Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

In multi-criteria optimization problems, several objective functions have to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as the optimal solution. Instead, the aim is to compute a so-called Pareto curve of solutions. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We design a deterministic polynomial-time algorithm for multi-criteria g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate Pareto curves for all 1/2<=g<=1. In particular, we obtain a (2+eps)-approximation for multi-criteria metric STSP. We also present two randomized approximation algorithms for multi-criteria g-metric STSP that achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1 +3g -4g^2) + eps, respectively. Moreover, we present randomized approximation algorithms for multi-criteria g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do this, we design randomized approximation schemes for multi-criteria cycle cover and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006

On Approximating Multi-Criteria TSP

We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP). First, we devise randomized approximation algorithms for multi-criteria maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP, where the edge weights have to be symmetric, we devise an algorithm with an approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps. Our algorithms work for any fixed number k of objectives. Furthermore, we present a deterministic algorithm for bi-criteria Max-STSP that achieves an approximation ratio of 7/27. Finally, we present a randomized approximation algorithm for the asymmetric multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full version, where some proofs are simplifie

Minimum-weight Cycle Covers and Their Approximability

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2-eps for certain sets L. For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where $n$ is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n). To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic Concepts in Computer Science (WG 2007). Minor change

Deterministic algorithms for multi-criteria TSP

We present deterministic approximation algorithms for the multi-criteria traveling salesman problem (TSP). Our algorithms are faster and simpler than the existing randomized algorithms.\ud First, we devise algorithms for the symmetric and asymmetric multi-criteria Max-TSP that achieve ratios of 1/2k − ε and 1/(4k − 2) − ε, respectively, where k is the number of objective functions. For two objective functions, we obtain ratios of 3/8 − ε and 1/4 − ε for the symmetric and asymmetric TSP, respectively. Our algorithms are self-contained and do not use existing approximation schemes as black boxes.\ud Second, we adapt the generic cycle cover algorithm for Min-TSP. It achieves ratios of 3/2 + ε, $\frac{1}{2} + \frac{\gamma^3}{1-3\gamma^2} +\varepsilon$, and $\frac{1}{2} + \frac{\gamma^2}{1-\gamma} +\varepsilon$ for multi-criteria Min-ATSP with distances 1 and 2, Min-ATSP with $\gamma$-triangle inequality and Min-STSP with $\gamma$-triangle inequality, respectively

A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time

We give the first constant-factor approximation for the Directed Latency problem in quasi-polynomial time. Here, the goal is to visit all nodes in an asymmetric metric with a single vehicle starting at a depot $r$ to minimize the average time a node waits to be visited by the vehicle. The approximation guarantee is an improvement over the polynomial-time $O(\log n)$-approximation [Friggstad, Salavatipour, Svitkina, 2013] and no better quasi-polynomial time approximation algorithm was known. To obtain this, we must extend a recent result showing the integrality gap of the Asymmetric TSP-Path LP relaxation is bounded by a constant [K\"{o}hne, Traub, and Vygen, 2019], which itself builds on the breakthrough result that the integrality gap for standard Asymmetric TSP is also a constant [Svensson, Tarnawsi, and Vegh, 2018]. We show the standard Asymmetric TSP-Path integrality gap is bounded by a constant even if the cut requirements of the LP relaxation are relaxed from $x(\delta^{in}(S)) \geq 1$ to $x(\delta^{in}(S)) \geq \rho$ for some constant $1/2 < \rho \leq 1$. We also give a better approximation guarantee in the special case of Directed Latency in regret metrics where the goal is to find a path $P$ minimize the average time a node $v$ waits in excess of $c_{rv}$, i.e. $\frac{1}{|V|} \cdot \sum_{v \in V} (c_v(P)-c_{rv})$

On Approximating Restricted Cycle Covers

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change