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A classic result of Lenstra [Math.~Oper.~Res.~1983] says that an integerlinear program can be solved in fixed-parameter tractable (FPT) time for theparameter being the number of variables. We extend this result by incorporatingnon-decreasing piecewise linear convex or concave functions to our (mixed)integer programs. This general technique allows us to establish parameterizedcomplexity of a number of classic computational problems. In particular, weprove that Weighted Set Multicover is in FPT when parameterized by the numberof elements to cover, and that there exists an FPT-time approximation schemefor Multiset Multicover for the same parameter. Further, we use our generaltechnique to prove that a number of problems from computational social choice(e.g., problems related to bribery and control in elections) are in FPT whenparameterized by the number of candidates. For bribery, this resolves a nearly10-year old family of open problems, and for weighted electoral control ofApproval voting, this improves some previously known XP-memberships toFPT-memberships.

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