The Parameterized Approximability of TSP with Deadlines

Modern algorithm theory includes numerous techniques to attack hard problems, such as approximation algorithms on the one hand and parameterized complexity on the other hand. However, it is still uncommon to use these two techniques simultaneously, which is unfortunate, as there are natural problems that cannot be solved using either technique alone, but rather well if we combine them. The problem to be studied here is not only natural, but also practical: Consider TSP, generalized as follows. We impose deadlines on some of the vertices, effectively constraining them to be visited prior to a given point of time. The resulting problem DlTSP (a special case of the well-known TSP with time windows) inherits its hardness from classical TSP, which is both well known from practice and renowned to be one of the hardest problems with respect to approximability: Within polynomial time, not even a polynomial approximation ratio (let alone a constant one) can be achieved (unless P = NP). We will show that DlTSP is even harder than classical TSP in the following sense. Classical TSP, despite its hardness, admits good approximation algorithms if restricted to metric (or near-metric) inputs. Not so DlTSP (and hence, neither TSP with time windows): We will prove that even for metric inputs, no constant approximation ratio can ever be achieved (unless P = NP). This is where parameterization becomes crucial: By combining methods from the field of approximation algorithms with ideas from the theory of parameterized complexity, we apply the concept of parameterized approximation. Thereby, we obtain a 2.5-approximation algorithm for DlTSP with a running time of k! · poly(|G|), where k denotes the number of deadlines. Furthermore, we prove that there is no fpt-algorithm with an approximation guarantee of 2-ε for any ε > 0, unless P = NP. Finally, we show that, unlike TSP, DlTSP becomes much harder when relaxing the triangle inequality. More precisely, for an arbitrary small violation of the triangle inequality, DlTSP does not admit an fpt-algorithm with approximation guarantee ((1-ε)/2)|V| for any ε > 0, unless P = N

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks

Reusing optimal TSP solutions for locally modified input instances : Extended abstract

Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification- U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U. Here, we give non-trivial examples both of problems where this is and problems where this is not the case4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

On the Approximability of TSP on Local Modifications of Optimally Solved Instances

Given an instance of TSP together with an optimal solution, we consider the scenario in which this instance is modified locally, where a local modification consists in the alteration of the weight of a single edge. More generally, for a problem U, let LM-U (local-modification-U) denote the same problem as U, but in LM-U, we are also given an optimal solution to an instance from which the input instance can be derived by a local modification. The question is how to exploit this additional knowledge, i.e., how to devise better algorithms for LM-U than for U. Note that this need not be possible in all cases: The general problem of LM-TSP is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β-triangle inequality (LM-Δβ-TSP) remains NP-hard for all β>½. However, for LM-Δ-TSP (i.e., metric LM-TSP), we will present an efficient 1.4-approximation algorithm. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. Similarly, for all 1<β<3.34899, we achieve a better approximation ratio for LM-Δ-TSP than for Δβ-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs

Approximation algorithms for the traveling salesman problem

We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2-OPT, one of the most-known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2-OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 and traveling salesman with distances 1 and 2 within better than 741/742. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now