27,143 research outputs found
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems
Given a graph G=(V, E) and k source-sink pairs (s1, t1), …, (sk, tk) with each si, ti V, the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the source-sink pairs with minimized length of the longest path. We show that the weighted Min-Sum Disjoint Paths problem is FPNP-complete in general graphs, and the unweighted Min-Sum Disjoint Paths problem and the unweighted Min-Max Disjoint Paths problem cannot be approximated within m(m1-1) for any constant > 0 even in planar graphs, assuming P P NP, where m is the number of edges in G. We give for the first time a simple bicriteria approximation algorithm for the unweighted Min-Max Edge-Disjoint Paths problem and the weighted Min-Sum Edge-Disjoint Paths problem, w
Finding Disjoint Paths on Edge-Colored Graphs: More Tractability Results
The problem of finding the maximum number of vertex-disjoint uni-color paths
in an edge-colored graph (called MaxCDP) has been recently introduced in
literature, motivated by applications in social network analysis. In this paper
we investigate how the complexity of the problem depends on graph parameters
(namely the number of vertices to remove to make the graph a collection of
disjoint paths and the size of the vertex cover of the graph), which makes
sense since graphs in social networks are not random and have structure. The
problem was known to be hard to approximate in polynomial time and not
fixed-parameter tractable (FPT) for the natural parameter. Here, we show that
it is still hard to approximate, even in FPT-time. Finally, we introduce a new
variant of the problem, called MaxCDDP, whose goal is to find the maximum
number of vertex-disjoint and color-disjoint uni-color paths. We extend some of
the results of MaxCDP to this new variant, and we prove that unlike MaxCDP,
MaxCDDP is already hard on graphs at distance two from disjoint paths.Comment: Journal version in JOC
A Local Search Modeling for Constrained Optimum Paths Problems (Extended Abstract)
Constrained Optimum Path (COP) problems appear in many real-life
applications, especially on communication networks. Some of these problems have
been considered and solved by specific techniques which are usually difficult
to extend. In this paper, we introduce a novel local search modeling for
solving some COPs by local search. The modeling features the compositionality,
modularity, reuse and strengthens the benefits of Constrained-Based Local
Search. We also apply the modeling to the edge-disjoint paths problem (EDP). We
show that side constraints can easily be added in the model. Computational
results show the significance of the approach
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