Structural parameters, tight bounds, and approximation for (k, r)-center

Abstract In ( k , r ) - Center  we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: • For any r ≥ 1 , we show an algorithm that solves the problem in O ∗ ( ( 3 r + 1 ) cw ) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm’s performance. As a corollary, for r = 1 , this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw . • We strengthen previously known FPT lower bounds, by showing that ( k , r ) - Center  is W[1]-hard parameterized by the input graph’s vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. • We show that the complexity of the problem parameterized by tree-depth is 2 Θ ( td 2 ) , by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k , r . In particular, we give algorithms which, for any ϵ > 0 , run in time O ∗ ( ( tw ∕ ϵ ) O ( tw ) ) , O ∗ ( ( cw ∕ ϵ ) O ( cw ) ) and return a ( k , ( 1 + ϵ ) r ) -center if a ( k , r ) -center exists, thus circumventing the problem’s W-hardness.

[1]  Dániel Marx,et al.  Parameterized Complexity and Approximation Algorithms , 2008, Comput. J..

[2]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[3]  Geevarghese Philip,et al.  Hardness of r-dominating set on Graphs of Diameter (r + 1) , 2013, IPEC.

[4]  Dániel Marx,et al.  Efficient Approximation Schemes for Geometric Problems? , 2005, ESA.

[5]  Hans L. Bodlaender,et al.  Treewidth: Characterizations, Applications, and Computations , 2006, WG.

[6]  Hans L. Bodlaender,et al.  The algorithmic theory of treewidth , 2000, Electron. Notes Discret. Math..

[7]  Kazuhisa Makino,et al.  Parameterized Edge Hamiltonicity , 2014, WG.

[8]  Egon Wanke,et al.  The Tree-Width of Clique-Width Bounded Graphs Without Kn, n , 2000, WG.

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

[10]  Michael Lampis,et al.  Parameterized Approximation Schemes Using Graph Widths , 2013, ICALP.

[11]  Glencora Borradaile,et al.  Optimal dynamic program for r-domination problems over tree decompositions , 2015, IPEC.

[12]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree , 1995, J. Algorithms.

[13]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[14]  Samir Khuller,et al.  Fault tolerant K-center problems , 2000, Theor. Comput. Sci..

[15]  Peter J. Slater,et al.  R-Domination in Graphs , 1976, J. ACM.

[16]  Peter Rossmanith,et al.  Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution , 2009, ESA.

[17]  Sven Oliver Krumke,et al.  On a Generalization of the p-Center Problem , 1995, Inf. Process. Lett..

[18]  Rolf Niedermeier,et al.  Improved Tree Decomposition Based Algorithms for Domination-like Problems , 2002, LATIN.

[19]  Torben Hagerup,et al.  Parallel Algorithms with Optimal Speedup for Bounded Treewidth , 1995, SIAM J. Comput..

[20]  A. Brandstädt,et al.  A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs , 1998 .

[21]  David Eisenstat,et al.  Approximating k-center in planar graphs , 2014, SODA.

[22]  Jaroslav Nesetril,et al.  Tree-depth, subgraph coloring and homomorphism bounds , 2006, Eur. J. Comb..

[23]  Evripidis Bampis,et al.  Parameterized Power Vertex Cover , 2016, WG.

[24]  Bruno Courcelle,et al.  Upper bounds to the clique width of graphs , 2000, Discret. Appl. Math..

[25]  Yoshiko Wakabayashi,et al.  The k-hop connected dominating set problem: hardness and polyhedra , 2015, Electron. Notes Discret. Math..

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

[27]  Feodor F. Dragan,et al.  Parameterized Approximation Algorithms for Some Location Problems in Graphs , 2017, COCOA.

[28]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[29]  Tomás Feder,et al.  Optimal algorithms for approximate clustering , 1988, STOC '88.

[30]  David B. Shmoys,et al.  A unified approach to approximation algorithms for bottleneck problems , 1986, JACM.

[31]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width , 2000, Theory of Computing Systems.

[32]  Gerhard J. Woeginger,et al.  Scheduling of pipelined operator graphs , 2012, J. Sched..

[33]  Judit Bar-Ilan,et al.  How to Allocate Network Centers , 1993, J. Algorithms.

[34]  Jan Arne Telle,et al.  Algorithms for Vertex Partitioning Problems on Partial k-Trees , 1997, SIAM J. Discret. Math..

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

[36]  Erik Jan van Leeuwen,et al.  Faster Algorithms on Branch and Clique Decompositions , 2010, MFCS.

[37]  Russell Impagliazzo,et al.  On the Complexity of k-SAT , 2001, J. Comput. Syst. Sci..

[38]  David Peleg,et al.  The fault-tolerant capacitated K-center problem , 2015, Theor. Comput. Sci..

[39]  Andreas Emil Feldmann,et al.  Fixed-Parameter Approximations for k-Center Problems in Low Highway Dimension Graphs , 2015, Algorithmica.

[40]  Pankaj K. Agarwal,et al.  Exact and Approximation Algortihms for Clustering , 1997 .

[41]  Samir Khuller,et al.  The Capacitated K-Center Problem , 2000, SIAM J. Discret. Math..

[42]  Erik D. Demaine,et al.  Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs , 2005, TALG.

[43]  Sigve Hortemo Sæther,et al.  Faster algorithms for vertex partitioning problems parameterized by clique-width , 2013, Theor. Comput. Sci..

[44]  Dana Moshkovitz,et al.  The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover , 2012, Theory Comput..

[45]  Rina Panigrahy,et al.  An O(log*n) approximation algorithm for the asymmetric p-center problem , 1996, SODA '96.