View in EDS

HOLOGRAPHIC ALGORITHMS.

Saved in:
Bibliographic Details
Title: HOLOGRAPHIC ALGORITHMS.
Authors: VALIANT, LESLIE G.
Source: SIAM Journal on Computing; 2007, Vol. 37 Issue 5, p1565-1594, 30p, 8 Diagrams, 2 Charts
Subject Terms: HOLOGRAPHY, ALGORITHM research, POLYNOMIAL time algorithms, COMPUTATIONAL complexity, MANY-valued logic, COMBINATORIAL enumeration problems, COMPUTATIONAL mathematics, MATCHING theory
Abstract: Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in one-to-one, or possibly one-to-many, fashion. In this paper we propose a new kind of reduction that allows for gadgets with many-to-many correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #P-complete problem, then polynomial systems can be obtained, the solvability of which would imply P#P = NC2. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Complementary Index
Description
Abstract:Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in one-to-one, or possibly one-to-many, fashion. In this paper we propose a new kind of reduction that allows for gadgets with many-to-many correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #P-complete problem, then polynomial systems can be obtained, the solvability of which would imply P<sup>#P</sup> = NC2. [ABSTRACT FROM AUTHOR]
ISSN:00975397
DOI:10.1137/070682575