The Longest Path Problem has a Polynomial Solution on Interval Graphs

The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871–883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n4) time, where n is the number of vertices of the input graph.

[1]  C. P. Rangan,et al.  A Unified Approach to Domination Problems on Interval Graphs , 1988, Inf. Process. Lett..

[2]  Rajeev Motwani,et al.  Finding large cycles in Hamiltonian graphs , 2005, SODA '05.

[3]  Ryuhei Uehara,et al.  Longest Path Problems on Ptolemaic Graphs , 2008, IEICE Trans. Inf. Syst..

[4]  Alan A. Bertossi,et al.  Finding Hamiltonian Circuits in Proper Interval Graphs , 1983, Inf. Process. Lett..

[5]  Ryuhei Uehara,et al.  Efficient Algorithms for the Longest Path Problem , 2004, ISAAC.

[6]  Haiko Müller,et al.  Hamiltonian circuits in chordal bipartite graphs , 1996, Discret. Math..

[7]  Harold N. Gabow,et al.  Finding paths and cycles of superpolylogarithmic length , 2004, STOC '04.

[8]  Harold N. Gabow,et al.  Finding Long Paths, Cycles and Circuits , 2008, ISAAC.

[9]  C. Pandu Rangan,et al.  Linear Algorithm for Optimal Path Cover Problem on Interval Graphs , 1990, Inf. Process. Lett..

[10]  Katerina Asdre,et al.  The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs , 2008, Algorithmica.

[11]  Sundar Vishwanathan,et al.  An approximation algorithm for finding a long path in Hamiltonian graphs , 2000, SODA '00.

[12]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[13]  Haim Kaplan,et al.  Four Strikes Against Physical Mapping of DNA , 1995, J. Comput. Biol..

[14]  J. Mark Keil.  Finding Hamiltonian Circuits in Interval Graphs , 1985, Inf. Process. Lett..

[15]  David S. Johnson,et al.  The Planar Hamiltonian Circuit Problem is NP-Complete , 1976, SIAM J. Comput..

[16]  Giri Narasimhan,et al.  A Note on the Hamiltonian Circuit Problem on Directed Path Graphs , 1989, Inf. Process. Lett..

[17]  Peter Damaschke,et al.  The Hamiltonian Circuit Problem for Circle Graphs is NP-Complete , 1989, Inf. Process. Lett..

[18]  Peter Damaschke,et al.  Paths in interval graphs and circular arc graphs , 1993, Discret. Math..

[19]  Zhao Zhang,et al.  Algorithms for long paths in graphs , 2007, Theor. Comput. Sci..

[20]  Sheng-Lung Peng,et al.  Deferred-query: An efficient approach for some problems on interval graphs , 1999, Networks.

[21]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..

[22]  D. Kratsch,et al.  Finding Hamiltonian paths in cocomparability graphs using the bump number algorithm , 1991 .

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

[24]  Ryuhei Uehara,et al.  Linear structure of bipartite permutation graphs and the longest path problem , 2007, Inf. Process. Lett..

[25]  A. J. M. van Gasteren,et al.  On computing a longest path in a tree , 2002, Inf. Process. Lett..

[26]  David R. Karger,et al.  On approximating the longest path in a graph , 1997, Algorithmica.

[27]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .