We consider the problem of minimizing the makespan(Cmax) on m identical parallel batch processing machines. The batch processing machine can process up to B jobs simultaneously. The jobs that are processed together fo...
详细信息
We consider the problem of minimizing the makespan(Cmax) on m identical parallel batch processing machines. The batch processing machine can process up to B jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. For a batch of jobs, the processing time of the batch is equal to the largest processing time among the jobs in the batch. In this paper, we design a fully polynomial time approximation scheme (FPTAS) to solve the bounded identical parallel batch scheduling problem Pm |B < n|Cmax when the number of identical parallel batch processing machines m is constant.
Given a sets, we want to choose exactly k sets such that the cardinality of their intersection is maximized. This is the so-called MAX-k-INTERSECT problem. We prove that MAX-k-INTERSECT cannot be approximated within a...
详细信息
Given a sets, we want to choose exactly k sets such that the cardinality of their intersection is maximized. This is the so-called MAX-k-INTERSECT problem. We prove that MAX-k-INTERSECT cannot be approximated within an absolute error of 1/2n(1-2 epsilon) + O(n(1-3 epsilon)) unless P = NP. This answers an open question about its hardness. We also give a correct proof of an inapproximable result by Clifford and Papa (2011) [3] by proving that MAX-INTERSECT problem is equivalent to the MAX-CLIQUE problem. (C) 2012 Elsevier B.V. All rights reserved.
Given a graph G and a demand function p: V(G) -> N, a proper n-[p]coloring is a mapping f : V(G) -> 2([1.....n]) such that vertical bar f (v)vertical bar >= p(v) for every vertex v epsilon V(G) and f(v) boole...
详细信息
Given a graph G and a demand function p: V(G) -> N, a proper n-[p]coloring is a mapping f : V(G) -> 2([1.....n]) such that vertical bar f (v)vertical bar >= p(v) for every vertex v epsilon V(G) and f(v) boolean AND f(u) = emptyset for any two adjacent vertices u and v. The least integer n for which a proper n-[p]coloring exists, chi(p)(G), is called multichrornatic number of G. Finding multichromatic number of induced subgraphs of the triangular lattice (called hexagonal graphs) has applications in cellular networks. Weighted clique number of a graph G, omega(p)(G), is the maximum weight of a clique in G, where the weight of a clique is the total demand of its vertices. McDiarmid and Reed (2000) [8] conjectured that chi(p)(G) <= (9/8)omega(p)(G) + C for triangle-free hexagonal graphs, where C is some absolute constant. In this article, we provide an algorithm to find a 7-[3]coloring of triangle-free hexagonal graphs (that is, when p(v) = 3 for all v epsilon V (G)), which implies that chi(p)(G) <= (7/6)omega(p)(G) + C. Our result constitutes a shorter alternative to the inductive proof of Havet (2001) [5] and improves the short proof of Sudeep and Vishwanathan (2005) [17], who proved the existence of a 14-[6]coloring. (It has to be noted, however, that our proof makes use of the 4-color theorem.) All steps of our algorithm take time linear in vertical bar V(G)vertical bar, except for the 4-coloring of an auxiliary planar graph. The new techniques may shed some light on the conjecture of McDiarmid and Reed (2000) [8]. (C) 2011 Elsevier B.V. All rights reserved.
We consider the problem of finding a spanning tree with maximum number of leaves. A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex h...
详细信息
We consider the problem of finding a spanning tree with maximum number of leaves. A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). The problem is known to be APX-hard in general, and NP-hard for cubic graphs. We show that it is also APX-hard for cubic graphs. The APX-hardness of the related problem Minimum Connected Dominating Set for cubic graphs follows. (C) 2011 Elsevier B.V. All rights reserved.
For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard...
详细信息
For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k-way cut is to iteratively increase the number of components of the graph by h - 1, where 2 <= h <= k, until the graph has k components. The approximation ratio of this algorithm is known for h <= 3 but is open for h >= 4. In this paper, we consider a general algorithm that successively increases the number of components of the graph by h(i) - 1, where 2 <= h(1) <= h(2) <= ... <= h(q) and Sigma(q)(i=1) (h(i) - 1) = k - 1. We prove that the approximation ratio of this general algorithm is 2 - (Sigma(q)(i=1) ((hi)(2)))/((k)(2)), which is tight. Our result implies that the approximation
Given a set A = {a(1) ,..., a(n)} of n image points and a set B = {b(1) ,..., b(n)} of n model points, the problem is to find a transformation matching (a one-to-one mapping) each point a is an element of A to some po...
详细信息
Given a set A = {a(1) ,..., a(n)} of n image points and a set B = {b(1) ,..., b(n)} of n model points, the problem is to find a transformation matching (a one-to-one mapping) each point a is an element of A to some point b is an element of B such that the length of the longest edge in the matching is minimised (so-called bottleneck distance). The geometric transformations we allow are translation, rotation, reflexion and scaling. In this paper, we give (1 + epsilon)- approximation algorithms for the case when the points are given in R-2, two of which run in O(n(3.5)/e(4) logn) and O(n(2.5)/e(4) logn log diam(B)/d(opt)) time, respectively, where diam(B) is the diameter of B and dopt is the bottleneck distance in an optimal matching. (C) 2012 Elsevier B.V. All rights reserved.
One approach to guarantee the performance of underwater acoustic sensor networks is to deploy multiple Surface-level Gateways (SGs) at the surface. This paper addresses the connected (or survivable) Constrained Surfac...
详细信息
One approach to guarantee the performance of underwater acoustic sensor networks is to deploy multiple Surface-level Gateways (SGs) at the surface. This paper addresses the connected (or survivable) Constrained Surface-level Gateway Placement (C-SGP) problem for 3-D underwater acoustic sensor networks. Given a set of underwater sensor nodes (USNs) which are floated at different depths to perform collaborative monitoring tasks over a given region, and a set of candidate locations where SGs may be placed, our objective is to place minimum number of SGs at a subset of candidate locations such that it is connected (or k-connected) from any USN to the base station. We first propose a general algorithm for the connected C-SGP problem and prove its approximation ratio. We also give a constant ratio approximation algorithm for the problem. Second, for the survivable C-SGP problem we also propose a general algorithm and prove its approximation ratio. Finally, we give a constant ratio approximation algorithm for the 2-connected C-SGP problem. (c) 2011 Elsevier B.V. All rights reserved.
In this paper we propose an approximation algorithm for scheduling malleable tasks with precedence constraints. Based on an interesting model for malleable tasks with continuous processor allotments by Prasanna and Mu...
详细信息
In this paper we propose an approximation algorithm for scheduling malleable tasks with precedence constraints. Based on an interesting model for malleable tasks with continuous processor allotments by Prasanna and Musicus (1991. 1994, 1996) vertical bar 23-25 vertical bar, we define two natural assumptions for malleable tasks: the processing time of any malleable task is non-increasing in the number of processors allotted, and the speedup is concave in the number of processors. We show that under these assumptions the work function of any malleable task is non-decreasing in the number of processors and is convex in the processing time. Furthermore, we propose a two-phase approximation algorithm for the scheduling problem. In the first phase we solve a linear program to obtain a fractional allotment for all tasks. By rounding the fractional solution, each malleable task is assigned a number of processors. In the second phase a variant of the list scheduling algorithm is employed. By choosing appropriate values of the parameters, we show (via a nonlinear program) that the approximation ratio of our algorithm is at most 100/63 + 100(root 6469 + 13)/5481 approximate to 3.291919. We also show that our result is asymptotically tight. (C) 2011 Elsevier Inc. All rights reserved.
Computing the edit distance between two genomes under certain operations is a basic problem in the study of genome evolution. The double-cut-and-join (DCJ) model has formed the basis for most algorithmic research on r...
详细信息
Computing the edit distance between two genomes under certain operations is a basic problem in the study of genome evolution. The double-cut-and-join (DCJ) model has formed the basis for most algorithmic research on rearrangements over the last few years. The edit distance under the DCJ model can be easily computed for genomes without duplicate genes. In this paper, we study the edit distance for genomes with duplicate genes under a model that includes DCJ operations, insertions and deletions. We prove that computing the edit distance is equivalent to finding the optimal cycle decomposition of the corresponding adjacency graph, and give an approximation algorithm with an approximation ratio of 1.5 + L.
The problem of correlated data gathering in wireless sensor networks is studied in this paper. For the sake of efficiency, tree transmission structures are often used for data gathering. Previously, the problem of min...
详细信息
The problem of correlated data gathering in wireless sensor networks is studied in this paper. For the sake of efficiency, tree transmission structures are often used for data gathering. Previously, the problem of minimizing the total communication cost with a single-tree transmission structure was shown to be NP-hard. However, when the explicit communication approach is used, the total communication cost can be further reduced, provided a double-tree transmission structure is used and inverse links are allowed. This motivates us to devise a double-tree routing scheme in which two trees are used for data transmission, one carrying raw data and the other carrying encoded data. We show that with the double-tree routing scheme, the problem of minimizing the total communication cost remains NP-hard. A distributed algorithm for solving it is suggested. We show that under the simple correlation model, the algorithm has an approximation ratio of two. Extensive simulations are conducted to verify the effectiveness of the double-tree routing scheme.
暂无评论