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Yoshida Y.,Preferred Infrastructure | Zhou Y.,Carnegie Mellon University
ITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science | Year: 2014

We consider approximation schemes for the maximum constraint satisfaction problems and the maximum assignment problems. Though they are NP-Hard in general, if the instance is "dense" or "locally dense", then they are known to have approximation schemes that run in polynomial time or quasi-polynomial time. In this paper, we give a unified method of showing these approximation schemes based on the Sherali-Adams linear programming relaxation hierarchy. We also use our linear programming-based framework to show new algorithmic results on the optimization version of the hypergraph isomorphism problem. Copyright 2014 ACM. Source


Yoshida Y.,Preferred Infrastructure
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2016

Let {fi : Fip → {0, 1}} be a sequence of functions, where p is a fixed prime and Fp is the finite field of order p. The limit of the sequence can be syntactically defined using the notion of ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of function sequences, and study properties of it. One application of this metric is that it provides a simpler characterization of affine-invariant parameters of functions that are constant-query estimable than the previous one obtained by Yoshida (STOC'14). Using this characterization, we show that the property of being a function of a constant number of low-degree polynomials and a constant number of factored polynomials (of arbitrary degrees) is constant-query testable if it is closed under blowing-up. Examples of this property include the property of having a constant spectral norm and degree-structural properties with rank conditions. © Copyright (2016) by SIAM: Society for Industrial and Applied Mathematics. Source


Yoshida Y.,Preferred Infrastructure
Computational Complexity | Year: 2014

In the ListH-Homomorphism Problem, for a graph H that is a parameter of the problem, an instance consists of an undirected graph G with a list constraint (Formula presented.) for each variable (Formula presented.), and the objective is to determine whether there is a list H-homomorphism (Formula presented.), that is, (Formula presented.) for every (Formula presented.) and (Formula presented.) whenever (Formula presented.).We consider the problem of testing list H-homomorphisms in the following weighted setting: An instance consists of an undirected graph G, list constraints L, weights imposed on the vertices of G, and a map (Formula presented.) given as an oracle access. The objective is to determine whether f is a list H-homomorphism or far from any list H-homomorphism. The farness is measured by the total weight of vertices (Formula presented.) for which f(v) must be changed so as to make f a list H-homomorphism. In this paper, we classify graphs H with respect to the number of queries to f required to test the list H-homomorphisms. Specifically, we show that (i) list H-homomorphisms are testable with a constant number of queries if and only if H is a reflexive complete graph or an irreflexive complete bipartite graph and (ii) list H-homomorphisms are testable with a sublinear number of queries if and only if H is a bi-arc graph. © 2014 Springer Basel Source


Yoshida Y.,Preferred Infrastructure
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2011

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it. In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the following: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are unconditional. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all. A CSP instance is called ε-far from satisfiability if we must remove at least an ε-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from ε-far instances with probability at least 2/3. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model. © 2011 ACM. Source


Yoshida Y.,Preferred Infrastructure
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2014

Let P be a property of function Fp n → {0; 1} for a fixed prime p. An algorithm is called a tester for P if, given a query access to the input function f, with high probability, it accepts when f satisfies P and rejects when f is "far" from satisfying P. In this paper, we give a characterization of affine-invariant properties that are (two-sided error) testable with a constant number of queries. The characterization is stated in terms of decomposition theorems, which roughly claim that any function can be decomposed into a structured part that is a function of a constant number of polynomials, and a pseudo-random part whose Gowers norm is small. We first give an algorithm that tests whether the structured part of the input function has a specific form. Then we show that an affine-invariant property is testable with a constant number of queries if and only if it can be reduced to the problem of testing whether the structured part of the input function is close to one of a constant number of candidates. © 2014 ACM. Source

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