Traveling Salesman Path Research Articles

An LP-based approximation algorithm for the generalized traveling salesman path problem

The traveling salesman problem (TSP) is one of the classic research topics in the field of operations research, graph theory and computer science. In this paper, we propose a generalized model of traveling salesman problem, denoted by generalized traveling salesman path problem. Let G=(V,E,c) be a weighted complete graph, in which c is a nonnegative metric cost function on edge set E, i.e., c:E→R+. The traveling salesman path problem aims to find a Hamiltonian path in G with minimum cost. Compared to the traveling salesman path problem, we are given extra vertex subset V′ and edge subset E′ in the problem proposed in this paper; its goal is to construct a path which traverses all the edges in E′ while only needs to visit each vertex in V′ exactly once. Based on integer programming, we give a mathematical model of the problem, and design a 1+52-approximation algorithm for the problem by combining linear programming rounding strategy and a special graph structure.

The asymmetric traveling salesman path LP has constant integrality ratio

We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If $\rho_{\text{ATSP}}$ and $\rho_{\text{ATSPP}}$ denote the integrality ratios for the asymmetric TSP and its path version, then $\rho_{\text{ATSPP}}\le 4\rho_{\text{ATSP}}-3$. We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is $\rho_{\text{ATSP}}^{\text{NW}}$, then the integrality ratio for ATSPP on node-weighted instances is at most $2\rho_{\text{ATSP}}^{\text NW}-1$. Moreover, we show that for ATSP node-weighted instances and unweighted digraph instances are almost equivalent. From this we deduce a lower bound of 2 on the integrality ratio of unweighted digraph instances.

Minimum Bend Shortest Rectilinear Route Discovery for a Moving Sink in a Grid Based Wireless Sensor Network

In a rectilinear route, a moving sink is restricted to travel either horizontally or vertically along the connecting edges. We present a new algorithm that finds the shortest round trip rectilinear route covering the specified nodes in a grid based Wireless Sensor Network. The proposed algorithm determines the shortest round trip travelling salesman path in a two-dimensional grid graph. A special additional feature of the new path discovery technique is that it selects that path which has the least number of corners (bends) when more than one equal length shortest round trip paths are available. This feature makes the path more suitable for moving objects like Robots, drones and other types of vehicles which carry the moving sink. In the prosed scheme, the grid points are the vertices of the graph and the lines joining the grid points are the edges of the graph. The optimal edge set that forms the target path is determined using the binary integer programming.

An Energy-Efficient Multiobjective Scheduling Model for Monitoring in Internet of Things

To ensure robustness in wireless networks, monitoring the network state, performance and functioning of the nodes and links is crucial, especially for critical applications. This paper targets Internet of Things (IoT) networks. In the IoT, devices (things) are vulnerable due to security risks from the Internet. Moreover, they are resource-constrained and connected via lossy links. This paper addresses the optimized scheduling of the monitoring role between the embedded devices in IoT networks. The objective is to minimize energy consumption and communication overhead of monitoring, for each node. Several subsets of the potential monitoring nodes are generated by solving a minimal vertex cover (VC) problem with constraint generation. Assuming periodical functioning, VCs are optimally assigned to time periods in order to distribute the monitoring role throughout the entire network. The assignment of VCs to periods is modeled as a multiobjective generalized assignment problem. To further optimize the energy consumption of the monitors, they are sequenced across time periods to minimize the state transitions of nodes. This part of the problem is modeled as a traveling salesman path problem. The proposed model is tested on randomly generated instances and the experimental results illustrate its effectiveness to optimize the scheduled monitoring for fault tolerance in IoT networks.

Bounds for the traveling salesman paths of two-dimensional modular lattices

We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi---Perron type algorithms.

An upper bound for the length of a traveling salesman path in the Heisenberg group

We show that a sufficient condition for a subset E in the Heisenberg group (endowed with the Carnot–Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones’s geometric lemma. Our estimates improve on those of [6], allowing for any power r < 4 to replace the power 2 of the Jones- \beta -number. This complements in an open ended way our work in [13], where we showed that such an estimate was necessary, however with the power r = 4.

New Bounds for the Traveling Salesman Constant

Let X1, X2, …, Xn be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X1, …, Xn) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that limn→∞n−1/2L(X1, …, Xn) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve both upper and lower bounds.

New Bounds for the Traveling Salesman Constant

Let X 1, X 2, …, X n be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X 1, …, X n ) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that lim n→∞ n −1/2 L(X 1, …, X n ) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve both upper and lower bounds.

Reconstruction of curves from point clouds using fuzzy logic and ant colony optimization

A new approach based on fuzzy logic and ant colony optimization is presented for the reconstruction of curves from a set of unorganized points. Fuzzy clustering is used to reduce the number of points to cluster centres. Ant colony optimization is used to construct a travelling salesman path which is a closed curve. Extra edges are deleted and new edges are added using the fuzzy membership function. The algorithm presented in this paper has been used for reconstructing open as well as closed curves. The results obtained for multiple and self-intersecting curves are also good. Various examples for open, closed, multiple and intersecting curves with complicated shapes are shown to illustrate the significance of the presented algorithm.

An Efficient Heuristic for Peak Capture Power Minimization During Scan-Based Test

IR-Drop induced timing failures during testing can be avoided by minimizing the peak capturepower. This paper models the Capture-Power minimization problem as an instance of the Bottleneck Traveling Salesman Path Problem (BTSPP). The solution for the BTSPP implies an ordering on the input test vectors, which when followed during testing minimizes the Peak Capture-Power. The paper also presents a methodology for estimating a lower bound on the peak capture-power. Applying the proposed technique on ITC'99 benchmarks yielded optimal (equal to the estimated lower bound) results for all circuits. Interestingly, the technique also significantly reduced the average power consumed during testing when compared with commercial state-of-the-art tools

Asymmetric Traveling Salesman Path and Directed Latency Problems

We study integrality gaps and approximability of three closely related problems on directed graphs with edge lengths that satisfy the triangle inequality. Given two specified vertices $s$ and $t$, two of these problems ask to find an $s$-$t$ path in the graph visiting all other vertices. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total length of this path. In the directed latency problem, the objective is to minimize the sum of the latencies of the vertices, where the latency of a vertex $v$ is the distance from $s$ to $v$ along the path. The third problem that we study is the $k$-person ATSPP, in which the goal is to find $k$ paths from $s$ to $t$, of minimum total length, such that every vertex is on at least one of these paths. All of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio $O(\log n)$ [C. Chekuri and M. Pal, Theory Comput., 3 (2007), pp. 197--209], [U. Feige and M. Singh, “Improved approximation ratios for traveling salesperson tours and paths in directed graphs,” in Proceedings of the 10th APPROX, 2007, pp. 107--118] until the recent result that improves it to $O(\log n/\log \log n)$ [A. Asadpour et al., “An $O(\log n/\log \log n)$-approximation algorithm for the asymmetric traveling salesman problem,” in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, 2010, pp. 379--389], [U. Feige and M. Singh, “Improved approximation ratios for traveling salesperson tours and paths in directed graphs,” in Proceedings of the 10th APPROX, 2007, pp. 107--118]. However, the best known bound on the integrality gap of any linear programming relaxation for ATSPP is only $O(\sqrt{n})$. For directed latency, the best previously known approximation algorithm has a guarantee of $O(n^{1/2+\epsilon})$ for any constant $\epsilon>0$ [V. Nagarajan and R. Ravi, “The directed minimum latency problem,” in Proceedings of the 11th APPROX, 2008, pp. 193--206]. We present a new algorithm for the ATSPP problem that has an approximation ratio of $O(\log n)$, but whose analysis also upper bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed in [C. Chekuri and M. Pal, Theory Comput., 3 (2007), pp. 197--209]. We then pursue a deeper study of this linear program and its variations, which leads to an $O(\log n)$-approximation for the directed latency problem, a significant improvement over previously known results. Our result for $k$-person ATSPP is an $O(k^2 \log n)$-approximation that bounds the integrality gap of a linear programming relaxation by the same factor. We are not aware of any previous work on this problem.

Formula: see text]-Approximation for Graphic TSP.

The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of frac{3}{2}, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only frac{4}{3}. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of frac{13}{9} on the approximation factor, as well as a bound of frac{19}{12}+varepsilon for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

Approximation Algorithms for the Directed k-Tour and k-Stroll Problems

We consider two natural generalizations of the Asymmetric Traveling Salesman problem: the k-Stroll and the k-Tour problems. The input to the k-Stroll problem is a directed n-vertex graph with nonnegative edge lengths, an integer k, as well as two special vertices s and t. The goal is to find a minimum-length s-t walk, containing at least k distinct vertices (including the endpoints s,t). The k-Tour problem can be viewed as a special case of k-Stroll, where s=t. That is, the walk is required to be a tour, containing some pre-specified vertex s. When k=n, the k-Stroll problem becomes equivalent to Asymmetric Traveling Salesman Path, and k-Tour to Asymmetric Traveling Salesman.Our main result is a polylogarithmic approximation algorithm for the k-Stroll problem. Prior to our work, only bicriteria (O(log2 k),3)-approximation algorithms have been known, producing walks whose length is bounded by 3OPT, while the number of vertices visited is Ω(k/log2 k). We also show a simple O(log2 n/loglogn)-approximation algorithm for the k-Tour problem. The best previously known approximation algorithms achieved min(O(log3 k),O(log2 n⋅logk/loglogn)) approximation in polynomial time, and O(log2 k) approximation in quasipolynomial time.

Routing strategies for integrating forward distribution and reverse collection

Due to the significant growth in the demand for logistics services and the ongoing trend towards the outsourcing of logistics activities, logistics service providers have concurrently gained a significant profit-making opportunity and faced the unprecedented challenge of surviving in the midst of increasingly fierce market competition. This research thus aims to propose routing strategies that optimally integrate forward distribution and reverse collection to help logistics service providers reduce their operating costs when providing transportation services, and thus improve their market competitive power. The integrated routing problem is formulated as the mixed pickup-delivery asymmetric traveling salesman path problem. The problem is NP-hard and new in the literature. Therefore, the objectives of this research are to first model the problem and then develop an efficient and effective heuristic solution algorithm to the problem.

On the LP Relaxation of the Asymmetric Traveling Salesman Path Problem

This is a comment on the article An O(logn) Approximation Ratio for the Asymmetric Traveling Salesman Path by C. Chekuri and M. PTheory of Com- puting 3(2007), 197-209. We observe that the LP relaxation for the Asymmetric Traveling Salesman Path Problem suggested in Section 5 of that paper is not accurate, and state a corrected linear relaxation for the problem. The inaccuracy occurs in the statement of an open problem and does not affect the validity of any of the results in the Chekuri-Ppaper.

What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs

Given a graph and degree upper bounds on vertices, the BDMST problem requires us to find a minimum cost spanning tree respecting the given degree bounds. This problem generalizes the Travelling Salesman Path Problem (TSPP), even in unweighted graphs, and so we expect that it is necessary to relax the degree constraints to get efficient algorithms. Konemann and Ravi (Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing, pp. 537–546, 2000; Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, pp. 389–395, 2003) give bicriteria approximation algorithms for the problem using local search techniques of Fischer (Technical Report 14853, Cornell University, 1993). Their algorithms find solutions which make a tradeoff of the approximation factor for the cost of the resulting tree against the factor by which degree constraints are violated. In particular, they give an algorithm which, for a graph with a spanning tree of cost C and degree B, and for parameters b,w>1, produces a tree whose cost is at most wC and whose degree is at most $\frac{w}{w-1}bB+\log_{b}n.$ A primary contribution of Konemann and Ravi is to use a Lagrangean relaxation to formally relate the BDMST problem to what we call the MDMST problem, which is the problem of finding an MST of minimum degree in a graph. In their solution to the MDMST problem, they make central use of a local-search approximation algorithm of Fischer. In this paper, we give the first approximation algorithms for the BDMST problem—both our algorithms find trees of optimal cost. We achieve this improvement using a primal-dual cost bounding methodology from Edmonds’ weighted matching algorithms which was not previously used in this context. In order to follow Edmonds’ approach, we develop algorithms for a variant of the MDMST problem in which there are degree lower bound requirements. This variant may be of independent interest; in particular, our results extend to a generalized version of the BDMST problem in which both upper and lower degree bounds are given. First we give a polynomial-time algorithm that finds a tree of optimal cost and with maximum degree at most $\frac{b}{2-b}B+O(\log_{b}n)$for any b∈(1,2). We also give a quasi-polynomial-time approximation algorithm which produces a tree of optimal cost C and maximum degree at most B+O(log n/log log n). That is, the error is additive as well as restricted just to the degree. This further improvement in degree is obtained by using augmenting-path techniques that search over a larger solution space than Fischer’s local-search algorithm.

An O(log n ) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem

Given an arc-weighted directed graph G = (V,A,l) and a pair of vertices s,t, we seek to find an s-twalk of minimum length that visits all the vertices in V. If l satisfies the asymmetric triangle inequality, the problem is equivalent to that of finding an s-tpath of minimum length that visits all the vertices. We refer to this problem as ATSPP. When s = t this is the well known asymmetric traveling salesman tour problem (ATSP). Although an O(logn) approximation ratio has long been known for ATSP, the best known ratio for ATSPP is $O(\sqrt{n})$. In this paper we present a polynomial time algorithm for ATSPP that has approximation ratio of O(logn). The algorithm generalizes to the problem of finding a minimum length path or cycle that is required to visit a subset of vertices in a given order.

Traveling salesman path problems

In the traveling salesman path problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. In this paper, we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem, in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively. We also address the asymmetric traveling salesman path problem (ATSPP) and give a factor $$O(\sqrt{n})$$-approximation algorithm for this problem.

A [formula omitted]-approximation algorithm for the clustered traveling salesman tour and path problems

We consider the ordered cluster traveling salesman problem (OCTSP). In this problem, a vehicle starting and ending at a given depot must visit a set of n points. The points are partitioned into K, K⩽n , prespecified clusters. The vehicle must first visit the points in cluster 1, then the points in cluster 2,… , and finally the points in cluster K so that the distance traveled is minimized. We present a 5 3 -approximation algorithm for this problem which runs in O( n 3) time. We show that our algorithm can also be applied to the path version of the OCTSP: the ordered cluster traveling salesman path problem (OCTSPP). Here the (different) starting and ending points of the vehicle may or may not be prespecified. For this problem, our algorithm is also a 5 3 -approximation algorithm.

A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2−min{(d(v)−2)/(degT(v)−2):degT(v)>2}, where degT(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousLpnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.