Satisfying More Than Half of a System of Linear Equations Over GF(2): A Multivariate Approach

In the parameterized problem MaxLin2-AA[k ], we are given a system with variables x1,…,xnx1,…,xn consisting of equations of the form ∏i∈Ixi=b∏i∈Ixi=b, where xi,b∈{−1,1}xi,b∈{−1,1} and I⊆[n]I⊆[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+kW/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0k=0). We show that MaxLin2-AA[k ] has a kernel with at most View the MathML sourceO(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1)2O(klogk)(nm)O(1). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,rk,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove that Max-r-Lin2-AA[k,rk,r] has a kernel with at most (2k−1)r(2k−1)r variables

Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$,$[n]=\{1,2,...,n\}$, and$\chi_I(x)=\prod_{i\in I}x_i$and$\hat{f}(I)$are non-zero reals. The degree of$f$is$\max \{|I|:\ I\in {\cal F}\}$and the width of$f$is the minimum integer$\rho$such that every$i\in [n]$appears in at most$\rho$sets in$\cal F$. For$i\in [n]$, let$\mathbf{x}_i$be a random variable taking values 1 or -1 uniformly and independently from all other variables$\mathbf{x}_j$,$j\neq i.$Let$\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The$p$-norm of$f$is$||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$for any$p\ge 1$. It is well-known that$||f||_q\ge ||f||_p$whenever$q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on$f$: if$f$is of degree$d$and$q> p>1$then$ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$This inequality is called the Hypercontractive Inequality. We show that one can replace$d$by$\rho$in the Hypercontractive Inequality for each$q> p\ge 2$as follows:$ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$where$r=\lceil q/2\rceil$. For the case$q=4$and$p=2$, which is important in many applications, we prove a stronger inequality:$ ||f||_4\le (2\rho+1)^{1/4}||f||_2.

Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzik (1986) proved that every connected oriented graph G on n vertices and m arcs contains an acyclic subgraph with at least m/2+(n-1)/4 arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does G have an acyclic subgraph with least m/2 + (n-1)/4 + k arcs, where k is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime (12k)!n^{O(1)}. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size O(k^2)

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency