Maximum edge disjoint paths and minimum unweighted multicut problems in grid graphs

Abstract : Let G=(V,E) be an undirected graph and let L be a list of K pairs (source si, sink ti) of terminal vertices of G (or nets). The maximum edge disjoint paths problem (MaxEDP) consists in maximizing the number of nets linked by edge disjoint paths. The related minimum multicut problem (MinUWMC) is to find a minimum set of edges whose removal separates si from ti for each i in an augmented graph (i.e., where each terminal vertex is linked to the graph by a unique edge). Both problems are NP-hard even in planar graphs. MaxEDP defined in rectilinear grids where any vertex can be a terminal is also NP-hard. However, A. Frank gives in [Frank82] necessary and sufficient conditions for the existence of K edge disjoint paths when the terminals are two-sided (i.e. they are all distinct and lie on the uppermost and lowermost lines of the grid): thus, solving MaxEDP is equivalent to removing the minimum number of nets in order to fulfill Frank's conditions. We prove that this can be done by solving a polynomial number of linear programs having totally unimodular constraints matrices. Then, we show that, in two-sided augmented grids, MinUWMC is polynomial time solvable via linear programming, by using a duality relationship with a continuous multiflow problem. As a by-product, the gap between the optimal values of MaxEDP and MinUWMC is proved to be at most one.
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Contributor : Laboratoire Cedric <>
Submitted on : Friday, March 6, 2015 - 10:57:24 AM
Last modification on : Saturday, February 9, 2019 - 1:24:37 AM

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Marie-Christine Costa, Frédéric Roupin, Cédric Bentz. Maximum edge disjoint paths and minimum unweighted multicut problems in grid graphs. Contibuted talk, Proceedings Graph Theory (GT'04), Paris, Jan 2004, X, France. pp.23. ⟨hal-01125059⟩

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