Property Testing of LP-Type Problems

TL;DR

This paper introduces a property testing framework for LP-Type problems, enabling efficient constraint evaluation with query complexities independent of the total number of constraints, bridging optimization and property testing.

Contribution

It develops the first systematic connection between LP-Type problems and property testing, providing query-efficient testers for these problems with tight bounds.

Findings

Query complexity is independent of the number of constraints.

Established tight bounds for testing clusterability.

Connected LP-Type problems with property testing systematically.

Abstract

Given query access to a set of constraints $S$, we wish to quickly check if some objective function $\varphi$ subject to these constraints is at most a given value $k$. We approach this problem using the framework of property testing where our goal is to distinguish the case $\varphi(S) \le k$ from the case that at least an $\epsilon$ fraction of the constraints in $S$ need to be removed for $\varphi(S) \le k$ to hold. We restrict our attention to the case where $(S, \varphi)$ are LP-Type problems which is a rich family of combinatorial optimization problems with an inherent geometric structure. By utilizing a simple sampling procedure which has been used previously to study these problems, we are able to create property testers for any LP-Type problem whose query complexities are independent of the number of constraints. To the best of our knowledge, this is the first work that connects the area of LP-Type problems and property testing in a systematic way. Among our results is a tight upper bound on the query complexity of testing clusterability with one cluster considered by Alon, Dar, Parnas, and Ron (FOCS 2000). We also supply a corresponding tight lower bound for this problem and other LP-Type problems using geometric constructions.