Multi-Criteria TSP: Min and Max Combined

We present randomized approximation algorithms for multicriteria traveling salesman problems (TSP), where some objective functions should be minimized while others should be maximized. For the symmetric multi-criteria TSP (STSP), we present an algorithm that computes (2/3−ε, 4+ε) approximate Pareto curves. Here, the first parameter is the approximation ratio for the objectives that should be maximized, and the second parameter is the ratio for the objectives that should be minimized. For the asymmetric multi-criteria TSP (ATSP), we present an algorithm that computes (1/2 − ε, log2 n + ε) approximate Pareto curves. In order to obtain these results, we simplify the existing approximation algorithms for multi-criteria Max-STSP and Max-ATSP. Finally, we give algorithms with improved ratios for some special cases. 1 Multi-Criteria TSP 1.1 Traveling Salesman Problems The traveling salesman problem (TSP) is a basic problem in combinatorial optimization. An instance of Max-TSP is a complete graph G = (V,E) with edge weights w : E → Q+. The goal is to find a Hamiltonian cycle (also called a tour) of maximum weight, where the weight of a Hamiltonian cycle is the sum of its edge weights. (The weight of an arbitrary set of edges is analogously defined.) If G is undirected, then we speak of Max-STSP (symmetric TSP). If G is directed, we have Max-ATSP (asymmetric TSP). Min-TSP is similarly defined, but now the edge weights d : E → Q+ are required to fulfil the triangle inequality: d(u, v) ≤ d(u, x) + d(x, v) for all u, v, x ∈ V (without the triangle inequality, approximating the problem is impossible). The aim is to find a Hamiltonian cycle of minimum weight. Min-STSP is the symmetric variant, where G is undirected, while Min-ATSP is the asymmetric variant. All four variants of TSP are NP-hard and APX-hard. Thus, we are in need of approximation algorithms. Christofides’ algorithm [14] achieves a ratio of 3/2 for Min-STSP. Min-ATSP can be approximated with a factor of 23 · log2 n, where n is the number of vertices of the instance [8]. The currently best approximation algorithm for Max-STSP achieves an approximation ratio of 7/9 [12], and the currently best algorithm for Max-ATSP achieves a ratio of 2/3 [9]. Cycle covers are one of the main tools for designing approximation algorithms for the TSP [3, 8, 9, 12]. A cycle cover of a graph is a set of vertex-disjoint c © Springer – 7th Workshop on Approximation and Online Alg. (WAOA 2009) cycles such that every vertex is part of exactly one cycle. Hamiltonian cycles are special cases of cycle covers that consist just of a single cycle. The general idea is to compute an initial cycle cover, and then we join the cycles to obtain a Hamiltonian cycle. 1.2 Multi-Criteria Optimization In many optimization problems, there is more than one objective function. This is also the case for the TSP: We might want to minimize travel time, expenses, number of flight changes, etc., while a taxi driver might want to maximize his profit, or we want to maximize, for instance, our profit along the way. This gives rise to multi-criteria TSP, where Hamiltonian cycles are sought that optimize several objectives simultaneously. However, as far as we are aware, multi-criteria TSP has only been considered in a restricted setting, where either all objectives should be minimized or all objectives should be maximized. In this paper, we consider the general setting with both types of objectives at the same time. If k objectives are to be maximized and ` objectives are to be minimized, then we have k-Max-`-Min-ATSP and k-Max-`-Min-STSP. If the number of criteria does not matter, we will also speak of MC-ATSP and MC-STSP. If ` = 0 or k = 0, then we obtain the special cases k-Max-ATSP and k-Max-STSP as well as `-Min-ATSP and `-Min-STSP. Analogously, if the number of criteria is unimportant, we have MC-Max-ATSP and so on. With respect to a single objective function, the notion of an optimal solution is well-defined. But if we have more than one objective function, there is no natural notion of a best choice. Instead, we have to content ourselves with trade-off solutions. The goal of multi-criteria optimization is to deal with this dilemma. In order to transfer the notion of an optimal solutions to multi-criteria optimization problems, Pareto curves (also known as Pareto sets or efficient sets) were introduced introduced (cf. Ehrgott [6]). A Pareto curve is a set of solutions that are potential optimal choices. An instance of k-Max-`-Min-ATSP is a directed complete graph G = (V,E) with edge weights w1, . . . , wk : E → Q+ and d1, . . . , d` : E → Q+. The functions w1, . . . , wk should be maximized while d1, . . . , d` should be minimized. We call w1, . . . , wk the max objectives and d1, . . . , d` the min objectives. For convenience, let w = (w1, . . . , wk) and d = (d1, . . . , d`). Inequalities of vectors are meant component-wise. A Hamiltonian cycle H dominates another Hamiltonian cycle H ′ if w(H) ≥ w(H ′) and d(H) ≤ d(H ′) and at least one of these inequalities is strict. This means that H is strictly preferable to H ′. A Pareto curve of solutions contains all solutions that are not dominated by another solution. For other optimization problems, multi-criteria variants are defined analogously. Unfortunately, Pareto curves cannot be computed efficiently in many cases: First, they are often of exponential size. Second, they are NP-hard to compute even for otherwise easy optimization problems. Third, TSP is NP-hard already with a single objective function, and optimization problems do not become easier with more objectives involved. Therefore, we have to be satisfied with approximate Pareto curves. A set P of Hamiltonian cycles is called an (α, β) approximate Pareto curve for the instance (G,w, d) if the following holds: For every Hamiltonian cycle H ′, there exists an H ∈ P with w(H) ≥ αw(H ′) and d(H) ≤ βd(H ′). We have α ≤ 1, β ≥ 1, and a (1, 1) approximate Pareto curve is a Pareto curve. An algorithm is called an (α, β) approximation algorithm if, given G and w, it computes an (α, β) approximate Pareto curve. It is called a randomized (α, β) approximation if its success probability is at least 1/2. This success probability can be amplified to 1 − 2−m by executing the algorithm m times and taking the union of all sets of solutions. A fully polynomial time approximation scheme (FPTAS) for a multi-criteria optimization problem computes (1 − ε, 1 + ε) approximate Pareto curves in time polynomial in the size of the instance and 1/ε for all ε > 0. Multi-criteria matching admits a randomized FPTAS [13], i. e., the algorithm succeeds in computing a (1− ε, 1 + ε) approximate Pareto curve with a probability of at least 1/2. This randomized FPTAS yields also a randomized FPTAS for the multi-criteria cycle cover problem [11].