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.
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.
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 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.
We consider the problem of packing squares into bins which are unit squares, where the goal is to minimize the number of bins used. We present an algorithm for this problem with an absolute worst-case ratio of 2, whic...
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We consider the problem of packing squares into bins which are unit squares, where the goal is to minimize the number of bins used. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P not equal NP. (C) 2004 Elsevier B.V. All rights reserved.
We study the NP-hard problem of labeling points with maximum-radius circle pairs: given n point sites in the plane, find a placement for 2n interior-disjoint uniform circles, such that each site touches two circles an...
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We study the NP-hard problem of labeling points with maximum-radius circle pairs: given n point sites in the plane, find a placement for 2n interior-disjoint uniform circles, such that each site touches two circles and the circle radius is maximized. We present a new approximation algorithm for this problem that runs in O(n log n + n log epsilon\1) time and O(n) space and achieves an approximation factor of (2 + root 3 + 2 root 4 + root 3)/(4 + root 3) + epsilon (approximate to 1.486 + epsilon), which improves the previous best bound of 1.491 + epsilon. (c) 2006 Elsevier B.V. All rights reserved.
We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges forms a planar graph. By planar duality, this is e...
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We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges forms a planar graph. By planar duality, this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.
We improve the approximation ratio for MAXIMUM WEIGHT SERIES-PARALLEL SUBGRAPH from 1 / 2 to 1 / 2 + 1 / 60. (c) 2023 Elsevier B.V. All rights reserved.
We improve the approximation ratio for MAXIMUM WEIGHT SERIES-PARALLEL SUBGRAPH from 1 / 2 to 1 / 2 + 1 / 60. (c) 2023 Elsevier B.V. All rights reserved.
Cloud radio access networks (C-RANs) are proposed as promising architecture to improve the capacity and enhance the coverage of mobile communication systems. In this letter, we study the baseband unit (BBU) pools plan...
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Cloud radio access networks (C-RANs) are proposed as promising architecture to improve the capacity and enhance the coverage of mobile communication systems. In this letter, we study the baseband unit (BBU) pools planning problem in the C-RAN, where we try to minimize the total deployment cost while satisfying the traffic demands of remote radio heads connected to the BBU pools, the processing capacity of each BBU pool, and the latency requirements of the C-RAN. Our problem formulation leads a mixed integer linear programming problem that is NP-hard. We introduce an approximation algorithm to address it efficiently. Numerical results show that our proposed algorithm performs much better than other heuristic ones that are popular to deal with such kind of optimization tasks. Moreover, our proposal also yields a performance guaranteed scheme for the BBU pools planning problem in the C-RAN.
Emerging applications with low-latency requirements such as real-time analytics, immersive media applications, and intelligent virtual assistants have rendered Edge Computing as a critical computing infrastructure. Ex...
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Emerging applications with low-latency requirements such as real-time analytics, immersive media applications, and intelligent virtual assistants have rendered Edge Computing as a critical computing infrastructure. Existing studies have explored the cloudlet placement problem in a homogeneous scenario with different goals such as latency minimization, load balancing, energy efficiency, and placement cost minimization. However, placing cloudlets in a highly heterogeneous deployment scenario considering the next-generation 5G networks and IoT applications is still an open challenge. The novel requirements of these applications indicate that there is still a gap in ensuring low-latency service guarantees when deploying cloudlets. Furthermore, deploying cloudlets in a cost-effective manner and ensuring full coverage for all users in edge computing are other critical conflicting issues. In this article, we address these issues by designing a bifactor approximation algorithm to solve the heterogeneous cloudlet placement problem to guarantee a bounded latency and placement cost, while fully mapping user applications to appropriate cloudlets. We first formulate the problem as a multi-objective integer programming model and show that it is a computationally NP-hard problem. We then propose a bifactor approximation algorithm, ACP, to tackle its intractability. We investigate the effectiveness of ACP by performing extensive theoretical analysis and experiments on multiple deployment scenarios based on New York City OpenData. We prove that ACP provides a (2,4)-approximation ratio for the latency and the placement cost. The experimental results show that ACP obtains near-optimal results in a polynomial running time making it suitable for both short-term and long-term cloudlet placement in heterogeneous deployment scenarios.
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