arXiv subjects

Sandor P. Fekete

Publications and source records attributed to Sandor P. Fekete.

48 records · Page 3Linked to original sources

arXiv2004

Online Searching with an Autonomous Robot

We discuss online strategies for visibility-based searching for an object hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task is closely related to a number of well-studied problems. Our robot uses a three-dimensional laser scanner in a stop, scan, plan, go fashion for building a virtual three-dimensional environment. Besides planning trajectories and avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We derive a practically useful and asymptotically optimal strategy that guarantees a competitive ratio of 2, which differs remarkably from the well-studied scenario without the need of stopping for surveying the environment. Our strategy is used by Kurt3D, documented in a separate video.

cs.RO↗

arXiv2004

A General Framework for Bounds for Higher-Dimensional Orthogonal Packing Problems

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. In the context of a branch-and-bound framework for solving these packing problems to optimality, it is of crucial importance to have good and easy bounds for an optimal solution. Previous efforts have produced a number of special classes of such bounds. Unfortunately, some of these bounds are somewhat complicated and hard to generalize. We present a new approach for obtaining classes of lower bounds for higher-dimensional packing problems; our bounds improve and simplify several well-known bounds from previous literature. In addition, our approach provides an easy framework for proving correctness of new bounds.

cs.DS↗

arXiv2003

Maximum dispersion and geometric maximum weight cliques

We consider a facility location problem, where the objective is to ``disperse'' a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.

cs.DS↗

arXiv2003

Characterizing Matchings as the Intersection of Matroids

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function mu(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, mu(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether mu(G)<= k. Using combinatorial arguments, we prove that mu(n)is in Omega(loglog n). On the other hand, we establish that mu(n) is in O(log n / loglog n). Finally, we prove that mu(n)=4 for n=5,...,12, and mu(n)=5 for n=13,14,15.

math.CO↗

arXiv2003

A combinatorial characterization of higher-dimensional orthogonal packing

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for instances in two- or higher-dimensional space. We present a new approach for modeling packings, using a graph-theoretical characterization of feasible packings. Our characterization allows it to deal with classes of packings that share a certain combinatorial structure, instead of having to consider one packing at a time. In addition, we can make use of elegant algorithmic properties of certain classes of graphs. This allows our characterization to be the basis for a successful branch-and-bound framework. This is the first in a series of papers describing new approaches to higher-dimensional packing.

cs.DS↗

arXiv2003

On the continuous Fermat-Weber problem

We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the ``continuous k-median (Fermat-Weber) problem'' where the goal is to select one or more center points that minimize the average distance to a set of points in a demand region. In such problems, the average is computed as an integral over the relevant region, versus the usual discrete sum of distances. The resulting facility location problems are inherently geometric, requiring analysis techniques of computational geometry. We provide polynomial-time algorithms for various versions of the L1 1-median (Fermat-Weber) problem. We also consider the multiple-center version of the L1 k-median problem, which we prove is NP-hard for large k.

cs.CG↗

arXiv2003

The Geometric Maximum Traveling Salesman Problem

We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n^{f-2} log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. Our approach can be extended to the more general case of quasi-norms with not necessarily symmetric unit ball, where we get a complexity of O(n^{2f-2} log n). For the special case of two-dimensional metrics with f=4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with O(n) running time. The algorithm does not use any indirect addressing, so its running time remains valid even in comparison based models in which sorting requires Omega(n \log n) time. The basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms. Complementing the results on simplicity for polyhedral norms, we prove that for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard. This sheds new light on the well-studied difficulties of Euclidean distances.

cs.DS↗

arXiv2002

Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.

cs.DS↗

arXiv2002

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. We first give algorithms for the single-door case (i.e., k=1), analyzing the algorithms both theoretically and experimentally. We prove that our algorithms have optimal makespan 2A-1, where A is the area of R. We next give an algorithm for the multi-door case (k>1), based on a wall-following version of the leader-follower strategy. We prove that our strategy is O(log(k+1))-competitive, and that this bound is tight for our strategy and other related strategies.

cs.RO↗

arXiv2002

Traveling Salesmen in the Presence of Competition

We propose the ``Competing Salesmen Problem'' (CSP), a 2-player competitive version of the classical Traveling Salesman Problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does. In particular, we consider the situation where players take turns, moving along one edge at a time within a graph G=(V,E). The set of customers is given by a subset V_C V of the vertices. At any given time, both players know of their opponent's position. A player wins if he is able to reach a majority of the vertices in V_C before the opponent does. We prove that the CSP is PSPACE-complete, even if the graph is bipartite, and both players start at distance 2 from each other. We show that the starting player may lose the game, even if both players start from the same vertex. For bipartite graphs, we show that the starting player always can avoid a loss. We also show that the second player can avoid to lose by more than one customer, when play takes place on a graph that is a tree T, and V_C consists of leaves of T. For the case where T is a star and V_C consists of n leaves of T, we give a simple and fast strategy which is optimal for both players. If V_C consists not only of leaves, the situation is more involved.

cs.CC↗

arXiv2002

On the Reflexivity of Point Sets

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.

cs.CG↗

arXiv2002

An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.

cs.CG↗