Traversing combinatorial 0/1-polytopes via optimization
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10476060" target="_blank" >RIV/00216208:11320/23:10476060 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1109/FOCS57990.2023.00076" target="_blank" >https://doi.org/10.1109/FOCS57990.2023.00076</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1109/FOCS57990.2023.00076" target="_blank" >10.1109/FOCS57990.2023.00076</a>
Alternative languages
Result language
angličtina
Original language name
Traversing combinatorial 0/1-polytopes via optimization
Original language description
In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets and polytopes. Our method relies on a simple and versatile algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope conv(X), where X is a subset of {0,1}^n. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem min {w . x ; x is from X}, and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is only by a factor of log n larger than the running time of the optimization algorithm. When X encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope conv(X) along a Hamilton path corresponds to listing the combinatorial objects by local change operations, i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all (c-optimal) bases and independent sets in a matroid; (c-optimal) spanning trees, forests, matchings, maximum matchings, and c-optimal matchings in a general graph; vertex covers, minimum vertex covers, c-optimal vertex covers, stable sets, maximum stable sets and c-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, c-optimal antichains, and c-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope and order polytope, respectively. As another corollary from our framework, we obtain an O(t_LP log n) delay algorithm for the vertex enumeration problem on 0/1-polytopes {x is is from R^n l A x <= b}, where A is from R^{m x n} and b is from R^m, and t_LP is the time needed to solve the linear program min {w . x ; A x <= b}. This improves upon the 25-year old O(t_LP n) delay algorithm due to Bussieck and Lübbecke.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/GA22-15272S" target="_blank" >GA22-15272S: Principles of combinatorial generation</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
Proceedings - IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS
ISBN
979-8-3503-1894-4
ISSN
2575-8454
e-ISSN
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Number of pages
10
Pages from-to
1282-1291
Publisher name
IEEE Computer Society
Place of publication
Santa Cruz
Event location
Santa Cruz, USA
Event date
Nov 6, 2023
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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