In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constant-facto...
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ISBN:
(纸本)9781450300162
In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constant-factor approximation algorithm for planar TSPN with pairwise-disjoint connected neighborhoods of any size or shape. Prior approximation bounds were O(log n), except in special cases.
In this paper, we study the squared metric k-facility location problem, which generalizes the k-means problem in that each facility has a specific cost of opening it in the solution. The current best approximation gua...
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ISBN:
(纸本)9783030926816;9783030926809
In this paper, we study the squared metric k-facility location problem, which generalizes the k-means problem in that each facility has a specific cost of opening it in the solution. The current best approximation guarantee for the squared metric k-facility location problem is a ratio of 44.473 + epsilon based on a local search algorithm. We give a (36.343 + epsilon)-approximation for the problem using the techniques of primal-dual and Lagrangian relaxation. We propose a new rounding approach that exploits the properties of the squared metric, which is the crucial step in getting the improved approximation ratio.
In the Multicast k-Tree Routing Problem, a data copy is sent from the source node to at most k destination nodes in every transmission. The goal is to minimize the total cost of sending data to all destination nodes, ...
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ISBN:
(纸本)9783642020254
In the Multicast k-Tree Routing Problem, a data copy is sent from the source node to at most k destination nodes in every transmission. The goal is to minimize the total cost of sending data to all destination nodes, which is measured as the sum of the costs of all routing trees. This problem was formulated Out of optical networking and has applications in general multicasting. Several approximation algorithms, with increasing performance, have been proposed in the last several years;The most recent ones are heavily relied on a tree partitioning technique. In this paper, we present a further improved approximation algorithm along the line. The algorithm has a worst case performance ratio of 5/4 rho + 3/2, where p denotes the best approximation ratio for the Steiner Minimum Tree problem. The proofs of the technical routing lemmas also provide some insights on why such a performance ratio could be the best possible that one can get using this tree partitioning technique.
Link scheduling is crucial in improving the throughput in wireless networks and it has been widely studied under various interference models. In this paper, we study the link scheduling problem under physical interfer...
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ISBN:
(纸本)9781605585239
Link scheduling is crucial in improving the throughput in wireless networks and it has been widely studied under various interference models. In this paper, we study the link scheduling problem under physical interference model where all senders of the links transmit at a given power P and a link can transmit successfully if and only if the Signal-to-Interference-plus-Noise-Ratio (SINR) at the corresponding receiver is at least a certain threshold. The link scheduling problem is to find a maximum -independent set" (MIS) of links, i.e., the maximum number of links that can transmit successfully in one time-slot, given a set of input links. This problem has been shown to be with a constant approximation ratio for arbitrary background noise N >= 0. When each link l has a weight w(l) > 0, we propose a method for weighted MIS with approximation ratio O(min(log maxl is an element of L w(l)/minl is an element of L w(l), log maxl is an element of L parallel to l parallel to/minl is an element of L parallel to l parallel to)), where parallel to l parallel to is the Euclidean length of a link l.
We consider the classical Minimum Crossing Number problem: given an n-vertex graph G, compute a drawing of G in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamenta...
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ISBN:
(纸本)9781450392648
We consider the classical Minimum Crossing Number problem: given an n-vertex graph G, compute a drawing of G in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamental and extensively studied problem, whose approximability status is widely open. In all currently known approximation algorithms, the approximation factor depends polynomially on. - the maximum vertex degree in G. The best current approximation algorithm achieves an O(n(1/2)-is an element of poly(Delta. logn))-approximation, for a small fixed constant is an element of, while the best negative result is APX-hardness, leaving a large gap in our understanding of this basic problem. In this paper we design a randomized O (2(O((log n)7/8 log log n)) . poly(Delta)) -approximation algorithm for Minimum Crossing Number. This is the first approximation algorithm for the problem that achieves a subpolynomial in n approximation factor (albeit only in graphs whose maximum vertex degree is subpolynomial in n). In order to achieve this approximation factor, we design a new algorithm for a closely related problem called Crossing Number with Rotation System, in which, for every vertex upsilon is an element of V (G), the circular ordering, in which the images of the edges incident to v must enter the image of v in the drawing is fixed as part of input. Combining this result with the recent reduction of [Chuzhoy, Mahabadi, Tan '20] immediately yields the improved approximation algorithm for Minimum Crossing Number.
Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional axis parallel boxes in R-k. Equivalently, it is the minimum number of interval graphs on t...
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ISBN:
(纸本)9783642222993
Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional axis parallel boxes in R-k. Equivalently, it is the minimum number of interval graphs on the vertex set V such that the intersection of their edge sets is E. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below O(n(0.5-epsilon))-factor, for any epsilon > 0 in polynomial time unless NP = ZPP. Till date, there is no well known graph class of unbounded boxicity for which even an n(epsilon)-factor approximation algorithm for computing boxicity is known, for any epsilon < 1. In this paper, we study the boxicity problem on Circular Arc graphs intersection graphs of arcs of a circle. We give a (2+1/k)-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where k >= 1 is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is O(mn+n(2)) in both these cases and in O(mn+kn(2)) which is at most O(n(3)) time we also get their corresponding box representations, where a is the number of vertices of the graph and m is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time.
The problem of minimizing the maximum delivery times while scheduling jobs on the single processor is a classical combinatorial optimization problem. This problem is denoted by 1|r(j), q(j)|C-max, has many application...
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ISBN:
(纸本)9788395918384
The problem of minimizing the maximum delivery times while scheduling jobs on the single processor is a classical combinatorial optimization problem. This problem is denoted by 1|r(j), q(j)|C-max, has many applications, and it is NP-hard in strong sense. The goal of this paper is to propose a new 3/2-approximation algorithm, which runs in O(n log n) time. We proved that the bound of 3/2 is tight. To check the efficiency of the algorithm we tested it on random generated problems of up to 5000 jobs.
This paper presents a simple and fast polynomial time algorithm, the clever, steady strategy algorithm (CSSA). The proposed algorithm consists of three stages which produces optimal or approximate vertex cover for any...
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ISBN:
(纸本)9781509003297
This paper presents a simple and fast polynomial time algorithm, the clever, steady strategy algorithm (CSSA). The proposed algorithm consists of three stages which produces optimal or approximate vertex cover for any unknown un-weighted and undirected G = (V, E). In the first step the degree of each node of the given graph is calculated. In second step the minimum degree node(s) is find and also the adjacent nodes of minimum degree nodes are find. In the third stage the minimum degree node in all adjacent nodes of minimum degree is searched out and is selected as a candidate for MVC and all its edges are deleted. These three steps are processed repeatedly until no edge remains in the graph. The CSSA is tested on small as well as on large benchmark instances. The experimental results and comparative analysis show that the CSSA yields better and fast solution than those approximation algorithms found in literature for solving minimum vertex cover problem.
Given a set S of n disjoint line segments in R-2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can...
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ISBN:
(纸本)9783319426341;9783319426334
Given a set S of n disjoint line segments in R-2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can trivially be solved in logarithmic query time using O(n(4)) preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a trade-off between the space and the query time. They answer any query in O epsilon(n (1-alpha)) with O epsilon (n(2+2 alpha)) of preprocessing time and space, where alpha is a constant 0 <= alpha <= 1, epsilon > 0 is another constant that can be made arbitrarily small, and O epsilon(f(n)) = O(f( n)n(epsilon)). In this paper, we propose a randomized approximation algorithm for VCP with a tradeoff between the space and the query time. We will show that for an arbitrary constants 0 <= beta <= 2/3 and 0 < delta < 1, the expected preprocessing time, the expected space, and the query time of our algorithm are O( n(4-3 beta) log n), O(n(4-3 beta)), and O(1/delta(3)n(beta) log n), respectively. The algorithm computes the number of visible segments from p, or mp, exactly if m(p) <= (1)/(3)(delta)n(beta) log ***, it computes a ( 1+ delta)-approximation m'(p) with the probability of at least 1- 1/log n, where m(p) <= m'(p) <= ( 1 + delta) m(p).
We study a version of the graph 2-clustering problem. In this version, for a given undirected graph, one has to find a nearest 2-cluster graph, i.e., the graph on the same vertex set with exactly 2 non-empty connected...
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ISBN:
(纸本)9783030226299;9783030226282
We study a version of the graph 2-clustering problem. In this version, for a given undirected graph, one has to find a nearest 2-cluster graph, i.e., the graph on the same vertex set with exactly 2 non-empty connected components each of which is a complete graph. The distance between two graphs is the number of noncoinciding edges. The problem under consideration is NP-hard. In 2004, Bansal, Blum, and Chawla presented a simple polynomial time 3-approximation algorithm for the similar correlation clustering problem in which the number of clusters doesn't exceed 2. In 2008, Coleman, Saunderson, and Wirth presented a 2-approximation algorithm for this problem applying local search to every feasible solution obtained by the 3-approximation algorithm of Bansal, Blum, and Chawla. Unfortunately, the method of proving the performance guarantee of the Coleman, Saunderson, and Wirth's algorithm is not suitable for the graph 2-clustering. Coleman, Saunderson, and Wirth used switching technique that allows to reduce clustering any graph to the equivalent problem whose optimal solution is the complete graph, i.e., the cluster graph consisting of the single cluster. In the graph 2-clustering problem any optimal solution has to consist of exactly 2 clusters, so we need another approximation algorithm and another method of proving a bound on its worst-case behaviour. We present a polynomial time 2-approximation algorithm for the 2-clustering problem on general graphs. In contrast to the proof of Coleman, Saunderson, and Wirth, our proof of the performance guarantee of this algorithm doesn't use switchings.
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