1,735 research outputs found

    Single Source - All Sinks Max Flows in Planar Digraphs

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    Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

    Exact Localisations of Feedback Sets

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    The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph G=(V,E)G=(V,E) into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs Gel(e)G_{\mathrm{el}}(e), Gsi(e)G_{\mathrm{si}}(e) of all elementary cycles or simple cycles running through some arc eEe \in E, can be computed in O(E2)\mathcal{O}\big(|E|^2\big) and O(E4)\mathcal{O}(|E|^4), respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in O(VE3)\mathcal{O}(|V||E|^3). We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times O(2mE4log(V))\mathcal{O}\big(2^{m}|E|^4\log(|V|)\big) and O(2nΔ(G)4V4log(E))\mathcal{O}\big(2^{n}\Delta(G)^4|V|^4\log(|E|)\big), where mEV+1m \leq |E|-|V| +1, n(Δ(G)1)VE+1n \leq (\Delta(G)-1)|V|-|E| +1, respectively. The parameters m,nm,n can be computed in O(E3)\mathcal{O}(|E|^3), O(Δ(G)3V3)\mathcal{O}(\Delta(G)^3|V|^3), respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, m,nm,n equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER

    Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism

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    In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs GG and HH and a list of allowed vertices of HH for every vertex of GG, the question is whether there exists a homomorphism from GG to HH respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph GG, a non-negative integer \ell, and a set TT of rr terminals, the question is whether we can partition the vertices of GG into rr parts such that (a) each part contains one terminal and (b) there are at most \ell edges with only one endpoint in this part. We parameterize \textsc{List Digraph Homomorphism} by the number ww of edges of GG that are mapped to non-loop edges of HH and we give a time 2O(logh+2log)n4logn2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n algorithm, where hh is the order of the host graph HH. We also prove that \textsc{Min-Max Multiway Cut} can be solved in time 2O((r)2logr)n4logn2^{O((\ell r)^2\log \ell r)}\cdot n^{4}\cdot \log n. Our approach introduces a general problem, called {\sc List Allocation}, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an {\sf FPT}-algorithm for the {\sc List Allocation} problem that is designed using a suitable adaptation of the {\em randomized contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC), Patras, Greece, September 201
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