Strongly refuting all semi-random Boolean CSPs

09/17/2020
by   Jackson Abascal, et al.
0

We give an efficient algorithm to strongly refute semi-random instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best-known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean k-XOR problem on n variables that have O(n^k/2) constraints. (In a semi-random k-XOR instance, the equations can be arbitrary and only the right-hand sides are random.) One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work. Our approach involves taking an instance that does not satisfy this property (i.e., is not pseudorandom) and reducing it to a partitioned collection of 2-XOR instances. We analyze these subinstances using a carefully chosen quadratic form as a proxy, which in turn is bounded via a combination of spectral methods and semidefinite programming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernstein inequality. Even for the purely random case, this leads to a shorter proof compared to the ones in the literature that rely on problem-specific trace-moment computations.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

09/09/2021

Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"

We present an algorithm for strongly refuting smoothed instances of all ...
02/24/2021

Approximability of all Boolean CSPs in the dynamic streaming setting

A Boolean constraint satisfaction problem (CSP), Max-CSP(f), is a maximi...
04/14/2018

The threshold for SDP-refutation of random regular NAE-3SAT

Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has th...
12/24/2018

Sherali--Adams Strikes Back

Let G be any n-vertex graph whose random walk matrix has its nontrivial ...
05/12/2021

Sufficient reasons for classifier decisions in the presence of constraints

Recent work has unveiled a theory for reasoning about the decisions made...
06/24/2021

Certifying solution geometry in random CSPs: counts, clusters and balance

An active topic in the study of random constraint satisfaction problems ...
05/24/2013

Integrating tabu search and VLSN search to develop enhanced algorithms: A case study using bipartite boolean quadratic programs

The bipartite boolean quadratic programming problem (BBQP) is a generali...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.