Esther M. Arkin

Computational complexity of uncapacitated multi-echelon production planning problems

Recently there has been a flurry of research in the area of production planning for multi-echelon production-distribution systems with deterministic non-stationary demands and no capacity constraints. A variety of algorithms have been proposed to optimally solve these problems, with varying success. This paper investigates the issue of computational complexity of the problem for all commonly studied product structures, i.e. the single item, the serial system, the assembly system, the one-warehouse-N-retailer system, the distribution system, the joint replenishment system, and the general production-distribution system. Polynomial time algorithms are available for the single-item, serial and assembly systems. We prove that the remaining problems are NP-complete.

Approximation algorithms for lawn mowing and milling

Abstract We study the problem of finding shortest tours/paths for “lawn mowing” and “milling” problems: Given a region in the plane, and given the shape of a “cutter” (typically, a circle or a square), find a shortest tour/path for the cutter such that every point within the region is covered by the cutter at some position along the tour/path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3+e) -approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 6 5 -approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11 5 -approximation algorithm for milling simple rectilinear polygons.

Approximations for minimum and min-max vehicle routing problems

We consider a variety of vehicle routing problems. The input to a problem consists of a graph G = (N, E) and edge lengths l(e), e ∈ E. Customers located at the vertices have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and λ the longest distance traveled by a vehicle. We consider two types of problems. (1) Given a bound λ on the length of each path, find a minimum sized collection of paths that cover all the vertices of the graph, or all the edges from a given subset of edges of the input graph. We also consider a variation where it is desired to cover N by a minimum number of stars of length bounded by λ. (2) Given a number k find a collection of k paths that cover either the vertex set of the graph or a given subset of edges. The goal here is to minimize λ, the maximum travel distance. For all these problems we provide constant ratio approximation algorithms and prove their NP-hardness.

On Local Search for Weighted K -Set Packing

Given a collection of sets of cardinality at most k, with weights for each set, the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight. We study the worst case behavior of the t-local search heuristic for this problem proving a tight bound of k - 1 + 1/t . As a consequence, for any given r < 1/(k -1) we can compute in polynomial time a solution whose weight is at least r times the optimal.,

Resource-constrained geometric network optimization

WC study a variety of geometric network optimization prob lcms on a set of points, in which we are given a resource bound, a, on the total length of the network, and our ob jcctivc is to maximize the number of points visited (or the total “value” of points visited), In particular, we resolve the well-publicized open problem on the approximabiity of the rooted “orienteering problem” for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2approximation for this problem, We also obtain approximation algorithms for variants of this problem in which the network required is a tree (S-approximation) or a path Q-approximation). No prior approximation bounds were known for any of these problems, We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2-approximation for both the cycle and tree versions of the problem on points in the plane, as well as a G-approximation for the tree version in edge-weighted graphs, E’urther, we study generalizations of the basic orienteering problem, to the case of multiple roots, sites that are polygonnl regions, etc., where we again give the first known approximation results. Our methods are based on some new tools which may be of interest in their own right: (1) some new results on m-

Optimal Covering Tours with Turn Costs

We give the first algorithmic study of a class of “covering tour” problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (“pocket”), in order to minimize a cost that depends not only on the length of the tour but also on the number of turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (“postman”) problems with turn penalties. We prove lower bounds (NP-completeness of minimum-turn milling) and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

Minimum-cost coverage of point sets by disks

We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t_j)</i> and radii <i>(r_j)</i> that cover a given set of demand points <i>Y∈R</i>² at the smallest possible cost. We consider cost functions of the form Ε<i>_jf(r_j)</i>, where <i>f(r)=r</i>^α is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers <i>t_j</i>, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points <i>t_j</i> on a given line in order to cover demand points <i>Y∈R</i>²; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R²</i> and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

Algorithms for Rapidly Dispersing Robot Swarms in Unknown Environments

We develop and analyze algorithms for dispersing a swarm of primitive robots in an unknown environment, R. The primary objective is to minimize the makespan, that is, the time to fill the entire region. An environment is composed of pixels that form a connected subset of the integer grid. There is at most one robot per pixel and robots move horizontally or vertically at unit speed. Robots enter R by means of k ≥ 1 door pixels. Robots are primitive finite automata, only having local communication, local sensors, and a constant-sized memory.

Approximating the tree and tour covers of a graph

Abstract The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices from a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namely the weighted vertex cover, traveling salesman, and Steiner tree problems.

Processor allocation on Cplant: achieving general processor locality using one-dimensional allocation strategies

The Computational Plant or Cplant is a commodity-based supercomputer under development at Sandia National Laboratories. This paper describes resource-allocation strategies to achieve processor locality for parallel jobs in Cplant and other supercomputers. Users of Cplant and other Sandia supercomputers submit parallel jobs to a job queue. When a job is scheduled to run, it is assigned to a set of processors. To obtain maximum throughput, jobs should be allocated to localized clusters of processors to minimize communication costs and to avoid bandwidth contention caused by overlapping jobs. This paper introduces new allocation strategies and performance metrics based on space-filling curves and one dimensional allocation strategies. These algorithms are general and simple. Preliminary simulations and Cplant experiments indicate that both space-filling curves and one-dimensional packing improve processor locality compared to the sorted free list strategy previously used on Cplant. These new allocation strategies are implemented in the new release of the Cplant System Software, Version 2.0, phased into the Cplant systems at Sandia by May 2002.