A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance between the two players is two. We show th...
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A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance between the two players is two. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best-known result (3.9546, 4.3931). To achieve the improved bounds, we call upon results from absorbing Markov chain theory and mathematical programming theory-particularly fractional quadratic programming and semidefinite programming. Moreover, we also establish some important properties of this problem, which could be of independent interest and useful for resolving this problem completely. Finally, we conjecture that the symmetric rendezvous value is asymptotically equal to 4.25 based on our numerical calculations.
single machine scheduling with release dates and job delivery, jobs are processed on a single machine and then delivered by a capacitated vehicle to a single customer. Only one vehicle is employed to deliver these job...
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single machine scheduling with release dates and job delivery, jobs are processed on a single machine and then delivered by a capacitated vehicle to a single customer. Only one vehicle is employed to deliver these jobs. The vehicle can deliver at most c jobs at a shipment. The delivery completion time of a job is defined as the time at which the delivery batch containing the job is delivered to the customer and the vehicle returns to the machine. The objective is to minimize the makespan, i.e., the maximum delivery completion time of the jobs. When preemption is allowed to all jobs, we give a polynomial-time algorithm for this problem. When preemption is not allowed, we show that this problem is strongly NP-hard for each fixed c >= 1. We also provide a 5/3-approximation algorithm for this problem, and the bound is tight. (C) 2008 Published by Elsevier B.V.
We consider the problem of pricing (digital) items in order to maximize the revenue obtainable from a set of bidders. We suggest a natural monotonicity constraint on bundle prices, show that the problem remains NP-har...
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We consider the problem of pricing (digital) items in order to maximize the revenue obtainable from a set of bidders. We suggest a natural monotonicity constraint on bundle prices, show that the problem remains NP-hard, and we derive a PTAS. We also briefly discuss the highway pricing problem. (c) 2008 Elsevier B.V. All rights reserved.
We consider the minimum diameter spanning free problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload Cost problems: Minimum diameter spanning tree, Discret...
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We consider the minimum diameter spanning free problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload Cost problems: Minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85]. In this model an undirected edge-coloured graph G is given, together with a nonnegative symmetrical integer matrix R specifying the costs of changing from a colour to another one. The reload cost of a path in G arises at its internal nodes, when passing from the colour of one incident edge to the colour of the other. We prove that, unless P = NP, the problem of finding a spanning free of G having a minimum diameter with respect to reload costs, when restricted to graphs with maximum degree 4, cannot be approximated within any constant alpha < 2 if the reload costs are unrestricted, and cannot be approximated within any constant beta < 5/3 if the reload costs satisfy the triangle inequality. This solves a problem left open by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: minimum diameter spanning tree. Discrete Appl. Math. 113 (2001) 73-85]. (C) 2008 Elsevier B.V. All rights reserved.
Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let rho >= 1 be a real number. Distances in each face of this subdivision are measured by a convex ...
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Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let rho >= 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius 1/rho. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For any epsilon epsilon (0, 1) and for any two points upsilon(s) and upsilon(d), we give an algorithm that finds a path from upsilon(s) to upsilon(d) whose cost is at most (1 + epsilon) times the optimal. Our algorithm runs in O(rho(2) log rho/epsilon(2) n(3) log(rho n/epsilon)) time. This bound does not depend on any other parameters;in particular it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1,rho] boolean OR {infinity}, the time bound of our algorithm improves to O(rho log rho/epsilon n(3) log(rho n/epsilon)).
Given a directed graph G = (V, A) with a non-negative weight (length) function on its arcs w : A -> R+ and two terminals s, t epsilon V, our goal is to destroy all short directed paths from s to t in G by eliminati...
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Given a directed graph G = (V, A) with a non-negative weight (length) function on its arcs w : A -> R+ and two terminals s, t epsilon V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v epsilon V a fixed number k(upsilon) of out-going arcs can be removed. Our results indicate that the latter sub-case is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra's algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c < 2 the maximum s-t distance d (s, t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor c < 10 root 5-21 approximate to 1.36 the minimum number of arcs which has to be removed to guarantee d(s, t) >= d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.
We present a polynomial time approximation algorithm for unit time precedence constrained scheduling. Our algorithm guarantees schedules which are at most (2 - 7/3p +1) factor as long as the optimal, where p > 3 is...
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We present a polynomial time approximation algorithm for unit time precedence constrained scheduling. Our algorithm guarantees schedules which are at most (2 - 7/3p +1) factor as long as the optimal, where p > 3 is the number of processors. This improves upon a long standing bound of (2 - 2/p) due to Coffman and Graham. (C) 2008 Elsevier Inc. All rights reserved.
The minimum clique partitioning problem in weighted interval graphs (MCPI) is defined as follows. Given an interval graph with nonnegative node weights, the problem is to partition the nodes into a set of cliques such...
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The minimum clique partitioning problem in weighted interval graphs (MCPI) is defined as follows. Given an interval graph with nonnegative node weights, the problem is to partition the nodes into a set of cliques such that the sum of node weights in each clique is no more than a given bound. The objective of the problem is to minimize the number of cliques. Recently, Chen et al. [M. Chen, J. Li, J. Li, W. Li, and L. Wang, Some approximation algorithms for the clique partitioning problem in weighted interval graphs, Theoretical Computer Science 381 (2007), 124-133] proposed three approximation algorithms having constant factors 3, 2.5 and 2, and a linear time optimal algorithm for the case with identical weights. In this paper, we show that their factor 2 algorithm does not achieve the expected approximation ratio and the linear time algorithm cannot give an optimal solution for the identical weights case. We also develop an approximation algorithm with factor 2 for the variable weights case and an exact algorithm for the identical weights case. (C) 2008 Elsevier B.V. All rights reserved.
We consider the d-dimensional bin-packing problem, the most relevant generalization of classical bin packing, and show a general result about the asymptotic worst-case ratio of a wide class of approximation algorithms...
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We consider the d-dimensional bin-packing problem, the most relevant generalization of classical bin packing, and show a general result about the asymptotic worst-case ratio of a wide class of approximation algorithms that construct solutions in d stages, containing many heuristics previously considered in the literature. Moreover, we give the exact value of the asymptotic worst-case ratio between the optimal solution and the best solution obtainable by packing the items in d stages, showing how to achieve such a ratio efficiently. The key property behind our result is the asymptotic optimality of the fractional relaxation of (one-dimensional) bin packing, an important by-product of the approximation schemes for the problem from the 1980s, which, to the best of our knowledge, is used here for the first time only for the sake of the analysis. For the two-dimensional case, we push the approximablity threshold below 2 after more than 20 years (reaching 1.691...), but also improve and widely simplify the analysis of the previous best method from the 1980s. Moreover, for d >= 3 we improve the previous approximability threshold by a factor of 1.691..., a notable improvement when d is small.
In this paper we consider the single-machine parallel-batching scheduling problem with family jobs under on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the ...
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In this paper we consider the single-machine parallel-batching scheduling problem with family jobs under on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the existence of any job until its arrival. Our objective is to minimize the maximum completion time of the jobs (makespan). We deal with two variants of the problem: the unbounded model in which the machine can handle infinite number of jobs simultaneously and the bounded model. For the unbounded case, we provide an on-line algorithm with a worst-case ratio of 2 and prove that there exists no on-line algorithm with a worst-case ratio less than 2. For the bounded case, we also present an on-line algorithm with a worst-case ratio of 2. (c) 2007 Elsevier B.V. All rights reserved.
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