Some Approximation Algorithms for Minimum Vertex Cover in a Hypergraph

Abstract-In this paper we consider a hub location problem in a real multimodal public transportation network. This problem is also known as the park-and-ride problem. Hubs stations are special facilities that serve as switches in such a network. In practice the set of hubs has a strategic importance, because all of the traffic that passes through the network can be controlled by these elements. From the theoretical point of view, the minimal hub problem is NP hard. Two different approaches to this problem are presented. The first group of methods bases on the greedy algorithms. In the second group the evolutionary strategy is used. The computational results for these algorithms proved a significant efficiency, what can be clearly expressed in terms of an input data reduction and also in quality measure values for the obtained solutions of this problem. I. INTRODUCTIONET US consider a real public transportation network. This network can be described as a graph [3], [8], [9]. In this graph, each tram/bus stop corresponds to one vertex and any two vertices are adjacent iff they belong to at least one common public transport line. Hence, the set of vertices along a route forms a connected subgraph of the entire graph. Each vertex in this graph is characterized/labeled by a set of numbers of tram/bus lines passing through it. LIn such a graph, a hub set is a subset of vertices, such that any two vertices are connected by a path whose vertices lie in it. Let us define the minimum hub set problem as the problem of finding the hub set of minimal cardinality for a given graph. This collection has a strategic importance for the transportation system, because all the traffic that passes through the network can be controlled by these vertices. Communication hubs are excellent candidates for park-andride (P&R, sometimes P+R) points ([4]) with good connections to the center and other parts of the city and this idea is a basis of presented work. This problem is well known and is commonly used in industry, in particular in such areas as transport, telecommunication, and distributed computing Let us observe that we can try to find the minimum hub set for a given network in two different ways. This problem would be directly reduced to the task of designation the minimum dominating set [6], [14], [15]. Alternatively, we have a possibility of creating a hypergraph, such that its set of vertices is the same as in the original network, and for each set of vertices, which correspond to stops lying along a tram/bus route, we form the hyperedge. In this case, we reduce the considered problem to the search for the minimal transversal in a constructed hypergraph. II. BASIC MATHEMATICAL DEFINITIONSThis section provides some basic notation. A graph is a representation of a set of objects, where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. More formally, a graph is an ordered pair G=(V,...

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“…algorithm 2 (MSBT) [12]. It seeks the vertices with the lowest degree and removes them from the set of vertices.…”

Section: Basic Mathematical Definitionsmentioning

confidence: 99%

Graph based approach to the minimum hub problem in transportation network

Owsiński

Stańczak

Barski

et al. 2015

Annals of Computer Science and Information Systems

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“…Note that the problem is directly equivalent to the Minimum hypergraph transversal problem as well [33].…”

Section: Conditions For Solving Minlfamentioning

confidence: 99%

“…The idea is that instead of working directly on the original network we rather solve the corresponding minimum set cover problems on the respective bipartite graph representations, as there are various well-tested heuristics available in the literature for this important class of combinatorial optimization problems. In particular, we discuss the Lovasz-JohnsonChvatal algorithm from [30], the SBT algorithm from [23], the RSBT, MSBT algorithms from [33] and a straightforward backtracking algorithm. An added benefit of using these heuristics is that they immediately provide solutions to the LFA improvement problem beside LFA graph extension; we only need to terminate the algorithms after a pre-defined number of iterations, i.e., after adding a pre-defined number of new links.…”

Section: Algorithmsmentioning

confidence: 99%

“…The Reverse SBT algorithm [33], as the name says, does the reverse of SBT in that in every iteration it chooses the node with the highest degree instead of the smallest degree. Consequently, the pseudo-code is the same as given in Algorithm 2 with the slight modification that instead of line 3 we write v ← argmax b∈B deg(b).…”

Section: The Rsbt Algorithmmentioning

confidence: 99%

“…The Modified SBT algorithm [33] applies a small optimization step to SBT. Similarly to the SBT algorithm, in each iteration we choose the node v ∈ B with the smallest degree and, if there are nodes in A covered only by v, we add v to B c .…”

Section: The Msbt Algorithmmentioning

confidence: 99%

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Optimization methods for improving IP-level fast protection for local shared risk groups with Loop-Free Alternates

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Lately, demand for fast failure recovery in IP networks has become compelling. The Loop-Free Alternates (LFA) specification is a simple IP Fast ReRoute (IPFRR) scheme proposed by the IETF that does not require profound changes to the network infrastructure before deployment. However, this simplicity comes at a severe price, because LFA does not provide complete protection for all possible failure cases in a general topology. This is even more so if network components are prone to fail jointly. In this paper, we study an important network optimization problem arising in this context, the so called LFA graph extension problem, which asks for augmenting the topology with new links in an attempt to improve the LFA failure case coverage. Unfortunately, this problem is NP-complete. The main contributions of the paper are a novel extension of the bipartite graph model for the LFA graph extension problem to the multiple-failure case using the model of Shared Risk Groups, and a suite of accompanying heuristics to obtain approximate solutions. We also compare the performance of the algorithms in extensive numerical studies and we conclude that the op-

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Some Approximation Algorithms for Minimum Vertex Cover in a Hypergraph

Cited by 2 publications

References 8 publications

Graph based approach to the minimum hub problem in transportation network

Graph based approach to the minimum hub problem in transportation network

Optimization methods for improving IP-level fast protection for local shared risk groups with Loop-Free Alternates

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