On the complexity of the edge-disjoint min-min problem in undirected graphs

The min-min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-complete and admits no K-approximation for any K > 1 in the general case [1]. In this paper, we show that Bhatia et al [2]'s NP-complete proof, a claim of correction to Xu et al's proof [1], for the edge-disjoint min-min problem in undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in Bhatia et al's proof [2]. We then give a correct proof of NP-completeness of this problem in undirected graphs.

[1]  Robert E. Tarjan,et al.  A quick method for finding shortest pairs of disjoint paths , 1984, Networks.

[2]  Paul D. Seymour.  Disjoint paths in graphs , 2006, Discret. Math..

[3]  Y. Ebihara.  Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[4]  Gang Liu,et al.  A*Prune: an algorithm for finding K shortest paths subject to multiple constraints , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[5]  Chunming Qiao,et al.  On the complexity of and algorithms for finding the shortest path with a disjoint counterpart , 2006, TNET.

[6]  Eric Bouillet,et al.  Lightpath Re-optimization in mesh optical networks , 2005, IEEE/ACM Transactions on Networking.

[7]  Chung-Lun Li,et al.  The complexity of finding two disjoint paths with min-max objective function , 1989, Discret. Appl. Math..

[8]  Mei Yang,et al.  Finding Minimum-Cost Paths with Minimum Sharability , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.

[9]  Chung-Lun Li,et al.  Finding disjoint paths with different path-costs: Complexity and algorithms , 1992, Networks.

[10]  Ramesh Bhandari,et al.  Optimal physical diversity algorithms and survivable networks , 1997, Proceedings Second IEEE Symposium on Computer and Communications.

[11]  Arunabha Sen,et al.  Survivability of lightwave networks - path lengths in WDM protection scheme , 2001, J. High Speed Networks.

[12]  José Coelho de Pina,et al.  Length-bounded disjoint paths in planar graphs , 2002, Discret. Appl. Math..

[13]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[14]  Murali S. Kodialam,et al.  Finding disjoint paths with related path costs , 2006, J. Comb. Optim..

[15]  J. W. Suuballe,et al.  Disjoint Paths in a Network , 2022 .

[16]  Alexander Schrijver,et al.  Finding k Disjoint Paths in a Directed Planar Graph , 1994, SIAM J. Comput..

[17]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.