A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10386781" target="_blank" >RIV/00216208:11320/18:10386781 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.4230/LIPIcs.ICALP.2018.85" target="_blank" >https://doi.org/10.4230/LIPIcs.ICALP.2018.85</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.85" target="_blank" >10.4230/LIPIcs.ICALP.2018.85</a>

Result language

angličtina

Original language name

A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

Original language description

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(n^{g(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)n^3L) 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(n^{g(A)}) arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that n-fold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f(A)n^6 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 coefficient |A |_infty 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.

Czech name

—

Czech description

—

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

<a href="/en/project/GA17-09142S" target="_blank" >GA17-09142S: Modern algorithms: New challenges of complex data sets</a><br>

Continuities

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

Publication year

2018

Confidentiality

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

Article name in the collection

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

ISBN

978-3-95977-076-7

ISSN

1868-8969

e-ISSN

neuvedeno

Number of pages

14

Pages from-to

1-14

Publisher name

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Place of publication

Dagstuhl

Event location

Praha

Event date

Jul 9, 2018

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

—