Cuts and Disjoint Paths in the Valley-Free Path Model

In the valley-free path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by BGP routing policies of autonomous systems in the Internet. Robustness considerations lead to the problem of computing a maximum number of disjoint paths between two nodes, and the minimum size of a cut that separates them. We study these problems in the valley-free path model. For the problem of computing a maximum number of edge- or vertex-disjoint valid paths between two given vertices s and t, we give a 2-approximation algorithm and show that no better approximation ratio is possible unless P = NP. For the problem of computing a minimum vertex cut that separates s and t with respect to all valid paths, we give a 2-approximation algorithm and prove that the problem is APX-hard. The corresponding problem for edge cuts is shown to be polynomial-time solvable. We present additional results for acyclic graphs.

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

[2]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[3]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[4]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[5]  Lixin Gao.  On inferring autonomous system relationships in the internet , 2001, TNET.

[6]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[7]  Mihalis Yannakakis,et al.  The complexity of multiway cuts (extended abstract) , 1992, STOC '92.

[8]  Lixin Gao,et al.  On the evaluation of AS relationship inferences [Internet reachability/traffic flow applications] , 2004, IEEE Global Telecommunications Conference, 2004. GLOBECOM '04..

[9]  Roger Wattenhofer,et al.  The impact of Internet policy and topology on delayed routing convergence , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[10]  Egon Wanke,et al.  Oriented Paths in Mixed Graphs , 2004, ISAAC.

[11]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[12]  Mihalis Yannakakis,et al.  Multiway Cuts in Directed and Node Weighted Graphs , 1994, ICALP.

[13]  G. Huston,et al.  Interconnection, Peering and Settlements , 2003 .

[14]  Joseph Naor,et al.  A 2-Approximation Algorithm for the Directed Multiway Cut Problem , 2001, SIAM J. Comput..

[15]  Ramesh Govindan,et al.  Internet path inflation due to policy routing , 2001, SPIE ITCom.

[16]  Thomas Erlebach,et al.  Classifying customer-provider relationships in the internet , 2002 .

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

[18]  Elliot Anshelevich,et al.  Strategic Network Formation through Peering and Service Agreements , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[19]  P. Baake,et al.  On the economics of Internet peering , 1999 .

[20]  Deborah Estrin,et al.  The impact of routing policy on Internet paths , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[21]  Madhav V. Marathe,et al.  Formal-Language-Constrained Path Problems , 1997, SIAM J. Comput..

[22]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[23]  Randy H. Katz,et al.  Characterizing the Internet hierarchy from multiple vantage points , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.