Two-Level Rectilinear Steiner Trees

Given a set $P$ of terminals in the plane and a partition of $P$ into $k$ subsets $P_1, ..., P_k$, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree $T_i$ connecting the terminals in each set $P_i$ ($i=1,...,k$) and a top-level tree $T_{top}$ connecting the trees $T_1, ..., T_k$. The goal is to minimize the total length of all trees. This problem arises naturally in the design of low-power physical implementations of parity functions on a computer chip. For bounded $k$ we present a polynomial time approximation scheme (PTAS) that is based on Arora's PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each $T_i$ and $T_{top}$ ($i=1,...,k$). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each $T_i$ and $T_{top}$ independently. This gives us a $2.37$-factor approximation with a running time of $\mathcal{O}(|P|\log|P|)$ suitable for fast practical computations. The approximation factor reduces to $1.63$ by applying Arora's approximation scheme in the plane

Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pair's Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d \ge 3$, the problem is APX-hard; it is known to admit, for any \eps > 0, an O(n^\eps)-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(\log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d \ge 2$ dimensions and an $O(\log n)$-approximation algorithm for 2D. We show that an existing $O(\log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.Comment: 14 pages, 5 figures; added appendix and figure

Fully dynamic maintenance of euclidean minimum spanning trees

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path

Equidistribution in All Dimensions of Worst-Case Point Sets for the TSP

Given a set S of n points in the unit square [0, 1]d , an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S ⊂ [0, 1]d , where |S| = n. An open problem is to determine the structure of S(n) . We show that for any rectangular parallelepiped R contained in [0, 1]d , the number of points in S(n) ∩ R is asymptotic to n times the volume of R. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n)

Equidistribution of Point Sets for the Traveling Salesman and Related Problems

Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary of this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n)

Probability and Problems in Euclidean Combinatorial Optimization

This article summarizes the current status of several streams of research that deal with the probability theory of problems of combinatorial optimization. There is a particular emphasis on functionals of finite point sets. The most famous example of such functionals is the length associated with the Euclidean traveling salesman problem (TSP), but closely related problems include the minimal spanning tree problem, minimal matching problems and others. Progress is also surveyed on (1) the approximation and determination of constants whose existence is known by subadditive methods, (2) the central limit problems for several functionals closely related to Euclidean functionals, and (3) analogies in the asymptotic behavior between worst-case and expected-case behavior of Euclidean problems. No attempt has been made in this survey to cover the many important applications of probability to linear programming, arrangement searching or other problems that focus on lines or planes

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.Comment: 24 pages, 6 figures; revised to appear in Journal of the ACM. (clarified some minor points, fixed typos