A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits

Abstract

We give a nontrivial algorithm for the satisfiability problem for threshold circuits of depth two with a linear number of wires which improves over exhaustive search by an exponential factor. The independently interesting problem of the feasibility of sparse 0-1 integer linear programs is a special case. To our knowledge, our algorithm is the first to achieve constant savings even for the special case of Integer Linear Programming. The key idea is to reduce the satisfiability problem to the Vector Domination problem, the problem of checking whether there are two vectors in a given collection of vectors such that one dominates the other component-wise.Our result generalizes to formulas of arbitrary constant depth. We also provide a satisfiability algorithm with constant savings for depth two circuits with symmetric gates where the total weighted fan-in is at most linear in the number of variables.One of our motivations is proving strong lower bounds for TC0 circuits, exploiting the connection (established by Williams) between satisfiability algorithms and lower bounds. Our second motivation is to explore the connection between the expressive power of the circuits and the complexity of the corresponding circuit satisfiability problem.

A Satisfiability Algorithm for Some Class of Dense Depth Two Threshold Circuits

“…Note that the statement of Theorem 3.1 improves the following previous result by Impagliazzo et al [13] in the sense that we can construct a nontrivial algorithm when we relax some condition on sparsity in input circuits. We first give a rough and qualitative sketch of the outline of the algorithm in [13].…”

“…Lets consider the following parameterized problem: for given depth two threshold circuit C of size n c which is sparse in independent gates such that there exists an unique maximal independent gate set I of size greater than k , compute YES iff C is satisfiable, where c is a constant and parameter k in this problem k is the maximum size of independent gate set of C except I . This improves the previous result in [13] in the sense that we can construct a nontrivial algorithm when we relax some condition on sparsity in input circuits. We can consider a family of circuits with high dependency which can compute all boolean functions.More details about this fact are mentioned in the later.…”

“…So understanding satisfiability of depth two threshold circuits could give us various view points on both theoretical and practical areas including the above three problems. In the paper [13], Impagliazzo et al constructed the first nontrivial algorithm with constant savings in the exponent over brute force search for the satisfiability of sparse depth two threshold circuits which has cn-wires for every constant c. As a consequence, they also got a similar result for linear-size ILP. Here we say an algorithm is nontrivial, if its running time is bounded above by 2 n /w(n) where n is the number of input variables and w(n) is a super-polynomial function in n. Note that 2 n is just the number of assignments to n input variables.…”

“…We first give a rough and qualitative sketch of the outline of the algorithm in [13]. For given depth two sparse threshold circuits, they give three reductions: the first one transforms an arbitrary ILP instance with small number of inequalities to an instance of vector domination problem, and the second one transforms any depth two circuit with small number of gates to an union of ILP instances with small number of inequalities, and the third one transforms a depth two threshold circuit with linear number of wires to an union of depth two circuits with small number of gates.…”

“…Some algorithms solving CNF-SAT and MAX-SAT with constant savings when the formula is linear size are given in [17] and [7]. Assuming the SETH, we can not to solve satisfiability problem for super linear size depth two threshold circuits with constant savings as a direct extension of the result in [13].…”