevier B.V.
We give a polynomial-time algorithm to decode multivariate polynomial codes of degree d up to half their minimum distance, when the evaluation points are an arbitrary product set S-m, for every d 0. Our result gives ...
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We give a polynomial-time algorithm to decode multivariate polynomial codes of degree d up to half their minimum distance, when the evaluation points are an arbitrary product set S-m, for every d < vertical bar S vertical bar. Previously known algorithms could achieve this only if the set S had some very special algebraic structure, or if the degree d was significantly smaller than vertical bar S vertical bar. We also give a near-linear-time randomized algorithm, based on tools from list-decoding, to decode these codes from nearly half their minimum distance, provided d < (1- epsilon) vertical bar S vertical bar for constant epsilon > 0. Our result gives an m-dimensional generalization of the well-known decoding algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic version of the Schwartz-Zippel lemma.
The fastest known randomized algorithms for several parameterized problems use reductions to the k-MLD problem: detection of multilinear monomials of degree k in polynomials presented as circuits. The fastest known al...
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The fastest known randomized algorithms for several parameterized problems use reductions to the k-MLD problem: detection of multilinear monomials of degree k in polynomials presented as circuits. The fastest known algorithm for k-MLD is based on 2(k) evaluations of the circuit over a suitable algebra. We use communication complexity to show that it is essentially optimal within this evaluation framework. On the positive side, we give additional applications of the method: finding a copy of a given tree on k nodes, a minimum set of nodes that dominate at least t nodes, and an m-dimensional k-matching. In each case, we achieve a faster algorithm than what was known before. We also apply the algebraic method to problems in exact counting. Among other results, we show that a variation of it can break the trivial upper bounds for the disjoint summation problem.
A constant locality function is one in which each output bit dependson just a constant number of input bits. Viola and Wigderson (2018)gave an explicit construction of bipartite degree-3 Ramanujangraphs such that each...
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A constant locality function is one in which each output bit dependson just a constant number of input bits. Viola and Wigderson (2018)gave an explicit construction of bipartite degree-3 Ramanujangraphs such that each neighbor of a vertex can be computed using aconstant locality function. In this work, we construct the firstexplicit local Ramanujan graph (bipartite) of degree q + 1,where q > 2 is any prime *** and Capalbo (2002) used 4-regular, 8-regular and44-regular Ramanujan graphs to construct unique-neighbor expandersthat were 3-regular, 4-regular, 6-regular and `bipartite'(respectively). Viola and Wigderson (2018) had asked if a localconstruction of such unique-neighbor expanders exists. Ourconstruction gives local 4-regular, 8-regular and 44-regularRamanujan graphs, which also solves the corresponding open problemof the construction of local unique-neighbor *** only known explicit construction of Ramanujan graphs exists fordegree q + 1, where q is a prime-power. In this paper, weessentially localize the explicit Ramanujan graphs for all these degrees. Our results use the explicit Ramanujan graphs byMorgenstern (1994) and a significant generalization of the ideasused in Viola and Wigderson (2018).
For normalized floating point division, digital computers can take advantage of a division process that uses an iterative multiplying operation instead of repeated subtractions. An improvement of this division process...
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For normalized floating point division, digital computers can take advantage of a division process that uses an iterative multiplying operation instead of repeated subtractions. An improvement of this division process by using accelerating constants in the overrelaxation has previously been proposed. Multiplication by a chosen accelerating constant accelerates the process of generating accurate digits of a quotient in division. We propose a further improvement by generalizing the accelerating constants in the overrelaxation method. Two benefits resulting from this improvement promise to yield faster division in digital computers. [ABSTRACT FROM AUTHOR]
An efficient algorithm is presented for the exact calculation of resultants of multivariate polynomials with integer coefficients. The algorithm applies modular homomorphisms and the Chinese remainder theorem, evaluat...
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A computing-time study is presented of several algorithms for the exact solution of dense systems of linear equations with integer or dense polynomial coefficients. The analytical computing times for rational Gauss el...
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