A parameterized strongly polynomial algorithm for block structured integer programs

Martin Koutecký, Asaf Levin, Shmuel Onn

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The theory of n-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an n-fold integer program (IP) consists of parameter A, dimension n, and numerical data of binary encoding length L. It was known for some time that such programs can be solved in polynomial time using O(ng(A)L) arithmetic operations where g is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable time using O(f(A)n3L) arithmetic operations for a single-exponential function f. This, and a faster algorithm for a special case of combinatorial n-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using O(ng(A)) arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that nfold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f(A)n6 log n) arithmetic operations. In fact, our results are much more general, briefly outlined as follows. There is a strongly polynomial algorithm for integer linear programming (ILP) whenever a so-called Graver-best oracle is realizable for it. Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in fixed-parameter tractable time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial. We show that ILP is fixed-parameter tractable parameterized by the largest coe cient A and the primal or dual treedepth of A, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.

Original languageEnglish
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
ISBN (Electronic)9783959770767
DOIs
StatePublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period9/07/1813/07/18

Keywords

  • Graver basis
  • Integer programming
  • N-fold integer programming
  • Parameterized complexity

All Science Journal Classification (ASJC) codes

  • Software

Fingerprint

Dive into the research topics of 'A parameterized strongly polynomial algorithm for block structured integer programs'. Together they form a unique fingerprint.

Cite this