This paper studies single machine scheduling with batch deliveries, where a common due window for all jobs has to be determined, not given in advance. The objective is to minimize the overall cost for the process and ...
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This paper studies single machine scheduling with batch deliveries, where a common due window for all jobs has to be determined, not given in advance. The objective is to minimize the overall cost for the process and delivery. Concretely, it includes the penalty of a job being early or tardy, the cost for holding and delivering a job, and the cost incurred by a late starting or a long duration of the common due window. Observing the NP-hardness, we provide an optimal algorithm of the problem and convert it to a fully polynomial time approximation scheme for a special case.
We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k(2). The approximation guarantee is six times th...
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We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k(2). The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.
In this paper, we consider the group prize-collecting Steiner tree problem with submodular penalties (GPCST-SP problem). In this problem, we are given an undirected connected graph G = (V, E) with a pre-specified root...
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In this paper, we consider the group prize-collecting Steiner tree problem with submodular penalties (GPCST-SP problem). In this problem, we are given an undirected connected graph G = (V, E) with a pre-specified root r and a partition V = {V-0, V-1, ..., V-k} of V with r is an element of V-0. Assume c : E -> R+ is an edge cost function and p : 2(V) -> R+ is a submodular penalty function, where R+ is the set of nonnegative real numbers. For a group V-i is an element of V, we say it is spanned by a tree if the tree contains at least one vertex of that group. The goal of the GPCST-SP problem is to find an r-rooted tree that minimizes the costs of the edges in the tree plus the penalty cost of the subcollection S containing these groups not spanned by the tree. Our main result is a 2I-approximation algorithm for the problem, where I = max{vertical bar V-i vertical bar vertical bar i = 0, 1, 2, ..., k}.
In the incremental version of the k-median problem, we find a sequence of facility sets F-1 subset of F-2 subset of center dot center dot center dot subset of F-n, where each F-k contains at most k facilities. This se...
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In the incremental version of the k-median problem, we find a sequence of facility sets F-1 subset of F-2 subset of center dot center dot center dot subset of F-n, where each F-k contains at most k facilities. This sequence is said to be delta-competitive if the cost of each F-k is at most delta times the optimum cost of k facilities. The best deterministic (randomized) algorithm available for the metric space has a competitive ratio of 8 (7.656). The best one for the one"dimensional problem finds a 5.828-competitive sequence. We give a 7.076-competitive solution for the high-dimensional Euclidean space. (C) 2016 Elsevier B.V. All rights reserved.
In this paper. we consider the asymmetric traveling salesman problem with the gamma- parameterized triangle inequality for gamma is an element of |1/2, 1|. That means the edge weights in the given complete graph G = (...
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In this paper. we consider the asymmetric traveling salesman problem with the gamma- parameterized triangle inequality for gamma is an element of |1/2, 1|. That means the edge weights in the given complete graph G = (V, E, omega) satisfy omega(u, w) <= gamma . (omega(u, v) + omega(v, w)) for all distinct nodes u, v, w is an element of V. L.S. Chandran and L.S. Ram gave the first constant factor approximation algorithm with polynomial running time for this problem. They achieve performance ratio. gamma/1-gamma. M. Blaser, B. Manthey and J. Sgall obtain a 1+gamma/2-gamma-gamma(3)-approximation algorithm. We devise an approximation algorithm with performance ratio max{1 + gamma(3)/1-gamma(2), gamma+gamma(2)+1/2 + gamma(3)/1(-)gamma(2)}, which is better than both 1+gamma/2-gamma-gamma(3) and gamma/1-gamma for almost all gamma is an element of |1/2, 1|. (c) 2008 Elsevier Inc. All rights reserved.
In this paper we discuss a minmax regret version of the single-machine scheduling problem with the total flow time criterion. Uncertain processing times are modeled by closed intervals. We show that if the determinist...
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In this paper we discuss a minmax regret version of the single-machine scheduling problem with the total flow time criterion. Uncertain processing times are modeled by closed intervals. We show that if the deterministic problem is polynomially solvable, then its minmax regret version is approximable within 2. (C) 2007 Elsevier B.V. All rights reserved.
We consider the problem of finding a minimum-size hitting set in a range space F=(Q, R) defined by a measure on a family of subsets of an infinite set R. We observe that, under reasonably general assumptions, the infi...
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We consider the problem of finding a minimum-size hitting set in a range space F=(Q, R) defined by a measure on a family of subsets of an infinite set R. We observe that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta> 0, a set of size O(alpha(F)z*(F)) that hits a (1 - delta)-fraction of the ranges in R(with respect to the given measure) in time proportional to log1/delta, where z*(F) Fis the value of the fractionaloptimal solution and aFis the integrality gap of the standard LP relaxation of the hitting set problem for the restriction of Fon a set of points of size O(z*(F)log1/delta). Some applications of this result in Computational Geometry are given. (c) 2023 Elsevier B.V. All rights reserved.
We study the feedback vertex set problem in tournaments from the polyhedral point of view, and in particular we show that performing just one round of the Sherali- Adams hierarchy gives a relaxation with integrality g...
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We study the feedback vertex set problem in tournaments from the polyhedral point of view, and in particular we show that performing just one round of the Sherali- Adams hierarchy gives a relaxation with integrality gap 7/3. This allows us to derive a 7/3-approximation algorithm for the feedback vertex set problem in tournaments that matches the best deterministic approximation guarantee due to Mnich, Williams, and Vegh, and is a simplification and runtime improvement of their approach. (c) 2023 Elsevier B.V. All rights reserved.
The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is 2-o(1). In this paper, we use a collection of simple graph transformations, each of which guarantees an ...
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The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is 2-o(1). In this paper, we use a collection of simple graph transformations, each of which guarantees an approximation ratio of 3/2, to find approximate vertex covers for a large collection of randomly generated graphs and test graphs from various sources. The graph reductions are extremely fast, and even though they by themselves are not guaranteed to find a vertex cover, we manage to find a 3/2-approximate vertex cover for almost every single graph in our collection. The few graphs that we cannot solve have specific structure: they are triangle-free, with a minimum degree of at least 3, a lower bound of n/2 on the optimal vertex cover, and are unlikely to have a large bipartite subgraph.
The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a...
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The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2-approximation algorithm for the Bandwidth problem that takes worst-case O(1.9797(n)) = O(3(0.6217n)) time and uses polynomial space. This improves both the previous best 2- and 3-approximation algorithms of Cygan et al. which have O*(3(n)) and O*(2(n)) worst-case running time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident to vertices in the same bucket or to vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a divide-and-conquer strategy along with dynamic programming to achieve the improved time bound. (C) 2013 Elsevier B.V. All rights reserved.
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