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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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].
We evaluate the performance of fast approximation algorithms for MAX SAT on the comprehensive benchmark sets from the SAT and MAX SAT contests. Our examination of a broad range of algorithmic techniques reveals that g...
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ISBN:
(数字)9783319388519
ISBN:
(纸本)9783319388502;9783319388519
We evaluate the performance of fast approximation algorithms for MAX SAT on the comprehensive benchmark sets from the SAT and MAX SAT contests. Our examination of a broad range of algorithmic techniques reveals that greedy algorithms offer particularly striking performance, delivering very good solutions at low computational cost. Interestingly, their relative ranking does not follow their worst case behavior. Johnson's deterministic algorithm is constantly better than the randomized greedy algorithm of Poloczek et al. [31], but in turn is outperformed by the derandomization of the latter: this 2-pass algorithm satisfies more than 99% of the clauses for instances stemming from industrial applications. In general it performs considerably better than non-oblivious local search, Tabu Search, WalkSat, and several state-of-the-art complete and incomplete solvers, while being much faster. But the 2-pass algorithm does not achieve the excellent performance of Spears' computationally intense simulated annealing. Therefore, we propose a new hybrid algorithm that combines the strengths of greedy algorithms and stochastic local search to provide outstanding solutions at high speed: in our experiments its performance is as good as simulated annealing, achieving an average loss with respect to the best known value of less that 0.5%, while its speed is comparable to the greedy algorithms.
The Maximum Planar Subgraph (MPS) problem asks for a planar subgraph with maximum edge cardinality of a given undirected graph. It is known to be MaxSNP-hard and the currently best known approximation algorithm achiev...
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ISBN:
(纸本)9783319445434;9783319445427
The Maximum Planar Subgraph (MPS) problem asks for a planar subgraph with maximum edge cardinality of a given undirected graph. It is known to be MaxSNP-hard and the currently best known approximation algorithm achieves a ratio of 4/9. We analyze the general limits of approximation algorithms for MPS, based either on planarity tests or on greedy inclusion of certain subgraphs. On the one hand, we cover upper bounds for the approximation ratios. On the other hand, we show NP-hardness for thereby arising subproblems, which hence must be approximated themselves. We also provide simpler proofs for two already known facts.
We provide a quasilinear time algorithm for the p-center problem with an additive error less than or equal to 3 times the input graph's hyperbolic constant. Specifically, for the graph G = (V, E) with n vertices, ...
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ISBN:
(数字)9783319497877
ISBN:
(纸本)9783319497877;9783319497860
We provide a quasilinear time algorithm for the p-center problem with an additive error less than or equal to 3 times the input graph's hyperbolic constant. Specifically, for the graph G = (V, E) with n vertices, m edges and hyperbolic constant d, we construct an algorithm for p-centers in time O(p( delta vertical bar 1)(n vertical bar m) log(2)(n)) with radius not exceeding r(p) + delta when p <= 2 and r(p) + 3 delta when p >= 3, where rp are the optimal radii. Prior work identified p-centers with accuracy r(p) + delta but with time complexity O((n(3) log(2) n + n(2)m) log(2)(diam( G))) which is impractical for large graphs.
We describe an algorithm that finds an is an element of-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the proble...
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ISBN:
(数字)9783319334615
ISBN:
(纸本)9783319334615;9783319334608
We describe an algorithm that finds an is an element of-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in 1/is an element of, provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless P = NP.
Scaffolding is one of the main stages in genome assembly. During this stage, we want to merge contigs assembled from the paired-end reads into bigger chains called scaffolds. For this purpose, the following graph-theo...
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ISBN:
(数字)9783319398174
ISBN:
(纸本)9783319398174;9783319398167
Scaffolding is one of the main stages in genome assembly. During this stage, we want to merge contigs assembled from the paired-end reads into bigger chains called scaffolds. For this purpose, the following graph-theoretical problem has been proposed: Given an edge-weighted complete graph G and a perfect matching D of G, we wish to find a Hamiltonian path P in G such that all edges of D appear in P and the total weight of edges in P but not in D is maximized. This problem is NP-hard and the previously best polynomial-time approximation algorithm for it achieves a ratio of 1/2. In this paper, we design a new polynomial-time approximation algorithm achieving a ratio of 5-5 epsilon/9-8 epsilon for any constant 0 < epsilon < 1. Several generalizations of the problem have also been introduced in the literature and we present polynomial-time approximation algorithms for them that achieve better approximation ratios than the previous bests. In particular, one of the algorithms answers an open question.
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