Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
On Some Combinatorial Optimization Problems: Algorithms and Complexity
Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, The Institute of Technology. (TCSLAB)
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.

A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.

The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.

Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.

In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 32 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1663
Keyword [en]
Computational complexity, optimization, constraint satisfaction problem
National Category
Computer Science
Identifiers
URN: urn:nbn:se:liu:diva-116859DOI: 10.3384/diss.diva-116859ISBN: 978-91-7519-072-3 (print)OAI: oai:DiVA.org:liu-116859DiVA: diva2:806491
Public defence
2015-05-21, Alan Turing, E-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Funder
CUGS (National Graduate School in Computer Science), 09.01
Available from: 2015-04-23 Created: 2015-04-07 Last updated: 2015-04-27Bibliographically approved
List of papers
1. Max-Sur-CSP on Two Elements
Open this publication in new window or tab >>Max-Sur-CSP on Two Elements
2012 (English)In: Principles and Practice of Constraint Programming 18th International Conference, CP 2012, Québec City, QC, Canada, October 8-12, 2012. Proceedings / [ed] Michela Milano, Springer Berlin Heidelberg , 2012, 38-54 p.Chapter in book (Refereed)
Abstract [en]

Max-Sur-CSP is the following optimisation problem: given a set of constraints, find a surjective mapping of the variables to domain values that satisfies as many of the constraints as possible. Many natural problems, e.g. Minimum k-Cut (which has many different applications in a variety of fields) and Minimum Distance (which is an important problem in coding theory), can be expressed as Max-Sur-CSPs. We study Max-Sur-CSP on the two-element domain and determine the computational complexity for all constraint languages (families of allowed constraints). Our results show that the problem is solvable in polynomial time if the constraint language belongs to one of three classes, and NP-hard otherwise. An important part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised minimum cut problem. This algorithm may be of independent interest.

Place, publisher, year, edition, pages
Springer Berlin Heidelberg, 2012
Series
Lecture Notes in Computer Science, ISSN 0302-9743 (print), 1611-3349 (online) ; 7514
National Category
Computer Science
Identifiers
urn:nbn:se:liu:diva-79515 (URN)10.1007/978-3-642-33558-7_6 (DOI)978-3-642-33557-0 (ISBN)978-3-642-33558-7 (ISBN)
Conference
18th International Conference on Principles and Practice of Constraint Programming (CP-2012), 8-12 October, Québec City, Canada
Available from: 2012-08-06 Created: 2012-08-06 Last updated: 2015-04-23Bibliographically approved
2. The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom
Open this publication in new window or tab >>The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom
2013 (English)In: Automata, Languages, and Programming 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I / [ed] Fedor V. Fomin, Rūsiņš Freivalds, Marta Kwiatkowska, David Peleg, Springer Berlin/Heidelberg, 2013, 804-815 p.Chapter in book (Refereed)
Abstract [en]

Thapper and Živný [STOC’13] recently classified the complexity of VCSP for all finite-valued constraint languages. However, the complexity of VCSPs for constraint languages that are not finite-valued remains poorly understood. In this paper we study the complexity of two such VCSPs, namely Min-Cost-Hom and Min-Sol. We obtain a full classification for the complexity of Min-Sol on domains that contain at most three elements and for the complexity of conservative Min-Cost-Hom on arbitrary finite domains. Our results answer a question raised by Takhanov [STACS’10, COCOON’10].

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2013
Series
Lecture Notes in Computer Science, ISSN 0302-9743 (print), 1611-3349 (online) ; 7965
National Category
Computer Science
Identifiers
urn:nbn:se:liu:diva-102664 (URN)10.1007/978-3-642-39206-1_68 (DOI)000342686600068 ()978-3-642-39205-4 (ISBN)978-3-642-39206-1 (ISBN)
Conference
40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013
Available from: 2013-12-18 Created: 2013-12-18 Last updated: 2015-04-23Bibliographically approved
3. Computational Complexity of the Minimum Cost Homomorphism Problem on Three-element Domains
Open this publication in new window or tab >>Computational Complexity of the Minimum Cost Homomorphism Problem on Three-element Domains
2014 (English)Conference paper, Published paper (Refereed)
Abstract [en]

In this paper we study the computational complexity of the extended minimum cost homomorphism problem (Min-Cost-Hom) as a function of a constraint language, i.e. a set of constraint relations and cost functions that are allowed to appear in instances. A wide range of natural combinatorial optimisation problems can be expressed as extended Min-Cost-Homs and a classification of their complexity would be highly desirable, both from a direct, applied point of view as well as from a theoretical perspective.

The extended Min-Cost-Hom can be understood either as a flexible optimisation version of the constraint satisfaction problem (CSP) or a restriction of the (general-valued) valued constraint satisfaction problem (VCSP). Other optimisation versions of CSPs such as the minimum solution problem (Min-Sol) and the minimum ones problem (Min-Ones) are special cases of the extended Min-Cost-Hom.

The study of VCSPs has recently seen remarkable progress. A complete classification for the complexity of finite-valued languages on arbitrary finite domains has been obtained Thapper and Živný [STOC’13]. However, understanding the complexity of languages that are not finitevalued appears to be more difficult. The extended Min-Cost-Hom allows us to study problematic languages of this type without having to deal with with the full generality of the VCSP. A recent classification for the complexity of three-element Min-Sol, Uppman [ICALP’13], takes a step in this direction. In this paper we generalise this result considerably by determining the complexity of three-element extended Min-Cost-Hom.

Place, publisher, year, edition, pages
Dagstuhl: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014
Series
Leibniz International Proceedings in Informatics, ISSN 1868-8969 ; 25
National Category
Computer Science
Identifiers
urn:nbn:se:liu:diva-112916 (URN)10.4230/LIPIcs.STACS.2014.651 (DOI)978-3-939897-65-1 (ISBN)
Conference
31st International Symposium on Theoretical Aspects of Computer Science (STACS-2014)
Available from: 2014-12-19 Created: 2014-12-19 Last updated: 2015-04-23
4. Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
Open this publication in new window or tab >>Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
2014 (English)In: Mathematical Foundations of Computer Science 2014: 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II / [ed] Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, Zoltán Ésik, Springer Berlin/Heidelberg, 2014, 408-419 p.Chapter in book (Refereed)
Abstract [en]

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Recent algebraic techniques introduced by Jonsson et al. (SODA 2013) show that the time complexity of the parameterized SAT(·) problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that SAT(R) can be solved at least as fast as any other NP-hard SAT(·) problem. In this paper we extend this method and show that such languages also exist for the max ones problem (Max-Ones(Γ)) and the Boolean valued constraint satisfaction problem over finite-valued constraint languages (VCSP(Δ)). With the help of these languages we relate Max-Ones and VCSP to the exponential time hypothesis in several different ways.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2014
Series
Lecture Notes in Computer Science, ISSN 0302-9743 (print), 1611-3349 (online) ; 8635
National Category
Computer and Information Science
Identifiers
urn:nbn:se:liu:diva-112902 (URN)10.1007/978-3-662-44465-8_35 (DOI)000358254600035 ()2-s2.0-84906261436 (Scopus ID)978-3-662-44464-1 (ISBN)978-3-662-44465-8 (ISBN)
Conference
39th International Symposium on Mathematical Foundations of Computer Science (MFCS-2014)
Available from: 2014-12-19 Created: 2014-12-19 Last updated: 2017-02-23Bibliographically approved

Open Access in DiVA

fulltext(790 kB)270 downloads
File information
File name FULLTEXT01.pdfFile size 790 kBChecksum SHA-512
530615ba4cb228f76e5b779247799d0c46e19db18f8ea64327b431fde555c839b7631465dfb42392085ef20f4014686df9405cd3ea178e90eb56d84afe0035d2
Type fulltextMimetype application/pdf
omslag(40 kB)14 downloads
File information
File name COVER01.pdfFile size 40 kBChecksum SHA-512
9a2395f8a64c3a810dd86e2e46944cf8fdf82312612425e4f67c5a9849b64cbfe8b34312c85df4b7596e78af2dd4f6f580ab3fe4882ec3c1aa219eed1ba773de
Type coverMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Uppman, Hannes
By organisation
Software and SystemsThe Institute of Technology
Computer Science

Search outside of DiVA

GoogleGoogle Scholar
Total: 270 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 2341 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf