Approximability of Minimum AND-Circuits

Abstract

Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than 1.0051 unless P = NP, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of 1.278. For the general problem, we achieve an approximation ratio of d − 3/2, where d is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the Minimum ANDCircuit problem and several problems from different areas.

Improved approximation algorithms for minimum AND-circuits problem via k-set cover

“…For Min-3-AC, without loss of generality, we can assume that all monomials have degree exactly three as Arpe and Manthey [1] have shown. If a given instance includes monomials with degree one or two, then we solve an instance with degree exactly three which is preprocessed as follows, instead of the original instance.…”

“…According to Charikar et al [2,Section VIII.B], this problem has been studied extensively in the context of automated circuit design, and no approximation guarantees were, however, known. Arpe and Manthey [1] were the first who studied the problem from a complexity theoretic standpoint, and showed some results including approximation algorithms, APX-hardness and fixed parameter tractability. For details of the results and the background, see their paper [1].…”

“…Arpe and Manthey [1] were the first who studied the problem from a complexity theoretic standpoint, and showed some results including approximation algorithms, APX-hardness and fixed parameter tractability. For details of the results and the background, see their paper [1].…”

“…1 Present affiliation: Interdisciplinary Faculty of Science and Engineering, Shimane University, Shimane 690-8504, Japan. at most three, is APX-hard and have given three approximation algorithms called Cover, Match and Greedy.…”

“…Arpe and Manthey [1] have shown that Min-3-AC, which is Min-AC with instances restricted to monomials of degree E-mail address: morizumi@cis.shimane-u.ac.jp. 1 Present affiliation: Interdisciplinary Faculty of Science and Engineering, Shimane University, Shimane 690-8504, Japan.…”