Mixed Integer Programming with Convex/Concave Constraints: Fixed-Parameter Tractability and Applications to Multicovering and Voting

TL;DR

This paper extends fixed-parameter tractability results for integer programming to include convex and concave functions, enabling efficient algorithms for problems in multicovering and electoral social choice.

Contribution

It introduces a general technique to establish FPT results for mixed integer programs with convex/concave constraints, solving longstanding open problems.

Findings

Weighted Set Multicover is FPT when parameterized by the number of elements.

An FPT approximation scheme exists for Multiset Multicover.

Problems in electoral bribery and control are FPT when parameterized by the number of candidates.

Abstract

A classic result of Lenstra [Math.~Oper.~Res.~1983] says that an integer linear program can be solved in fixed-parameter tractable (FPT) time for the parameter being the number of variables. We extend this result by incorporating non-decreasing piecewise linear convex or concave functions to our (mixed) integer programs. This general technique allows us to establish parameterized complexity of a number of classic computational problems. In particular, we prove that Weighted Set Multicover is in FPT when parameterized by the number of elements to cover, and that there exists an FPT-time approximation scheme for Multiset Multicover for the same parameter. Further, we use our general technique to prove that a number of problems from computational social choice (e.g., problems related to bribery and control in elections) are in FPT when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems, and for weighted electoral control of Approval voting, this improves some previously known XP-memberships to FPT-memberships.