On the complexity of solution extension of optimization problems
•Extending a partial solution of an optimization problem occurs as a subproblem in many areas.•We propose a general framework for dealing with these extension problems.•We present several problems from different areas which can be expressed in this framework.•We show NP-hardness results and paramete...
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| Vydané v: | Theoretical computer science Ročník 904; s. 48 - 65 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier B.V
15.02.2022
Elsevier |
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | •Extending a partial solution of an optimization problem occurs as a subproblem in many areas.•We propose a general framework for dealing with these extension problems.•We present several problems from different areas which can be expressed in this framework.•We show NP-hardness results and parameterized complexity results for the presented problems.
The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal vertex covers of a graph G=(V,E), one usually arrives at the problem to decide for a vertex set U⊆V (pre-solution), if there exists a minimal vertex cover S (i.e., a vertex cover S⊆V such that no proper subset of S is a vertex cover) with U⊆S (minimal extension of U). We propose a general, partial-order based formulation of such extension problems which allows to model parameterization and approximation aspects of extension, and also highlights relationships between extension tasks for different specific problems. As examples, we study a number of specific problems which can be expressed and related in this framework.
In particular, we discuss extension variants of the problems dominating set and feedback vertex/edge set. All these problems are shown to be NP-complete even when restricted to bipartite graphs of bounded degree, with the exception of our extension version of feedback edge set on undirected graphs which is shown to be solvable in polynomial time. For the extension variants of dominating and feedback vertex set, we also show NP-completeness for the restriction to planar graphs of bounded degree. As non-graph problem, we also study an extension version of the bin packing problem. We further consider the parameterized complexity of all these extension variants, where the parameter is a measure of the pre-solution as defined by our framework. |
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| AbstractList | •Extending a partial solution of an optimization problem occurs as a subproblem in many areas.•We propose a general framework for dealing with these extension problems.•We present several problems from different areas which can be expressed in this framework.•We show NP-hardness results and parameterized complexity results for the presented problems.
The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal vertex covers of a graph G=(V,E), one usually arrives at the problem to decide for a vertex set U⊆V (pre-solution), if there exists a minimal vertex cover S (i.e., a vertex cover S⊆V such that no proper subset of S is a vertex cover) with U⊆S (minimal extension of U). We propose a general, partial-order based formulation of such extension problems which allows to model parameterization and approximation aspects of extension, and also highlights relationships between extension tasks for different specific problems. As examples, we study a number of specific problems which can be expressed and related in this framework.
In particular, we discuss extension variants of the problems dominating set and feedback vertex/edge set. All these problems are shown to be NP-complete even when restricted to bipartite graphs of bounded degree, with the exception of our extension version of feedback edge set on undirected graphs which is shown to be solvable in polynomial time. For the extension variants of dominating and feedback vertex set, we also show NP-completeness for the restriction to planar graphs of bounded degree. As non-graph problem, we also study an extension version of the bin packing problem. We further consider the parameterized complexity of all these extension variants, where the parameter is a measure of the pre-solution as defined by our framework. |
| Author | Fernau, Henning Khosravian Ghadikolaei, Mehdi Monnot, Jérôme Casel, Katrin Sikora, Florian |
| Author_xml | – sequence: 1 givenname: Katrin surname: Casel fullname: Casel, Katrin email: katrin.casel@hpi.de organization: Hasso Plattner Institute, University of Potsdam, 14482 Potsdam, Germany – sequence: 2 givenname: Henning orcidid: 0000-0002-4444-3220 surname: Fernau fullname: Fernau, Henning email: fernau@informatik.uni-trier.de organization: Universität Trier, Fachbereich 4, Informatikwissenschaften, Theoretische Informatik, 54286 Trier, Germany – sequence: 3 givenname: Mehdi surname: Khosravian Ghadikolaei fullname: Khosravian Ghadikolaei, Mehdi email: m.khosravian@gmail.com organization: Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France – sequence: 4 givenname: Jérôme surname: Monnot fullname: Monnot, Jérôme organization: Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France – sequence: 5 givenname: Florian orcidid: 0000-0003-2670-6258 surname: Sikora fullname: Sikora, Florian email: florian.sikora@lamsade.dauphine.fr organization: Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France |
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| Snippet | •Extending a partial solution of an optimization problem occurs as a subproblem in many areas.•We propose a general framework for dealing with these extension... |
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| SubjectTerms | Computational Complexity Computer Science Data Structures and Algorithms Discrete Mathematics Extension problems NP-hardness Parameterized complexity |
| Title | On the complexity of solution extension of optimization problems |
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