52,115 research outputs found

    The hardness of routing two pairs on one face

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    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

    On disjoint paths in acyclic planar graphs

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    We give an algorithm with complexity O(f(R)k2k3n)O(f(R)^{k^2} k^3 n) for the integer multiflow problem on instances (G,H,r,c)(G,H,r,c) with GG an acyclic planar digraph and r+cr+c Eulerian. Here, ff is a polynomial function, n=V(G)n = |V(G)|, k=E(H)k = |E(H)| and RR is the maximum request maxhE(H)r(h)\max_{h \in E(H)} r(h). When kk is fixed, this gives a polynomial algorithm for the arc-disjoint paths problem under the same hypothesis

    Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

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    Given an undirected graph and two disjoint vertex pairs s1,t1s_1,t_1 and s2,t2s_2,t_2, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s1s_1 with t1t_1, and s2s_2 with t2t_2, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions. Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs

    On Routing Disjoint Paths in Bounded Treewidth Graphs

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    We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph GG and a collection of kk source-destination pairs M={(s1,t1),,(sk,tk)}\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset M\mathcal{M}' of the pairs is a collection P\mathcal{P} of paths such that, for each pair (si,ti)M(s_i, t_i) \in \mathcal{M}', there is a path in P\mathcal{P} connecting sis_i to tit_i. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph GG has capacities cap(e)\mathrm{cap}(e) on the edges and a routing P\mathcal{P} is feasible if each edge ee is in at most cap(e)\mathrm{cap}(e) of the paths of P\mathcal{P}. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP. In this paper we obtain an O(r3)O(r^3) approximation for MaxEDP on graphs of treewidth at most rr and a matching approximation for MaxNDP on graphs of pathwidth at most rr. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an O(r3r)O(r \cdot 3^r) approximation for MaxEDP

    Finding Disjoint Paths on Edge-Colored Graphs: More Tractability Results

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    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

    Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems

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    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

    On Hamilton decompositions of infinite circulant graphs

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    The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

    Disjoint paths in geometric graphs

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    Construction of the shortest paths connecting two given nodes in a geometric graph is the quintessential problem in computational geometry. We consider several variations of this problem with applications that include robotics, Geographic Information Systems, and sensor networks. The first problem we address is the development of an efficient algorithm for constructing a pair of short node-disjoint paths connecting start and target nodes. The second problem investigated is the development of efficient algorithms for constructing narrow and in-range broadcast corridors in triangulated geometric graphs. Finally, we consider the development of an approximation algorithm for constructing reduced overlap trees in three-colored geometric graphs. Theoretical analysis and a detailed experimental investigation of the proposed algorithms are also presented

    Topological infinite gammoids, and a new Menger-type theorem for infinite graphs

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    Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph GG with vertex sets AA and BB that if every finite subset of AA is linked to BB by disjoint paths, then the whole of AA can be linked to the closure of BB by disjoint paths or rays in a natural topology on GG and its ends. This latter theorem re-proves and strengthens the infinite Menger theorem of Aharoni and Berger for `well-separated' sets AA and BB. It also implies the topological Menger theorem of Diestel for locally finite graphs
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