TY - JOUR

T1 - Mixed integer programming with convex/concave constraints

T2 - Fixed-parameter tractability and applications to multicovering and voting

AU - Bredereck, Robert

AU - Faliszewski, Piotr

AU - Niedermeier, Rolf

AU - Skowron, Piotr

AU - Talmon, Nimrod

N1 - Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/4/24

Y1 - 2020/4/24

N2 - 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—this is our most technical result. 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.

AB - 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—this is our most technical result. 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.

KW - Max cover

KW - Multiset multicover

KW - Parameterized complexity

KW - Weighted set multicover

UR - http://www.scopus.com/inward/record.url?scp=85078936557&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2020.01.017

DO - 10.1016/j.tcs.2020.01.017

M3 - Article

AN - SCOPUS:85078936557

VL - 814

SP - 86

EP - 105

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -