Finding Optimal Solutions With Neighborly Help
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.
Critical Graphs
Computational Complexity
Structural Self-Reducibility
Minimality Problems
Colorability
Vertex Cover
Satisfiability
Reoptimization
Advice
Theory of computation~Problems, reductions and completeness
Theory of computation~Complexity classes
78:1-78:14
Regular Paper
A full version of this paper, which includes the appendices, is published on arXiv [Elisabet Burjons et al., 2019] and available at https://arxiv.org/abs/1906.10078.
We thank the anonymous referees and Hans-Joachim Böckenhauer, Rodrigo R. Gumucio Escobar, Lane Hemaspaandra, Juraj Hromkovič, Rastislav Královič, Richard Královič, Xavier Muñoz, Martin Raszyk, Peter Rossmanith, Walter Unger, and Koichi Wada for helpful comments and discussions.
Elisabet
Burjons
Elisabet Burjons
Department of Computer Science, ETH Zürich, Universitätstrasse 6, CH-8092 Zürich, Switzerland
Fabian
Frei
Fabian Frei
Department of Computer Science, ETH Zürich, Universitätstrasse 6, CH-8092 Zürich, Switzerland
Edith
Hemaspaandra
Edith Hemaspaandra
Department of Computer Science, Rochester Institute of Technology, Rochester, NY 14623, USA
Research done in part while on sabbatical at ETH Zürich.
Dennis
Komm
Dennis Komm
Department of Computer Science, ETH Zürich, Universitätstrasse 6, CH-8092 Zürich, Switzerland
David
Wehner
David Wehner
Department of Computer Science, ETH Zürich, Universitätstrasse 6, CH-8092 Zürich, Switzerland
10.4230/LIPIcs.MFCS.2019.78
Claudia Archetti, Luca Bertazzi, and Maria Grazia Speranza. Reoptimizing the Traveling Salesman Problem. Networks, 42(3):154-159, 2003.
Giorgio Ausiello, Bruno Escoffier, Jérôme Monnot, and Vangelis Paschos. Reoptimization of Minimum and Maximum Traveling Salesman’s Tours. In Proceedings of the 10th Scandinavian Workshop on Algorithm Theory (SWAT 2006), volume 4059 of Lecture Notes in Computer Science, pages 196-207. Springer-Verlag, 2006.
Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner. Finding Optimal Solutions With Neighborly Help. Technical Report arXiv:1906.10078 [cs.CC], arXiv.org, June 2019. URL: https://arxiv.org/abs/1906.10078.
https://arxiv.org/abs/1906.10078
Hans-Joachim Böckenhauer, Luca Forlizzi, Juraj Hromkovič, Joachim Kneis, Joachim Kupke, Guido Proietti, and Peter Widmayer. Reusing Optimal TSP Solutions for Locally Modified Input Instances. In Proceedings of the 4th IFIP International Conference on Theoretical Computer Science (IFIP TCS 2006), pages 251-270. Springer-Verlag, 2006.
Hans-Joachim Böckenhauer, Juraj Hromkovič, and Dennis Komm. Reoptimization of Hard Optimization Problems. In Teofilo F. Gonzalez, editor, AAM Handbook of Approximation Algorithms and Metaheuristics, volume 1, chapter 25, pages 427-454. CRC Press 2018, 2nd edition, 2018.
Jin-Yi Cai and Gabriele E. Meyer. Graph Minimal Uncolorability is D^P-Complete. SIAM Journal on Computing, 16(2):259-277, 1987.
Gabriel A. Dirac. Some Theorems on Abstract Graphs. Proceedings of the London Mathematical Society, s3-2(1):69-81, 1952.
Piotr Faliszewski and Mitsunori Ogihara. On the Autoreducibility of Functions. Theory of Computing Systems, 46(2):222-245, 2010.
Michael Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
Frank Harary. Graph Theory. Addison-Wesley, 1991.
Edith Hemaspaandra, Holger Spakowski, and Jörg Vogel. The complexity of Kemeny elections. Theoretical Computer Science, 349(3):382-391, 2005.
Gwenaël Joret. Entropy and Stability in Graphs. PhD thesis, Université Libre de Bruxelles, Faculté des Sciences, 2008.
Albert R. Meyer and Mike Paterson. With What Frequency Are Apparently Intractable Problems Difficult? Technical Report MIT/LCS/TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979.
Christos H. Papadimitriou and David Wolfe. The Complexity of Facets Resolved. Journal of Computer and System Sciences, 37(1):2-13, 1988.
Claus-Peter Schnorr. Optimal Algorithms for Self-Reducible Problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322-337. Edinburgh University Press, 1976.
Markus W. Schäffter. Scheduling with forbidden sets. Discrete Applied Mathematics, 72(1-2):155-166, 1997.
Karl Wagner. Bounded Query Classes. SIAM Journal on Computing, 19(5):833-846, 1990.
Klaus W. Wagner. More Complicated Questions About Maxima and Minima, and some Closures of NP. Theoretical Computer Science, 51(1-2):53-80, 1987.
Walter Wessel. Criticity with respect to properties and operations in graph theory. In László Lovász András Hajnal and Vera T. Sós, editors, Finite and Infinite Sets. (6th Hungarian Combinatorial Colloquium, Eger, 1981), volume 2 of Colloquia Mathematica Societatis Janos Bolyai, pages 829-837. North-Holland, 1984.
Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode