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Efficient Algorithms for k-Terminal Cuts on Planar Graphs

機(jī)譯：平面圖上k末端切割的高效算法

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The minimum k-terminal cut problem is of considerable theoretical interest and arises in several applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper, we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we give a min{O(n~2 log n log m), O(m~2n~(1.5) log~2 n+kn)} time algorithm with a (2 - 2/k)-approximation ratio (clearly, m ≤ k). For the case when all k terminals are covered by the boundary of one face of G, we give an O(nk~3 + (n log n)k~2) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when they are applied to planar graphs.
機(jī)譯：最小的k端切割問題具有相當(dāng)大的理論意義，并且出現(xiàn)在幾個(gè)應(yīng)用領(lǐng)域，例如并行和分布式計(jì)算，VLSI電路設(shè)計(jì)和聯(lián)網(wǎng)。在本文中，我們針對(duì)n頂點(diǎn)無向加權(quán)平面圖G給出了針對(duì)該問題的兩種新的逼近算法和精確算法。對(duì)于k個(gè)終端被G的m> 1個(gè)面的邊界覆蓋的情況，我們給出了一個(gè)最小值{O（n?2 log n log m），O（m?2n?（1.5）log?2 n + kn）}時(shí)間算法，近似比為（2/2 / k）（m≤k） 。對(duì)于所有k個(gè)終端都被G的一個(gè)面的邊界覆蓋的情況，我們給出O（nk?3 +（n log n）k?2）時(shí)間精確算法，或者如果k = 3，則給出線性時(shí)間精確算法，用于計(jì)算最佳的k端切割。我們的算法基于有趣的觀察，并在將它們應(yīng)用于平面圖時(shí)對(duì)其進(jìn)行了改進(jìn)。

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