Property Testing of LP-Type Problems

Abstract

Given query access to a set of constraints S, we wish to quickly check if some objective function ϕ 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 ϕ(S) ≤ k from the case that at least an ǫ fraction of the constraints in S need to be removed for ϕ(S) ≤ k to hold. We restrict our attention to the case where (S, ϕ) 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.