A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated verticesi¾ź$$s$$ andi¾ź$$t$$, the goal is to delete as few edges as possible in order to increase the length of the new shortest $$st$$-path as much as possible. This scenario has been mostly studied from the viewpoint of approximation algorithms and heuristics, while we particularly introduce a parameterized and multivariate point of view. We derive refined tractability as well as hardness results, and identify numerous directions for future research. Among other things, we show that increasing the shortest path length by at least one is much easier than to increase it by at least two.

[1]  Vladimir Gurvich,et al.  On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction , 2008, Theory of Computing Systems.

[2]  Rolf Niedermeier,et al.  Invitation to data reduction and problem kernelization , 2007, SIGA.

[3]  Carsten Lund,et al.  Hardness of approximations , 1996 .

[4]  R. Kevin Wood,et al.  Shortest‐path network interdiction , 2002, Networks.

[5]  A. K. Mittal,et al.  The k most vital arcs in the shortest path problem , 1990 .

[6]  Delbert Ray Fulkerson,et al.  Maximizing the minimum source-sink path subject to a budget constraint , 1977, Math. Program..

[7]  Samir Khuller,et al.  The complexity of finding most vital arcs and nodes , 1995 .

[8]  Weifa Liang.  Finding the k most vital edges with respect to minimum spanning trees for fixed k , 2001, Discret. Appl. Math..

[9]  Feng Pan,et al.  Interdiction problems on planar graphs , 2013, Discret. Appl. Math..

[10]  Daniel Vanderpooten,et al.  Critical edges/nodes for the minimum spanning tree problem: complexity and approximation , 2013, J. Comb. Optim..

[11]  Cristina Bazgan,et al.  Parameterized Approximability of Maximizing the Spread of Influence in Networks , 2013, COCOON.

[12]  R. Kevin Wood,et al.  Deterministic network interdiction , 1993 .

[13]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[14]  Michael R. Fellows,et al.  Review of: Fundamentals of Parameterized Complexity by Rodney G. Downey and Michael R. Fellows , 2015, SIGA.

[15]  Mathias Weller.  DISTANCE AT , 2013 .

[16]  Michael R. Fellows,et al.  Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity , 2013, Eur. J. Comb..

[17]  Daniel Vanderpooten,et al.  Efficient determination of the k most vital edges for the minimum spanning tree problem , 2012, Comput. Oper. Res..

[18]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[19]  Marcin Pilipczuk,et al.  A Fast Branching Algorithm for Cluster Vertex Deletion , 2014, CSR.

[20]  Jan Kratochvíl,et al.  Cluster Vertex Deletion: A Parameterization between Vertex Cover and Clique-Width , 2012, MFCS.

[21]  Stefan Kratsch,et al.  Recent developments in kernelization: A survey , 2014, Bull. EATCS.

[22]  Rolf Niedermeier,et al.  Reflections on Multivariate Algorithmics and Problem Parameterization , 2010, STACS.

[23]  Jiong Guo,et al.  Parameterized Complexity of Edge Interdiction Problems , 2014, COCOON.

[24]  Roberto Solis-Oba,et al.  Increasing the weight of minimum spanning trees , 1996, SODA '96.

[25]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[26]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[27]  Enrico Nardelli,et al.  A faster computation of the most vital edge of a shortest path , 2001, Inf. Process. Lett..

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

[29]  Christian Komusiewicz,et al.  New Races in Parameterized Algorithmics , 2012, MFCS.