Fixing Improper Colorings of Graphs

Abstract In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper r -coloring φ of a graph G . We investigate the problem of finding a proper r -coloring of G , which is “the most similar” to φ , i.e., the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any fixed r ≥ 3 , even for bipartite planar graphs. Moreover, it is W [ 1 ] -hard even for bipartite graphs, when parameterized by the number k of allowed recoloring transformations. On the other hand, the problem is fixed-parameter tractable, when parameterized by k and the number r of colors. We provide a 2 n ⋅ n O ( 1 ) algorithm for the problem and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the fixing number of a graph G . It is the minimum k such that k recoloring transformations are sufficient to transform any coloring of G into a proper one.

[1]  Arie M. C. A. Koster,et al.  Combinatorial Optimization on Graphs of Bounded Treewidth , 2008, Comput. J..

[2]  Paul S. Bonsma,et al.  Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances , 2007, Theor. Comput. Sci..

[3]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[4]  Liming Cai,et al.  On the existence of subexponential parameterized algorithms , 2003, J. Comput. Syst. Sci..

[5]  Jan van den Heuvel,et al.  Mixing 3-colourings in bipartite graphs , 2007, Eur. J. Comb..

[6]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[7]  Ton Kloks.  Treewidth, Computations and Approximations , 1994, Lecture Notes in Computer Science.

[8]  Dániel Marx,et al.  Can you beat treewidth? , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[9]  Reinhard Laue,et al.  Simple 8-(31, 12, 3080), 8-(40, 12, 16200) and 8-(40, 12, 16520) designs from PSL(3, 5) and PSL(4, 3) , 2008, Discret. Math..

[10]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[11]  Pawel Rzazewski,et al.  Fixing Improper Colorings of Graphs , 2015, SOFSEM.

[12]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[13]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  Dániel Marx,et al.  Lower bounds based on the Exponential Time Hypothesis , 2011, Bull. EATCS.

[16]  Marthe Bonamy,et al.  Recoloring bounded treewidth graphs , 2013, Electron. Notes Discret. Math..

[17]  Michal Pilipczuk,et al.  Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams , 2015, ESA.

[18]  Anna Zych,et al.  New Reoptimization Techniques applied to Steiner Tree Problem , 2011, Electron. Notes Discret. Math..

[19]  Jan van den Heuvel,et al.  Connectedness of the graph of vertex-colourings , 2008, Discret. Math..

[20]  Andreas Björklund,et al.  Set Partitioning via Inclusion-Exclusion , 2009, SIAM J. Comput..

[21]  J. K. Il.  Precoloring Extension with Fixed Color Bound , 1994 .

[22]  Jan van den Heuvel,et al.  Finding paths between 3‐colorings , 2011, IWOCA.

[23]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[24]  Stefan Felsner,et al.  Recoloring directed graphs , 2009 .

[25]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[26]  Dennis Komm,et al.  Reoptimization of the Shortest Common Superstring Problem , 2009, Algorithmica.

[27]  Stefan Kratsch,et al.  Cross-Composition: A New Technique for Kernelization Lower Bounds , 2011, STACS.

[28]  Baruch Schieber,et al.  A Theory and Algorithms for Combinatorial Reoptimization , 2012, Algorithmica.

[29]  Dániel Marx,et al.  Known algorithms on graphs of bounded treewidth are probably optimal , 2010, SODA '11.

[30]  Takehiro Ito,et al.  On the complexity of reconfiguration problems , 2008, Theor. Comput. Sci..

[31]  Vangelis Th. Paschos,et al.  Reoptimization of minimum and maximum traveling salesman's tours , 2009, J. Discrete Algorithms.

[32]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.