Mixed integer programming with convex/concave constraints: Fixed-parameter tractability and applications to multicovering and voting

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 parameterization by the number of variables. We extend this result by incorporating piecewise linear convex or concave functions to our (mixed) integer programs. This general technique allows us to analyze the parameterized complexity of a number of classic NP-hard 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.

Fair Knapsack

“…We will use the classic result of Lenstra (1983) which says that an integer linear program (ILP) can be solved in FPT time with respect to the number of integer variables. We will also use a recent result of Bredereck et al (2017) who proved that one can apply concave/convex transformations of certain variables in an ILP, and that such a modified program can be still solved in an FPT time. We construct an ILP as follows.…”

“…The above program uses concave transformations (logarithms) for the maximized expression, and convex transformations (functions f z ) in the left-hand sides of the constraints, so we can use the result of Bredereck et al (2017) and claim that this program can be solved in an FPT time with respect to the number of integer variables. This completes the proof.…”

Winner Determination and Strategic Control in Conditional Approval Voting

“…The reason is that this can be reduced to an FPT version of the SET MULTI-COVER. Proposition 1 (implied by [Bredereck et al, 2020]). CAV-1 and CDV-1 can be solved in polynomial time for Unconditional Minisum if the domain size of each issue is constant.…”

Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

“…Additionally, let us consider the Multiset Multicover Problem. This problem has received quite a lot of attention in recent papers [5,6,17,24,25,26]. An exact (c max + 1) n • poly(φ)-complexity algorithm for this problem, parameterized by the universe size n and the maximum coverage constraint number c max , is given in [24,25].…”