Multi-Criteria TSP: Min and Max Combined

Abstract

We present randomized approximation algorithms for multi-criteria 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, 3 + 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 obtain an approximation performance of (1/2, log 2 n + ε).

Deterministic Algorithms for Multi-criteria TSP

“…The following lemma says that, for any tour H, there is always a small set K of edges such that, if we contract these edges, the resulting tour H−K consists solely of edges that do not contribute too much to the weight of H−K with respect to any objective function. The proof is identical to the proof of the corresponding lemma for the (1/2 − ε) approximation for k-Max-ATSP [18,19]. In the algorithm, we will "guess" good sets K, compute Hamiltonian cycles on G −K , and add the edges of K to get a Hamiltonian cycle of G.…”

“…Table 1 shows the current approximation ratios for the different variants of multi-criteria TSP. Many of these approximation algorithms can be extended to the case where some objectives should be minimized and others should be maximized [19]. Unfortunately, no deterministic algorithms are known except for k-Min-STSP and 2-Max-STSP.…”

“…Small set means that |K| ≤ f (k, ε) for some function f that does not depend on the number n of vertices. We can choose [18,19] (we have log(1 + ε) = O(1/ε) by Taylor expansion). Moreover, we can choose K such that V −K contains an even number of vertices.…”

“…Unfortunately, it is impossible to contract edges in the same way as in directed graphs [18]. As already done for the randomized algorithms, we circumvent this by setting the weight along paths of sufficient length to 0 [18,19]. To do this formally, we need the following notation: Let H be a Hamiltonian cycle, and let K ⊆ H. Let L = L(K) = {v | ∃e ∈ K : v ∈ e} be the set of vertices that are adjacent to edges of K. Let T = T (K) = {e ∈ H | e is adjacent to K but not in K}.…”

Single Approximation for Biobjective Max TSP

“…We are interested in the existence and the computation in polynomial time of a single tour with simultaneous performance guarantees on the two objectives. Our work falls into a recent stream of research on the approximability of multiobjective optimization problems [22,21,19,10,5,12,3,1,6] where multiobjective TSP takes a prominent place [8,2,4,17,7,11,15,16].…”