18,054 research outputs found

    Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

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    We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) nn-vertex graph GG along with kk terminal pairs (s1,t1),(s2,t2),,(sk,tk)(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k). The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of kk. Lochet's result implies the existence of a polynomial-time ckck-approximation for Maximum Vertex-Disjoint Shortest Paths, where c1c \leq 1 is a constant. Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an o(k)o(k)-approximations within f(k)f(k) \cdot poly(nn) time for any function ff that only depends on kk. Our second result demonstrates the infeasibility of achieving an approximation ratio of n12εn^{\frac{1}{2}-\varepsilon} in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a \lceil\sqrt{\ell}\rceil-approximation, where \ell is the number of edges in all the paths of an optimal solution. Since n\ell \leq n, this underscores the tightness of the n12εn^{\frac{1}{2}-\varepsilon}-inapproximability bound. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by \ell but does not admit a polynomial kernel. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights

    Parameterizing Path Partitions

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    We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains \NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results also transfer to the case of covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of the CIAC 2023 conferenc

    Walking Through Waypoints

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    We initiate the study of a fundamental combinatorial problem: Given a capacitated graph G=(V,E)G=(V,E), find a shortest walk ("route") from a source sVs\in V to a destination tVt\in V that includes all vertices specified by a set WV\mathscr{W}\subseteq V: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

    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
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