Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F19%3A00334095" target="_blank" >RIV/68407700:21240/19:00334095 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2019.44" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.STACS.2019.44</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2019.44" target="_blank" >10.4230/LIPIcs.STACS.2019.44</a>

Result language

angličtina

Original language name

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Original language description

We consider the standard ILP FEASIBILITY problem: given an integer linear program of the form {Ax = b, x 0}, where A is an integer matrix with k rows and Ji columns, x is a vector of Ji variables, and b is a vector of k integers, we ask whether there exists x E Ilk that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP FEASIBILITY when both k, the number of constraints, and MAILD.0, the largest absolute value of an entry in A, are small. Papadimitriou [29] was the first to give a fixed-parameter algorithm for ILP FEASIBILITY under parameterization by the number of constraints that runs in time ((MAIL, +1113110.0) " k) lk2). This was very recently improved by Eisenbrand and Weismantel [9], who used the Steinitz lemma to design an algorithm with running time (142410.) (k) "11b112, which was subsequently improved by Jansen and Rohwedder [17] to 0(14241o)k " log MbIfx,. We prove that for {0, 1}-matrices A, the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time 2 (k log k) " 14 0Y(k) would contradict the Exponential Time Hypothesis (ETH). This improves previous non-tight lower bounds of Fomin et al. [10]. We then consider integer linear programs that may have many constraints, but they need to be structured in a "shallow" way. Precisely, we consider the parameter dual treedepth of the matrix A, denoted tdD (A), which is the treedepth of the graph over the rows of A, where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by KouteckST 0 (td (A)) et al. [24] that ILP FEASIBILITY can be solved in time 112,1112 D (k loglIblfx,)(9(1). We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {-1, 0, 1}, the existence of an algorithm with running time 2' tdp (A)) " (k + f)(9(1) would contradict the ETH.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany

ISBN

978-3-95977-100-9

ISSN

—

e-ISSN

1868-8969

Number of pages

15

Pages from-to

"44:1"-"44:15"

Publisher name

Schloss Dagstuhl - Leibniz Center for Informatics

Place of publication

Wadern

Event location

Berlín

Event date

Mar 13, 2019

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

000472795800043