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An FPT Algorithm for Planar Multicuts with Sources and Sinks on the Outer Face

機(jī)譯：外表面帶有源和匯的平面多切口的FPT算法

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摘要

Given a list of k source-sink pairs in an edge-weighted graph G, the minimum multicut problem consists in selecting a set of edges of minimum total weight in G, such that removing these edges leaves no path from each source to its corresponding sink. To the best of our knowledge, no non-trivial FPT result for special cases of this problem, which is APX-hard in general graphs for any fixed k3, is known with respect to k only. When the graph G is planar, this problem is known to be polynomial-time solvable if k=O(1), but cannot be FPT with respect to k under the Exponential Time Hypothesis. In this paper, we show that, if G is planar and in addition all sources and sinks lie on the outer face, then this problem does admit an FPT algorithm when parameterized by k (although it remains APX-hard when k is part of the input, even in stars). To do this, we provide a new characterization of optimal solutions in this case, and then use it to design a divide-and-conquer approach: namely, some edges that are part of any such solution actually define an optimal solution for a polynomial-time solvable multiterminal variant of the problem on some of the sources and sinks (which can be identified thanks to a reduced enumeration phase). Removing these edges from the graph cuts it into several smaller instances, which can then be solved recursively.
機(jī)譯：給定邊緣加權(quán)圖G中的k個(gè)源-匯對(duì)列表，最小多切問題在于選擇G中具有最小總權(quán)重的一組邊，這樣除去這些邊就不會(huì)留下從每個(gè)源到其相應(yīng)匯的路徑。 。據(jù)我們所知，對(duì)于此問題的特殊情況，對(duì)于任何固定的k3，在一般圖中，APX都是APX困難的，對(duì)于該問題的特殊情況，沒有非平凡的FPT結(jié)果是已知的。當(dāng)圖G是平面時(shí)，如果k = O（1），則已知該問題可通過多項(xiàng)式時(shí)間求解，但在指數(shù)時(shí)間假設(shè)下，相對(duì)于k而言，該問題不能為FPT。在本文中，我們表明，如果G是平面的，并且所有源和匯都位于外表面，則當(dāng)用k參數(shù)化時(shí)，此問題確實(shí)接受了FPT算法（盡管當(dāng)k是該參數(shù)的一部分時(shí)，它仍然是APX困難的）。輸入，甚至是星號(hào)）。為此，我們?cè)谶@種情況下提供了最優(yōu)解的新特征，然后用它來設(shè)計(jì)分治法：即，任何此類解的一部分邊緣實(shí)際上定義了多項(xiàng)式的最優(yōu)解。在某些源和匯上可以解決此問題的時(shí)間可解決的多端變體（由于枚舉階段的減少，可以識(shí)別出來）。從圖中刪除這些邊緣會(huì)將其切成幾個(gè)較小的實(shí)例，然后可以遞歸求解。

著錄項(xiàng)

  • 來源
    《Algorithmica》 |2019年第1期|224-237|共14頁
  • 作者

    Bentz Cedric;

  • 作者單位

    CNAM, 292 Rue St Martin, F-75003 Paris, France;

    CEDRIC Lab, 292 Rue St Martin, F-75003 Paris, France;

  • 收錄信息 美國(guó)《科學(xué)引文索引》(SCI);美國(guó)《工程索引》(EI);
  • 原文格式 PDF
  • 正文語種 eng
  • 中圖分類
  • 關(guān)鍵詞

    Multicuts; Planar graphs; FPT algorithms;

    機(jī)譯：多點(diǎn)切割;平面圖;FPT算法;

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