On the complexity of the edge-disjoint min-min problem in planar digraphs

The min-min problem of finding a disjoint path pair with the length of the shorter path minimized is known to be NP-complete (Xu et al., 2006) [1]. In this paper, we prove that in planar digraphs the edge-disjoint min-min problem remains NP-complete and admits no K-approximation for any K>1 unless P=NP. As a by-product, we show that this problem remains NP-complete even when all edge costs are equal (i.e., stronglyNP-complete). To our knowledge, this is the first NP-completeness proof for the edge-disjoint min-min problem in planar digraphs.

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

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

[3]  J. W. Suurballe.  Disjoint paths in a network , 1974, Networks.

[4]  LiChung-Lun,et al.  The complexity of finding two disjoint paths with min-max objective function , 1989 .

[5]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

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

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

[8]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

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

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

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

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

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

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

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