Satisfiability is quasilinear complete in NQL

Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.2

On semiring complexity of Schur polynomials

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables $k$ is fixed

Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP$_{{\mathbb R}}$ under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201

Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page

Nonnegative Rank Measures and Monotone Algebraic Branching Programs

Inspired by Nisan\u27s characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove a rather tight lower bound for the computation of elementary symmetric polynomials by algebraic branching programs in the monotone setting or, equivalently, in the homogeneous syntactically multilinear setting

Notes on Boolean Read-k and Multilinear Circuits

A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant of f, the polynomial contains at least one monomial with the same set of variables, each appearing with degree at most k. Every monotone circuit is a read-k circuit for some k. We show that already read-1 (OR,AND) circuits are not weaker than monotone arithmetic constant-free (+,x) circuits computing multilinear polynomials, are not weaker than non-monotone multilinear (OR,AND,NOT) circuits computing monotone Boolean functions, and have the same power as tropical (min,+) circuits solving combinatorial minimization problems. Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied Mathematic