Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Popis výsledku anglicky

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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

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

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název statě ve sborníku

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

Počet stran výsledku

15

Strana od-do

"44:1"-"44:15"

Název nakladatele

Schloss Dagstuhl - Leibniz Center for Informatics

Místo vydání

Wadern

Místo konání akce

Berlín

Datum konání akce

13. 3. 2019

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000472795800043