On the Path Avoiding Forbidden Pairs Polytope

Blanco, Marco; Borndörfer, Ralf; Brückner, Michael; Hoàng, Nam Dũng; Schlechte, Thomas

doi:10.1016/j.endm.2015.07.057

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…In the latter problem one is given a set of arc pairs A and has to identify whether some s-t-path avoids all pairs in A (meaning SFP asks for a set that makes the corresponding PAFP instance unsolvable). Originating from the field of automated software testing [27], PAFP also has applications in aircraft routing [7] and biology, for example in peptide sequencing [8] or predicting gene structures [26]. The PAFP is NP-complete [17] and various restrictions on the set of forbidden pairs have been considered.…”

Section: Introductionmentioning

confidence: 99%

“…The problem becomes solvable if the pairs satisfy certain symmetry properties [34] or if they have a hierarchical structure [24] while it remains NPhard even if the pairs have a halving structure [24] or no two pairs are nested [25]. The structure of the PAFP polytope has been analyzed [7] and Hajiaghayi et al [21] show that determining a path that uses a minimal amount of forbidden pairs cannot have a sublinear approximation algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Almost Disjoint Paths and Separating by Forbidden Pairs

Bachtler¹,

Bergner²,

Krumke³

2022

Preprint

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By Menger's theorem the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minumum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem. Moreover the max-flow and min-cut problems form a strongly dual pair. We relax the disjointedness requirement on the paths, allowing them to be almost disjoint, meaning they may share up to one arc. The resulting almost disjoint paths problem (ADP) asks for k s-t-paths such that any two of them are almost disjoint. The separating by forbidden pairs problem (SFP) is the corresponding dual problem and calls for a set of k arc pairs such that every s-t-path contains both arcs of at least one such pair. In this paper, we explore these two problems, showing that they have an unbounded duality gap in general and analyzing their complexity. We prove that ADP is NP-complete when k is part of the input and that SFP is Σ2P-complete, even for acyclic graphs. Furthermore, we efficiently solve ADP when k ≤ 2 is fixed and present a polynomial time algorithm based on dynamic programming for ADP when k is constant and the considered graphs are acyclic.

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