Local Search Approximation Algorithms for the Complement of the Min-k-Cut Problems

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1 Local Search Approximation Algorithms for the Complement of the Min-k-Cut Problems Wenxing Zhu, Chuanyin Guo Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou , China Abstract Min-k-cut is the problem of partitioning vertices of a given graph or hypergraph into k subsets such that the total weight of edges or hyperedges crossing different subsets is minimized. For the capacitated min-k-cut problem, each edge has a nonnegative weight, and each subset has a possibly different capacity that imposes an upper bound on its size. The objective is to find a partition that minimizes the sum of edge weights on all pairs of vertices that lie in different subsets. The min-k-cut problem is NP-hard for k 3, and the capacitated min-k-cut problem is also NP-hard for k 2. Min-k-cut has numerous applications, for example, VLSI circuit partitioning, which is a key step in VLSI CAD. Although many heuristics have been proposed for min-k-cut problems, there are few approximation results of min-k-cut problems. We study the equivalent complement problem of the min-k-cut problem, which attempts to maximize the total weight of edges in every subset. We extend the method proposed in [15] to present deterministic local search approximation algorithms for the complement problem of the min-k-cut problem, and the complement problem of the capacitated min-k-cut problem on graph with performance ratio 1 k. Key words: min-k-cut, approximation algorithm, local search. 1 Introduction The min-cut problem takes as input an undirected graph with non-negative edge weights. The objective of the problem is to find a partition of the vertices into two subsets, such that the sum of weights of edges between the two subsets is minimized. A prototypical problem in this family is the min-k-cut problem: given an input undirected graph G = (V, E) with edge weights d: E R + and an integer k, partition the graph vertices into k parts, so as to minimize the total weight of the edges connecting different parts. In particular, each subset has a possibly different capacity that imposes an upper bound on its size, and the objective is to find a partition that minimizes the sum of edge weights on all pairs of vertices that lie in different subsets. This is the capacitated min-k-cut problem. When partition the graph vertices into k parts of equal size to minimize the total weight of the edges connecting different parts, the problem is called the min-k-cut balanced problem. The special case k = 2 is the famous minimum bisection problem, which is already known to be NP-hard [14]. Consequently, a long line of research has been devoted to polynomialtime approximation of various graph partitioning problems. There are some approximation results for the minimum multiway cut problem: given a weighted graph G with a set of terminal nodes T and weights on edges, the goal is to find the smallest cut separating all the terminals. Let k = T denote the cardinality of the terminal set. When k = 2, the problem reduces to the simple min-cut problem and hence is polynomial-time solvable. For k 3, it is known that the problem is NP-Hard and also APX-Hard [7]. Dahlhaus et al. [7] provided a greedy approximation algorithm 1

2 for the minimum multiterminal cut problem and gave a tight analysis to show that it achieves an approximation factor of 2(1 1 k ). Although the analysis provided is tight, better algorithms exist for the minimum multiterminal cut problem. Calinescu et al. [5] gave a k approximation algorithm, which was subsequently improved to a approximation algorithm by Karger et al. [9] for a geometric embedding of the problem. For the capacitated min-k-cut problem, which to our knowledge there exist few approximation results. However, related work on the min-k-cut balanced problem has been marked by striking breakthroughs in designing approximation algorithms with provable performance guarantees. In fact, Andreev and Räcke [2] proved that bi-criteria approximation is inevitable in the sense that, unless P=NP, the min-k-cut balanced problem has no polynomial time approximation within any finite factor. Leighton, et al. [24] and Simon and Teng [29] designed approximation algorithms based on recursively finding a minimum bisection for the min-k-cut balanced problem. This algorithm achieves an O(log k log n) approximation, when using the minimum bisection algorithm due to [3]. Using a more direct linear programming (LP) relaxation of partitioning problems called spreading metrics, Even et al. [10] designed an algorithm for this problem, achieving an approximation factor of O(log n). Andreev and Räcke [2] used a more sophisticate recursive partitioning together with dynamic programming to obtain an O(ɛ 2 log 1.5 n) approximation for any fixed ε > 0. Krauthgamer et al. [22] presented a polynomial-time (bi-criteria) approximation algorithm achieving an approximation of O( log n log k). Works on approximation algorithms for the minimum bisection problem have proved to be extremely fruitful. It is best illustrated by briefly mentioning a few key results regarding minimum bisection: the O(log n) bi-criteria approximation by Leighton and Rao [25], as well as the improvement to O( log n) by Arora et al. [3]. They approximated minimum bisection via a closely related problem of sparsest cuts. Räcke [27] obtained a remarkable O(log n) approximation, which improves the previous O(log 1.5 n) by Feige and Krauthgamer [12]. The previous best-known results for this problem are based on linear programming relaxation [16] or SDP relaxation [13]. Srivastav and Wolf [30] gave a 0.48-approximation algorithm using a different SDP relaxation. In fact, Srivastav and Wolf have proved that the SDP relaxation used in their analysis yields at most 0.5 approximation of dense-n/2-subgraph. Crossing the 0.5 barrier, Ye and Zhang [31] presented an approximation guarantee of using a Goemans-Williamson-style algorithm [17] based on the semidefinite programming relaxation. Finally, different inapproximability results shown in a sequence of papers [6, 8, 11, 21, 23], relate minimum bisection to the unique games conjecture [20], and to refuting random 3SAT formulas. This makes it possible to investigate minimum bisection in another area. In this paper, we develop and investigate local search approaches for the complement problem of the min-k-cut problem on graph, and the complement problem of the capacitated min-k-cut problem of graph. Our method is based on the local search method proposed in [15] for the capacitated max-k-cut problem. Noting that the min-k-cut problem is not equivalent to the max-k-cut problem, and no approximation results have been established for local search methods for the general min-k-cut problem, we think the results presented in this paper are interesting. The paper is organized as follows. We first present in Section 2 a local search algorithm for the complement problem of the min-k-cut problem on graph, which provides an approximation guarantee 1 k. In Section 3, we study a local search algorithm for the complement problem of the capacitated min-k-cut problem on graph, and show that the solution obtained has an approximation ratio 1 k. Finally, conclusions are made in Section 4. 2

3 2 Complement of the min-k-cut problem on graph Let G(V, E) be a graph with V = n vertices. Let edge e E connect vertices u, v V, and let d(u, v) be a non-negative weight on edge e = (u, v). We consider the problem of partitioning the set V into k subsets, V 1,..., V k, such that the sum of weights of edges in every subset is as large as possible. This is an equivalent problem of the min-k-cut problem. We consider a local search method for it, which is a Lin-Kernighan like algorithm. 2.1 Local search algorithm Our proposed algorithm begins with an arbitrary partitioning of the vertices, and moves a vertex from one subset to another if the sum of weights of edges in every subset increases. The algorithm stops when no further improvements can be attained by all possible moving of one vertex. Let d(u, V i ) = v V i,v u d(u, v) be the sum of weights of the edges from vertex u to the vertices in the subset V i. We consider the performance of the following local-search algorithm: Initialization step Partition the vertices into k sets, V 1,..., V k, by arbitrarily assigning vertices to V i, for all i = 1,..., k. Iterative step Determine if there is a vertex u V i and subset V j, i j, such that d(u, V i ) < d(u, V j ). If such a vertex exists, move u from V i to V j, and repeat this step. Termination step When d(u, V i ) d(u, V j ), for all i, j, i j, and for all u V i, the algorithm ends. 2.2 Worst-case analysis of the algorithm Now we characterize the worst-case bound of the local-search algorithm for graph for all k 3. Let OP T denote the optimal value of an instance of the complement problem of the min-k-cut problem on graph. Let V 1,..., V k be a solution obtained using the local-search algorithm. Let ALG denote the objective value of the solution V 1,..., V k. Let d(v i, V i ) denote the sum of weights of edges with both ends in V i. Let d(v i, V j ) denote the sum of weights of edges that connect a vertex in V i to a vertex in V j, and let d cut denote the sum of weights of the cut edges. Thus, and k ALG = d(v i, V i ), i=1 d cut = d(v i, V j ), i=1 j=i+1 ALG + d cut = e E d(e). 3

4 Theorem 1. The solution obtained using the local-search algorithm has an objective value not smaller than 1 k of the optimal value. Proof When the local search algorithm stops, we have d(u, V i ) d(u, V j ), for all i, j, i j, and for all u V i. We sum both sides of the above inequality over all u V i and get d(u, V i ) d(u, V j ). (1) u V i u V i Since each edge weight is added two times while computing d(v i, V i ), we have Moreover, since inequality (1) leads to u V i d(u, V i ) = 2d(V i, V i ). u V i d(u, V j ) = d(v i, V j ), 2d(V i, V i ) d(v i, V j ). Summing both sides of the above inequality over all i, j = 1,, k, i j, gives Since 2 d(v i, V i ) d(v i, V j ). (2) inequality (2) results in, d(v i, V j ) = 2 d(v i, V j ), i=1 j=i+1 which follows that 2 d(v i, V i ) 2 d(v i, V j ), i=1 j=i+1 k (k 1) d(v i, V i ) i=1 The above inequality can be rewritten as d(v i, V j ). i=1 j=i+1 Therefore, and we get (k 1)ALG d cut. OP T d cut + ALG ALG + (k 1)ALG, ALG OP T 1 k. 4

5 2.3 Running time Now we analyze running time of the above local-search algorithm. Each iterative step involves examining at most V vertices and selecting one that decreases the cut value. This process takes at most O( V 2 ) time. Since we assume that edges have non-negative weights, the sum of weights of the edges in every subset is increased after each iteration. The maximum possible sum of weights of the edges in every subset is d(e), hence there are at most such number of iterations. So the overall time complexity of this algorithm is O( V 2 d(e)). For the special case of unweighted graphs (all weights equal to 1), the e E time complexity becomes O( V 4 ), which is strongly polynomial in the input size. 3 Complement of the capacitated min-k-cut problem on graph Let G(V, E) be a graph with V = n vertices. Let edge e E connect vertices u, v V, and let d(u, v) be a non-negative weight on edge e = (u, v). We consider the problem of partitioning the set V into k subsets, V 1,..., V k, where the i-th subset V k contains at most p i vertices, and V k i=1 p i, such that the sum of weights of edges in every subset is as large as possible. 3.1 Local search method Our proposed algorithm begins with an arbitrary partitioning of the vertices, with size of each subset not greater than the capacity constraint. It searches in a neighborhood comprising all pairs of vertices that lie in different subsets, and evaluates the value of a switch by comparing the sum of edge weights with and without flipping a pair of vertices. The algorithm stops when no further improvements can be attained the sum of weights of edges in every subset. Let d(u, V i ) = v V i,v u d(u, v) denote the sum of weights of the edges from vertex u to the vertices in the subset V i. We consider the following local-search algorithm: Initialization step Partition the vertices into k sets, V 1,..., V k, by arbitrarily assigning V i p i vertices to V i, i = 1,..., k. Iterative step Determine if there is a pair of vertices u V i and v V j, i j, such that V i d(u, V i ) + V j d(v, V j ) < V i d(u, V j ) + V j d(v, V i ). If such a pair of vertices exist, reassign vertex u to V j, vertex v to V i, and repeat this step. Termination step When V i d(u, V i ) + V j d(v, V j ) V i d(u, V j ) + V j d(v, V i ), for all i, j, i j, and for all u V i, the algorithm stops. 3.2 Running time In general, local-search algorithms that run in polynomial time have proved elusive for combinatorially-hard problems. If all edge weights are equal, especially, if the objective is to maximize the number of edges in all subsets, then the above local search procedure runs e E 5

6 in time O(n 3 m), where n is the number of vertices and m is the number of edges in the graph. Let us analyze running time of the above local search algorithm for this special case of problem. Each iterative step involves examining at most O(n 2 ) vertices and selecting two vertices that increases the number of edges lying in the same part when swaped. This process takes O(n 3 ) time. Since we assume that edges have equal weights, the number of edges lying in the same part is increased after each iteration. The maximum possible objective value is m, hence there are at most such number of iterations. The overall time complexity of this algorithm is thus O(n 3 m), which is strongly polynomial in the input size. Johnson et al. [18] introduced polynomial-time local search (PLS), a complexity class containing problems that are equally hard in the sense that, if a locally optimal solution can be found in polynomial time for one problem, then such a solution can be found for all problems in the class. Problems in this class are called PLS-complete. The complement problem of the capacitated Max-k-Cut problem is NP-Hard, and it appears unlikely that it admits a polynomial time local-search procedure. Orlin et al. [26] provided a locally optimal approximation scheme for every problem in PLS. By these results, the problem of obtaining such a locally optimal solution for the general weighted version of the capacitated min-k-cut problem is PLS-complete. 3.3 Worst-case analysis of the algorithm The following theorem characterizes the worst-case bound on the quality of the local-search algorithm for all k 2. Theorem 2. The solution obtained using the local search algorithm has a value not smaller than 1 k of the optimal value of the complement problem of the capacitated min-kcut problem. Proof As the solution returned by the local search algorithm satisfies, V i d(u, V i ) + V j d(v, V j ) V i d(u, V j ) + V j d(v, V i ), for all i, j, i j, and for all u V i. We sum both sides of the above expression over all u V i and get V i u V i d(u, V i ) + V i V j d(v, V j ) V i u V i d(u, V j ) + V i V j d(v, V i ). Next, we sum both sides of the above inequality over all v V j and get V i V j u V i d(u, V i ) + V i V j v V j d(v, V j ) V i V j u V i d(u, V j ) + V i V j v V j d(v, V i ). Note that, u V i d(u, V j ) = d(v i, V j ), v V j d(v, V i ) = d(v j, V i ). Since each edge weight is added two times when computing d(v i, V i ) and d(v j, V j ), we have (3) 6

7 Thus inequality (3) leads to u V i d(u, V i ) = 2d(V i, V i ), v V j d(v, V j ) = 2d(V j, V j ). 2 V i V j (d(v i, V i ) + d(v j, V j )) V i V j (d(v i, V j ) + d(v j, V i )). Since d(v i, V j ) = d(v j, V i ), we get d(v i, V i ) + d(v j, V j ) d(v i, V j ). Summing both sides of the above inequality over all i, j = 1,, k, i j, gives Since (d(v i, V i ) + d(v j, V j )) d(v i, V j ). (4) and by (4), we have d(v i, V j ) = 2 d(v i, V j ) = 2d cut, i=1 j=i+1 It follows that (d(v i, V i ) + d(v j, V j )) 2d cut. k k (k 1)( d(v i, V i ) + d(v j, V j )) 2d cut. i=1 j=1 The above inequality can be rewritten as By k 2(k 1) d(v i, V i ) 2d cut. i=1 we get ALG = k d(v i, V i ), i=1 Therefore, which results in (k 1)ALG d cut. OP T d cut + ALG ALG + (k 1)ALG, ALG OP T 1 k. 7

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