On Approximation Hardness of the Bandwidth Problem

The bandwidth problem is the problem of enumerating the vertices of a given graph G such that the maximum difference between the numbers of adjacent vertices is minimal. The problem has a long history and a number of applications and is known to be NP-hard. There is not much known though on approximation hardness of this problem. In this paper we show, that there are no efficient polynomial time approximation schemes for the bandwidth problem under some plausible assumptions. Furthermore we show that there are no polynomial time approximation algorithms with an absolute error guarantee of n^{1-epsilon} for any epsilon <0 unless P=NP.

[1]  Liming Cai,et al.  On fixed-parameter tractability and approximability of NP-hard optimization problems , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[2]  Norman E. Gibbs,et al.  The bandwidth problem for graphs and matrices - a survey , 1982, J. Graph Theory.

[3]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[4]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[5]  Dieter Kratsch,et al.  Approximating the Bandwidth for Asteroidal Triple-Free Graphs , 1995, J. Algorithms.

[6]  David S. Johnson,et al.  COMPLEXITY RESULTS FOR BANDWIDTH MINIMIZATION , 1978 .

[7]  Curt Jones,et al.  Finding Good Approximate Vertex and Edge Partitions is NP-Hard , 1992, Inf. Process. Lett..

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

[9]  Luca Trevisan,et al.  On the Efficiency of Polynomial Time Approximation Schemes , 1997, Inf. Process. Lett..

[10]  Marek Karpinski,et al.  Polynomial time approximation schemes for dense instances of NP-hard problems , 1995, STOC '95.

[11]  H IbarraOscar,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975 .

[12]  Wenceslas Fernandez de la Vega,et al.  MAX-CUT has a randomized approximation scheme in dense graphs , 1996, Random Struct. Algorithms.

[13]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[14]  Marek Karpinski,et al.  An Approximation Algorithm for the Bandwidth Problem on Dense Graphs , 1997, Electron. Colloquium Comput. Complex..

[15]  G. S. Lueker,et al.  Bin packing can be solved within 1 + ε in linear time , 1981 .

[16]  Michael R. Fellows,et al.  Fixed-parameter intractability , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[17]  Dorit S. Hochba,et al.  Approximation Algorithms for NP-Hard Problems , 1997, SIGA.

[18]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[19]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[20]  M. Fellows,et al.  Beyond NP-completeness for problems of bounded width: hardness for the W hierarchy , 1994, Symposium on the Theory of Computing.

[21]  Lawren Smithline.  Bandwidth of the complete k-ary tree , 1995, Discret. Math..

[22]  Alan M. Frieze,et al.  The regularity lemma and approximation schemes for dense problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[23]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.