The Balanced Edge Cover Problem

Harada, Yuta; Ono, Hiroyuki; Sadakane, Kunihiko; Yamashita, Masafumi

doi:10.1007/978-3-540-92182-0_24

Cited by 2 publications

(7 citation statements)

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“…• Balanced Edge Cover: (Section 4) We also present a reduction from the balanced edge cover problem to the unweighted semi-matching problem. This leads to an O( √ nm log n) algorithm for the problem, improving the previous O(nm) algorithm by Harada et al [17].…”

Section: Our Results and Techniquesmentioning

confidence: 54%

“…algorithm for the problem, improving the previous O(nm) algorithm by Harada et al [17]. The main idea is to identify the "center" vertices of all the clusters in the optimal solution.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…Running time analysis An edge cover F can be constructed from a maximum cardinality matching by adding one edge incident to each uncovered vertex [14,35]. The maximum cardinality matching in a bipartite graph can be found by Micali-Vazirani's algorithm [34] Correctness We prove the correctness by applying the algorithm BEC1 proposed in [17]. This algorithm starts from any minimum edge cover and keeps augmenting along a cost-reducing path until such path does not exist.…”

Section: Extension To Balanced Edge Cover Problemmentioning

confidence: 99%

“…(See [17] for the formal definition.) It was shown that BEC1 returns an optimal balanced edge cover.…”

Section: Extension To Balanced Edge Cover Problemmentioning

confidence: 99%

“…In some applications such as sensor networks, there are no jobs and machines but the sensor nodes have to be clustered and each cluster has to pick its own head node to gather information from other nodes in the cluster. Motivated by this, Harada, Ono, Sadakane and Yamashita [17] introduced the balanced edge cover problem 1 where the goal is to find an edge cover (set of edges incident to every vertex) that minimizes the total cost over all vertices. (The cost on each vertex is as previously defined.)…”

Section: Related Workmentioning

confidence: 99%

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Faster Algorithms for Semi-Matching Problems

Fakcharoenphol

Laekhanukit

Nanongkai

2014

ACM Trans. Algorithms

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We consider the problem of finding semi-matching in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted cases.For the weighted case, we give an O(nm log n)-time algorithm, where n is the number of vertices and m is the number of edges, by exploiting the geometric structure of the problem. This improves the classical O(n 3 )-time algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974]. For the unweighted case, the bound can be improved even further. We give a simple divideand-conquer algorithm which runs in O( √ nm log n) time, improving two previous O(nm)-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lovász and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the Balanced Edge Cover problem in O( √ nm log n) time, improving the previous O(nm)-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008]. * The preliminary version of this paper appeared as [11] in the Proceeding of the 37th International Colloquium on Automata, Languages and Programming, (ICALP) 2010. † Most of the work was done while all authors were at Kasetsart University.let n denote the number of vertices and m denote the number of edges in G. A set M ⊆ E is a semi-matching if each job u ∈ U is incident with exactly one edge in M . For any semi-matching M , we define the cost of M , denoted by cost(M ), as follows. First, for any machine v ∈ V , its cost with respect to a semi-matching M isare weights of the edges in M incident with v sorted increasingly. Intuitively, this is the total completion time of jobs assigned to v. Note that for the unweighted case (i.e., when w e = 1 for every edge e), the cost of a machine v is simply deg M (v) · (deg M (v) + 1)/2. Now, the cost of the semi-matching M is simply the summation of the cost over all machines:The goal is to find an optimal semi-matching, a semi-matching with minimum cost.Related works Although the name "semi-matching" was recently proposed by Harvey, Ladner, Lovász, and Tamir [20], the problem was studied as early as 1970s when an O(n 3 ) algorithm was independently developed by Horn in [21] and by Bruno, Coffman and Sethi in [6]. Since then no progress has been made on this problem except on its special cases and variations. For the special case of inclusive set restriction where, for each pair of jobs u 1 and u 2 , either all neighbors of u 1 are neighbors of u 2 or vice versa, a faster algorithm with O(n 2 ) running time was given by Spyropoulos and Evans [40]. Many variations of this problem were proved to be NP-hard, including the preemptive version [39], the case when there are deadlines [41], and the case of optimizing total weighted tardiness [29]. The variation where the objective is to minimize max v∈V cost M (v) was also considered [32,25]. The unweighted case of the semi-matching problem also received considerable attention in the ...

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Section: Our Results and Techniquesmentioning

confidence: 54%

“…algorithm for the problem, improving the previous O(nm) algorithm by Harada et al [17]. The main idea is to identify the "center" vertices of all the clusters in the optimal solution.…”

Section: Our Results and Techniquesmentioning

confidence: 99%