Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

TSP in a Simple Polygon

We study the Traveling Salesman Problem inside a simple polygon. In this problem, which we call tsp in a simple polygon, we wish to compute a shortest tour that visits a given set S of n sites inside a simple polygon P with m edges while staying inside the polygon. This natural problem has, to the best of our knowledge, not been studied so far from a theoretical perspective. It can be solved exactly in poly(n,m) + 2^O(?nlog n) time, using an algorithm by Marx, Pilipczuk, and Pilipczuk (FOCS 2018) for subset tsp as a subroutine. We present a much simpler algorithm that solves tsp in a simple polygon directly and that has the same running time

A grid-based ant colony algorithm for automatic 3D hose routing

Ant Colony Algorithms applied to difficult combinatorial optimization problems such as the traveling salesman problem (TSP) and the quadratic assignment problem. In this paper we propose a grid-based ant colony algorithm for automatic 3D hose routing. Algorithm uses the tessellated format of the obstacles and the generated hoses in order to detect collisions. The representation of obstacles and hoses in the tessellated format greatly helps the algorithm towards handling free-form objects and speed up the computations. The performance of the algorithm has been tested on a number of 3D models

Approximation of Euclidean k-size cycle cover problem

For a fixed natural number k, a problem of k collaborating salesmen servicing the same set of cities (nodes of a given graph) is studied. We call this problem the Minimumweight k-size cycle cover problem (or Min-k-SCCP) due to the fact that the problem has the following mathematical statement. Let a complete weighted digraph (with loops) be given; it is required to find a minimum-weight cover of the graph by k vertex-disjoint cycles. The problem is a simple generalization of the well-known Traveling Salesman Problem (TSP). We show that Min-k-SCCP is strongly NP-hard in the general case. Metric and Euclidean special cases of the problem are intractable as well. We also prove that the Metric Min-k-SCCP belongs to the APX class and has a 2-approximation polynomial-time algorithm. For the Euclidean Min-2-SCCP in the plane, we present a polynomial-time approximation scheme extending the famous result obtained by S. Arora for the Euclidean TSP. Actually, for any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution of the Euclidean Min-2-SCCP in O(n^3(log n)^O(c)) time

Interference Minimization in Asymmetric Sensor Networks

A fundamental problem in wireless sensor networks is to connect a given set of sensors while minimizing the \emph{receiver interference}. This is modeled as follows: each sensor node corresponds to a point in $\mathbb{R}^d$ and each \emph{transmission range} corresponds to a ball. The receiver interference of a sensor node is defined as the number of transmission ranges it lies in. Our goal is to choose transmission radii that minimize the maximum interference while maintaining a strongly connected asymmetric communication graph. For the two-dimensional case, we show that it is NP-complete to decide whether one can achieve a receiver interference of at most $5$. In the one-dimensional case, we prove that there are optimal solutions with nontrivial structural properties. These properties can be exploited to obtain an exact algorithm that runs in quasi-polynomial time. This generalizes a result by Tan et al. to the asymmetric case.Comment: 15 pages, 5 figure

An analysis of various elastic net algorithms

The Elastic Net Algorithm (ENA) for solving the Traveling Salesman Problem is analyzed applying statistical mechanics. Using some general properties of the free energy function of stochastic Hopfield Neural Networks, we argue why Simic's derivation of the ENA from a Hopfield network is incorrect. However, like the Hopfield-Lagrange method, the ENA may be considered a specific dynamic penalty method, where, in this case, the weights of the various penalty terms decrease during execution of the algorithm. This view on the ENA corresponds to the view resulting from the theory on `deformable templates', where the term stochastic penalty method seems to be most appropriate. Next, the ENA is analyzed both on the level of the energy function as well as on the level of the motion equations. It will be proven and shown experimentally, why a non-feasible solution is sometimes found. It can be caused either by a too rapid lowering of the temperature parameter (which is avoidable), or by a peculiar property of the algorithm,namely, that of adhering to equidistance of the elastic net points. Thereupon, an alternative, Non-equidistant Elastic Net Algorithm (NENA) is presented and analyzed. It has a correct distance measure and it is hoped to guarantee feasibility in a more natural way. For small problem instances, this conjecture is confirmed experimentally. However, trying larger problem instances, the pictures changes: our experimental results show that the elastic net points appear to become `lumpy' which may cause non-feasibility again. Moreover, in cases both algorithms yield a feasible solution, the quality of the solution found by the NENA is often slightly worse than the one found by the original ENA. This motivated us to try an Hybrid Elastic Net Algorithm (HENA), which starts using the ENA and, after having found a good approximate solution, switches to the NENA in order to guarantee feasibility too. In practice, the ENA and HENA perform more or less the same. Up till now, we did not find parameters of the HENA, which invariably guarantee the desired feasibility of solutions

A quasi-polynomial algorithm for well-spaced hyperbolic TSP

We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $-1$. Let $\alpha$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an $n^{O(\log^2 n)\max(1,1/\alpha)}$ algorithm for Hyperbolic TSP. This is quasi-polynomial time if $\alpha$ is at least some absolute constant, and it grows to $n^{O(\sqrt{n})}$ as $\alpha$ decreases to $\log^2 n/\sqrt{n}$. (For even smaller values of $\alpha$, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)Comment: SoCG 202

Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension

We study the Min-k-SCCP on the partition of a complete weighted digraph by k vertex-disjoint cycles of minimum total weight. This problem is the generalization of the well-known traveling salesman problem (TSP) and the special case of the classical vehicle routing problem (VRP). It is known that the problem Min-k-SCCP is strongly NP-hard and remains intractable even in the geometric statement. For the Euclidean Min-k-SCCP in Rd, we construct a polynomial-time approximation scheme, which generalizes the approach proposed earlier for the planar Min-2-SCCP. For any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution in time of O(nd+1(k log n)(O (√dc))d-1 2k). © 201