Computing optimal transport distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. Despite the recent introduction of several algorithms with go...
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We consider the problem of approximating the girth, g, of an unweighted and undirected graph G "pV, Eq with n nodes and m edges. A seminal result of Itai and Rodeh [SICOMP’78] gave an additive 1-approximation in...
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We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D, {cij}) with n = |D| points, and a non-increasing weight vector w ∈ Rn+, and...
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In the Maximum Duo-Preservation String Mapping problem we are given two strings and wish to map the letters of the former to the letters of the latter so as to maximise the number of duos. A duo is a pair of consecuti...
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We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) r-Domination problem and the (Connected) p-Center problem for unweighted and undirected graphs. Given a graph G, we...
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Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V (G) → R, find a minimum weight subset S ⊆ V (...
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The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a simple polygon P, is fully guarded. Most standard versions of this problem are known to be N...
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In this paper we consider two distance-based relaxed variants of the maximum clique problem (MAX CLIQUE), named MAX d-CLIQUE and MAX d-CLUB: A d-clique in a graph G is a subset S subset of V (G) of vertices such that ...
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ISBN:
(纸本)9781509026784
In this paper we consider two distance-based relaxed variants of the maximum clique problem (MAX CLIQUE), named MAX d-CLIQUE and MAX d-CLUB: A d-clique in a graph G is a subset S subset of V (G) of vertices such that for pairs of vertices u, v is an element of S, the distance between u and v is at most d in G. A d-club in a graph G is a subset S' subset of V (G) of vertices that induces a subgraph of G of diameter at most d. MAX d-CLIQUE and MAX d-CLUB ask to find a maximum d-clique and a maximum d-club in a given unweighted graph, respectively. MAX 1-CLIQUE and MAX 1-CLUB cannot be efficiently approximated within a factor of n(1-epsilon) for any epsilon > 0 unless P = NP since they are identical to MAX CLIQUE [1], [2]. Also, it is known [3], [4] that it is NP-hard to approximate MAX d-CLIQUE and MAX d-CLUB to within a factor of n(1/2-epsilon) for any fixed d >= 2 and any epsilon > 0. As for approximability of MAX d-CLIQUE and MAX d-CLUB, [3] proposes a polynomial-time algorithm, called ByFindStar(d), and proves that its approximation ratio is O(n(1/2)) and O(n(2/3)) for any even d >= 2 and for any odd d >= 3, respectively. Very recently, a polynomial-time algorithm, called ByFindStar2(d), achieving an optimal approximation ratio of O(n1/2) for MAX d-CLIQUE and MAX d-CLUB is designed for any odd d >= 3 in [4]. In this paper we implement those approximation algorithms and evaluate their quality empirically for random graphs G(n,p),which have n vertices and each edge appears with probability p. The experimental results show that (i) ByFindStar2(d) of approximation ratio O(n(1/2)) can find larger d-clubs (d-cliques) than ByFindStard of approximation ratio O(n(2/3)) for odd d, (ii) the size of d-clubs (d-cliques) output by ByFindStar(d) is the same as ones by ByFindStar2(d) for even d, and (iii) ByFindStar(d) can find the same size of d-clubs (d-cliques) much faster than ByFindStar2(d).
We consider the problem of scheduling an application on a parallel computational platform. The application is a particular task graph, either a linear chain of tasks, or a set of independent tasks. The platform is mad...
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We consider the problem of scheduling an application on a parallel computational platform. The application is a particular task graph, either a linear chain of tasks, or a set of independent tasks. The platform is made of identical processors, whose speed can be dynamically modified. It is also subject to failures: if a processor is slowed down to decrease the energy consumption, it has a higher chance to fail. Therefore, the scheduling problem requires us to re-execute or replicate tasks (i.e., execute twice the same task, either on the same processor, or on two distinct processors), in order to increase the reliability. It is a tri-criteria problem: the goal is to minimize the energy consumption, while enforcing a bound on the total execution time (the makespan), and a constraint on the reliability of each task. Our main contribution is to propose approximation algorithms for linear chains of tasks and independent tasks. For linear chains, we design a fully polynomial-time approximation scheme. However, we show that there exists no constant factor approximation algorithm for independent tasks, unless P=NP, and we propose in this case an approximation algorithm with a relaxation on the makespan constraint.
We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster ver...
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ISBN:
(纸本)9783319334615;9783319334608
We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster vertex deletion in the literature and admits a straightforward 3-approximation algorithm since it is a special case of the vertex cover problem on a 3-uniform hypergraph. Very recently, You et al. [14] described an efficient 5/2-approximation algorithm for the unweighted version of the problem. Our main result is a 7/3-approximation algorithm for arbitrary weights, using the local ratio technique. We further conjecture that the problem admits a 2-approximation algorithm and give some support for the conjecture. This is in sharp constrast with the fact that the similar problem of deleting vertices to eliminate all triangles in a graph is known to be UGC-hard to approximate to within a ratio better than 3, as proved by Guruswami and Lee [7].
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