Approximation algorithms for problems in scheduling with set-ups

In this paper, we present new approximation results for the offline problem of single machine scheduling with sequence-independent set-ups and item availability, where the jobs to be scheduled are independent (i.e., have no precedence constraints) and have a common release time.We present polynomial-time approximation algorithms for two versions of this problem. In the first version, the input includes a weight for each job, and the goal is to minimize the total weighted completion time. On any input, our algorithm produces a schedule whose total weighted completion time is within a factor 2 of optimal for that input.In the second version, the input includes a due date for each job, and the goal is to minimize the maximum lateness of any job. On any input, our algorithm produces a schedule with the following performance guarantee: the maximum lateness of a job is at most the maximum lateness of the optimal schedule on a machine that runs at half the speed of our machine.     

Approximation algorithms for scheduling problems with a modified total weighted tardiness objective

We study the approximability of minimum total weighted tardiness with a modified objective which includes an additive constant. This ensures the existence of a positive lower bound for the minimum value. Moreover the new objective has a natural interpretation in just-in-time production systems.     

Approximation algorithms for art gallery problems in polygons

In this paper, we present approximation algorithms for minimum vertex and edge guard problems for polygons with or without holes with a total of n vertices. For simple polygons, approximation algorithms for both problems run in O(n⁴) time and yield solutions that can be at most O(logn) times the optimal solution. For polygons with holes, approximation algorithms for both problems give the same approximation ratio of O(logn), but the running time of the algorithms increases by a factor of n to O(n⁵).     

Approximation algorithms for Hamming clustering problems

We study Hamming versions of two classical clustering problems. The Hamming radius p-clustering problem (HRC) for a set S of k binary strings, each of length n, is to find p binary strings of length n that minimize the maximum Hamming distance between a string in S and the closest of the p strings; this minimum value is termed the p-radius of S and is denoted by . The related Hamming diameter p-clustering problem (HDC) is to split S into p groups so that the maximum of the Hamming group diameters is minimized; this latter value is called the p-diameter of S.We provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever k is constant. We also observe that HDC admits straightforward polynomial-time solutions when k=O(logn) and p=O(1), or when p=2. Next, by reduction from the corresponding geometric p-clustering problems in the plane under the L₁ metric, we show that neither HRC nor HDC can be approximated within any constant factor smaller than two unless P=NP. We also prove that for any >0 it is NP-hard to split S into at most pk^1/7− clusters whose Hamming diameter does not exceed the p-diameter, and that solving HDC exactly is an NP-complete problem already for p=3. Furthermore, we note that by adapting Gonzalez' farthest-point clustering algorithm [T. Gonzalez, Theoret. Comput. Sci. 38 (1985) 293–306], HRC and HDC can be approximated within a factor of two in time O(pkn). Next, we describe a 2^O(p/)k^O(p/)n²-time (1+)-approximation algorithm for HRC. In particular, it runs in polynomial time when p=O(1) and =O(log(k+n)). Finally, we show how to find in

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time a set L of O(plogk) strings of length n such that for each string in S there is at least one string in L within distance (1+), for any constant 0<<1.     

Approximation algorithms from inexact solutions to semidefinite programming relaxations of combinatorial optimization problems

Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semidefinite relaxations to optimality. In this paper, we investigate the effect on the performance guarantees of an approximate solution to the semidefinite relaxation for MaxCut, Max2Sat, and Max3Sat. We show that it is possible to make simple modifications to the approximate solutions and obtain performance guarantees that depend linearly on the most negative eigenvalue of the approximate solution, the size of the problem, and the duality gap. In every case, we recover the original performance guarantees in the limit as the solution approaches the optimal solution to the semidefinite relaxation.     

Approximation algorithms for a vehicle routing problem

In this paper we investigate a vehicle routing problem motivated by a real-world application in cooperation with the German Automobile Association (ADAC). The general task is to assign service requests to service units and to plan tours for the units such as to minimize the overall cost. The characteristics of this large-scale problem due to the data volume involve strict real-time requirements. We show that the problem of finding a feasible dispatch for service units starting at their current position and serving at most k requests is NP-complete for each fixed k ≥ 2. We also present a polynomial time (2k − 1)-approximation algorithm, where again k denotes the maximal number of requests served by a single service unit. For the boundary case when k equals the total number |E| of requests (and thus there are no limitations on the tour length), we provide a -approximation. Finally, we extend our approximation results to include linear and quadratic lateness costs.     

On an inventory model for deteriorating items and time-varying demand

In this paper, we present an optimal procedure for finding the replenishment schedule for the inventory system in which items deteriorate over time and demand rates are increasing over a known and finite planning horizon.     

Improved approximation for the capacitated inventory access point problem

The Inventory Access Point (IAP) is the single-item lot-sizing problem where a single customer faces demands in a discrete planning horizon, and the goal is to find a replenishment policy that minimizes the total inventory and ordering costs. While the uncapacitated version is polynomial, only a 3-approximation is known for the capacitated case. We improve this factor to 2.619 and, as a byproduct, we also improve the best factor for SIRPFL, which is a variant with multiple depots and customers.     

Approximation algorithms for dynamic resource allocation

We consider a problem of allocating limited quantities of M types of resources among N independent activities that evolve over T epochs. In each epoch, we assign to each activity a task which consumes resources, generates utility, and determines the subsequent state of the activity. We study the complexity of, and approximation algorithms for, maximizing average utility.     

Approximation algorithms for constrained generalized tree alignment problem

In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows:

Constrained Generalized Tree Alignment Problem [S. Divakaran, Algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S.

In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2−2/k.     

Approximation algorithms for UET scheduling problems with exact delays

In this paper we consider coupled-task single-machine and two-machine flow shop scheduling problems with exact delays, unit processing times, and the makespan as an objective function. The main results of the paper are fast 7/4- and 3/2-approximation algorithms for solving the single- and two-machine problems, respectively.     

On two-echelon inventory systems with Poisson demand and lost sales

We consider a two-echelon, continuous review inventory system under Poisson demand and a one-for-one replenishment policy. Demand is lost if no items are available at the local warehouse, the central depot, or in the pipeline in between. We give a simple, fast and accurate approach to approximate the service levels in this system. In contrast to other methods, we do not need an iterative analysis scheme. Our method works very well for a broad set of cases, with deviations to simulation below 0.1% on average and below 0.36% for 95% of all test instances.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Online lot-sizing problems with ordering, holding and shortage costs

We study inventory management problems where demands are revealed incrementally and procurement decisions must be made before the demands are realized. There are no probabilistic distributions nor non-trivial bounds to characterize demands. We consider two cost minimization problems: (1) perishable products with lost sales and (2) durable products with backlogged demand. In both problems, costs are period dependent. These problems are analyzed by utilizing linear-fractional programming and duality theory. Structural results are proved and then developed into practical strategies.     

Approximation algorithms for group prize-collecting and location-routing problems

In this paper we develop approximation algorithms for generalizations of the following three known combinatorial optimization problems, the Prize-Collecting Steiner Tree problem, the Prize-Collecting Travelling Salesman Problem and a Location-Routing problem.Given a graph G=(V,E) on n vertices and a length function on its edges, in the grouped versions of the above mentioned problems we assume that V is partitioned into k+1 groups, {V₀,V₁,…,V_k}, with a penalty function on the groups. In the Group Prize-Collecting Steiner Tree problem the aim is to find S, a collection of groups of V and a tree spanning the rest of the groups not in S, so as to minimize the sum of the costs of the edges in the tree and the costs of the groups in S. The Group Prize-Collecting Travelling Salesman Problem, is defined analogously. In the Group Location-Routing problem the customer vertices are partitioned into groups and one has to select simultaneously a subset of depots to be opened and a collection of tours that covers the customer groups. The goal is to minimize the costs of the tours plus the fixed costs of the opened depots. We give a -approximation algorithm for each of the three problems, where I is the cardinality of the largest group.     

Approximation algorithms for the transportation problem with market choice and related models

Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them.