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%2F20%3A00342828" target="_blank" >RIV/68407700:21240/20:00342828 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1145/3397484" target="_blank" >https://doi.org/10.1145/3397484</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1145/3397484" target="_blank" >10.1145/3397484</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 ℓ columns, x is a vector of ℓ variables, and b is a vector of k integers, we ask whether there exists x element N ℓ 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 Ainfinity, the largest absolute value of an entry in A, are small. Papadimitriou was the first to give a fixed-parameter algorithm for ILP Feasibility under parameterization by the number of constraints that runs in time ((Ainfinity + binfinity) ⋅ k)O(k2). This was very recently improved by Eisenbrand and Weismantel, who used the Steinitz lemma to design an algorithm with running time (kAinfinity)O(k) ⋅ log binfinity, which was subsequently refined by Jansen and Rohwedder to O( kAinfinity)k ⋅ log ( Ainfinity + binfinity) ⋅ log Ainfinity. 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 2o(k log k) ⋅ (ℓ + binfinity)o(k) would contradict the exponential time hypothesis. This improves previous non-tight lower bounds of Fomin et al. 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 Koutecký et al. that ILP Feasibility can be solved in time Ainfinity2O(tdD(A)) ⋅ (k + ℓ + log binfinity)O(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 22o(tdD(A)) ⋅ (k + ℓ)O(1) would contradict the exponential time hypothesis.

Czech name

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Czech description

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Type

J_SC - Article in a specialist periodical, which is included in the SCOPUS database

CEP classification

—

OECD FORD branch

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

Project

<a href="/en/project/EF16_019%2F0000765" target="_blank" >EF16_019/0000765: Research Center for Informatics</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2020

Confidentiality

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

Name of the periodical

ACM Transactions on Computation Theory

ISSN

1942-3454

e-ISSN

1942-3462

Volume of the periodical

12

Issue of the periodical within the volume

3

Country of publishing house

US - UNITED STATES

Number of pages

19

Pages from-to

"19:1"-"19:19"

UT code for WoS article

000583677700005

EID of the result in the Scopus database

2-s2.0-85088097305