Parameterized Algorithms for Constraint Satisfaction Problems Above Average with Global Cardinality Constraints

TL;DR

This paper introduces a fixed-parameter tractable algorithm for solving CSPs with global cardinality constraints that exceed the average number of satisfied constraints, improving efficiency for certain problem instances.

Contribution

The paper presents a novel fixed-parameter algorithm for CSPs with global cardinality constraints, achieving efficient solutions above the average in parameterized time.

Findings

Algorithm finds solutions exceeding the average in time (2^{O(t^2)} + n^{O(d)})

CSP above average with global constraints is fixed-parameter tractable

Provides a new approach to solving constrained CSPs efficiently

Abstract

Given a constraint satisfaction problem (CSP) on $n$ variables, $x_1, x_2, \dots, x_n \in \{\pm 1\}$, and $m$ constraints, a global cardinality constraint has the form of $\sum_{i = 1}^{n} x_i = (1-2p)n$, where $p \in (\Omega(1), 1 - \Omega(1))$ and $pn$ is an integer. Let $AVG$ be the expected number of constraints satisfied by randomly choosing an assignment to $x_1, x_2, \dots, x_n$, complying with the global cardinality constraint. The CSP above average with the global cardinality constraint problem asks whether there is an assignment (complying with the cardinality constraint) that satisfies more than $(AVG+t)$ constraints, where $t$ is an input parameter. In this paper, we present an algorithm that finds a valid assignment satisfying more than $(AVG+t)$ constraints (if there exists one) in time $(2^{O(t^2)} + n^{O(d)})$. Therefore, the CSP above average with the global cardinality constraint problem is fixed-parameter tractable.