Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

TL;DR

This paper provides a theoretical analysis of the effectiveness and limitations of polynomial-time preprocessing in solving various AI-related combinatorial problems, highlighting when such preprocessing can or cannot significantly reduce problem size.

Contribution

It establishes the boundaries of polynomial kernelization for key AI problems, showing most cannot be reduced polynomially, except for the AtMost-NValue constraint consistency problem.

Findings

Most problems do not admit polynomial kernels under standard complexity assumptions.

The AtMost-NValue constraint consistency problem admits a quadratic polynomial kernel.

Provides worst-case guarantees for preprocessing algorithms in AI reasoning tasks.

Abstract

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.