An improved algorithm for the planar 3-cut problem
TL;DR: This paper presents an O ( n log n ) algorithm for finding a minimum 3-cut in planar graphs and improves the best previously known algorithm for the problem by an O( n logn) factor.
About: This article is published in Journal of Algorithms.The article was published on 1991-01-02. It has received 20 citations till now. The article focuses on the topics: Planar graph & Line graph.
Citations
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01 Sep 1991TL;DR: Two simple approximation algorithms are presented for the minimum k-cut problem, requiring a total of only n-1 maximum flow computations for finding a set of near-optimal k-cuts.
Abstract: Two simple approximation algorithms are presented for the minimum k-cut problem. Each algorithm finds a k-cut having weight within a factor of (2-2/k) of the optimal. One of the algorithms is particularly efficient, requiring a total of only n-1 maximum flow computations for finding a set of near-optimal k-cuts, one for each value of k between 2 and n. >
187 citations
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TL;DR: This survey reviews, classify and discusses several recent advances and results obtained for each variant, including theoretical complexity, exact solving algorithms, approximation schemes and heuristic approaches, and proves new complexity results and induce some solving algorithms through relationships established between different variants.
160 citations
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22 Oct 2011TL;DR: This work considers the minimum-k-way cut problem for unweighted undirected graphs with a size bound on the number of cut edges allowed, and shows that this problem is fixed-parameter tractable (FPT) with the standard parameterization in terms of the solution size $s$.
Abstract: We consider the minimum $k$-way cut problem for unweighted undirected graphs with a size bound $s$ on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into $k$ components, or report that this requires cutting more than $s$ edges. We show that this problem is fixed-parameter tractable (FPT) with the standard parameterization in terms of the solution size $s$. More precisely, for $s=O(1)$, we present a quadratic time algorithm. Moreover, we present a much easier linear time algorithm for planar graphs and bounded genus graphs. Our tractability result stands in contrast to known W[1] hardness of related problems. Without the size bound, Downey et al.~[2003] proved that the minimum $k$-way cut problem is W[1] hard with parameter $k$, and this is even for simple unweighted graphs. Downey et al.~asked about the status for planar graphs. We get linear time with fixed parameter $k$ for simple planar graphs since the minimum $k$-way cut of a planar graph is of size at most $6k$. More generally, we get FPT with parameter $k$ for any graph class with bounded average degree. A simple reduction shows that vertex cuts are at least as hard as edge cuts, so the minimum $k$-way vertex cut is also W[1] hard with parameter $k$. Marx [2004] proved that finding a minimum $k$-way vertex cut of size $s$ is also W[1] hard with parameter $s$. Marx asked about the FPT status with edge cuts, which we prove tractable here. We are not aware of any other cut problem where the vertex version is W[1] hard but the edge version is FPT, e.g., Marx [2004] proved that the $k$-terminal cut problem is FPT parameterized by the cut size, both for edge and vertex cuts.
65 citations
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TL;DR: The algorithm is a divide-and-conquer method based on a procedure that reduces an instance of the minimum $k$-way cut problem to $O(n^{2k-5})$ instances of theminimum $(\lfloor (k+\sqrt{k})/2\rfloor+1)$- way cut problem, and can be implemented to run in time.
Abstract: Let $G=(V,E)$ be an edge-weighted undirected graph with $n$ vertices and $m$ edges. We present a deterministic algorithm to compute a minimum $k$-way cut of $G$ for a given $k$. Our algorithm is a divide-and-conquer method based on a procedure that reduces an instance of the minimum $k$-way cut problem to $O(n^{2k-5})$ instances of the minimum $(\lfloor (k+\sqrt{k})/2\rfloor+1)$-way cut problem, and can be implemented to run in $O(n^{4k/(1-1.71/\sqrt{k}) -31} )$ time. With a slight modification, the algorithm can find all minimum $k$-way cuts in $O(n^{4k/(1-1.71/\sqrt{k}) -16} )$ time.
46 citations
Cites background from "An improved algorithm for the plana..."
...The case of unweighted planar graphs with k = 3 can be solved in O(n log n) time [15]....
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25 Sep 1996TL;DR: With the same time and processor resources, a tree-decomposition of width at most two can be built of a given series parallel graph, and hence, very efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs, including many well known problems, e.g., all problems that can be stated in monadic second order logic.
Abstract: In this paper, a parallel algorithm is given that, given a graph G=(V, E), decides whether G is a series parallel graph, and if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log¦E¦log*¦E¦) time and O(¦E¦) operations on an EREW PRAM, andO(log¦E¦) time and O(¦E¦) operations on a CRCW PRAM (note that if G is a simple series parallel graph, then ¦E¦=O(¦V¦)). With the same time and processor resources, a tree-decomposition of width at most two can be built of a given series parallel graph, and hence, very efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs, including many well known problems, e.g., all problems that can be stated in monadic second order logic. The results hold for undirected series parallel graphs graphs, as well as for directed series parallel graphs.
43 citations
Cites background or methods from "An improved algorithm for the plana..."
...He and Yesha [12] and He [11] gave parallel algorithms for recognizing directed and undirected series parallel graphs in O(log2n+ logm) time with O(n+m) processors on an EREW PRAM, and henceO((n+m)(log2n+ logm)) operations....
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...He [11] and Eppstein [10] have shown (using results from Duffin [9]) that this problem reduces in a direct way to the problem with specified vertices, as the following result holds....
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References
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01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher:
With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter.
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9,262 citations
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01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
7,497 citations
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TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.
1,312 citations
01 Oct 1977
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A,B,C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A,B,C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2\sqrt{2}\sqrt{2}$ vertices. We exhibit an algorithm which finds such a partition A,B,C in O(n) time.
1,264 citations
"An improved algorithm for the plana..." refers methods in this paper
...We will use a divide-and-conquer approach based on the Lipton-Tarjan separator theorem [6,7]....
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