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,$3+\varepsilon$)-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, $\log_2 n + \varepsilon$)

Manthey, Bodo

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Multi-criteria TSP: Min and max combined

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Operations Research Letters 40 (2012) 36–38Contents lists available at SciVerse ScienceDirectOperations Research Lettersjournal homepage: www.elsevier.com/locate/orlMulti-criteria TSP: Min and max combinedBodo MantheyUniversity of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The Netherlandsa r t i c l e i n f oArticle history:Received 13 July 2011Accepted 11 October 2011Available online 26 October 2011Keywords:Multi-criteria optimizationPareto optimizationApproximation algorithmsTraveling salesman problema b s t r a c tWe present randomized approximation algorithms for multi-criteria traveling salesman problems (TSP),where some objective functions should be minimized while others should be maximized. For thesymmetric multi-criteria TSP (STSP), we present an algorithm that computes (2/3, 3 + ε)-approximatePareto curves. Here, the first parameter is the approximation ratio for the objectives that should bemaximized, and the second parameter is the ratio for the objectives that should be minimized. For theasymmetric multi-criteria TSP (ATSP), we obtain an approximation performance of (1/2, log2 n+ ε).© 2011 Elsevier B.V. All rights reserved.1. Multi-criteria TSPThe traveling salesman problem (TSP) is a basic optimizationproblem. The goal is to find Hamiltonian cycles of maximum orminimum weight. An instance of Max-TSP is a complete graphG = (V , E) with edge weights w : E → Q+. The goal is to find aHamiltonian cycle (also called a tour) of maximum weight, wherethe 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 issimilarly defined, but now the edge weights d : E → Q+ arerequired to fulfill the triangle inequality: d(u, v) ≤ d(u, x)+d(x, v)for all u, v, x ∈ V . (Without the triangle inequality, Min-TSP doesnot allow for any approximation [2], and we are concerned withapproximation algorithms.) The aim is to find a Hamiltonian cycleof minimum weight. Min-STSP is the symmetric variant, where Gis undirected, whileMin-ATSP is the asymmetric variant.However, we often have more than one objective function: wemight want to minimize travel time, travel expenses, etc., whilemaximizing, e.g., our profit along the way. This gives rise to multi-criteria TSP. So far, multi-criteria TSP has only been considered inthe setting where either all objectives should be minimized or allobjectives should be maximized. In this paper, we consider thegeneral setting with both types of objectives simultaneously.If k objectives are to be maximized and ℓ objectives are to beminimized, then we have k-Max-ℓ-Min-ATSP and k-Max-ℓ-Min-STSP. For ℓ = 0 or k = 0, we have k-Max-ATSP and k-Max-STSPas well as ℓ-Min-ATSP and ℓ-Min-STSP, respectively.Unfortunately, if we have more than one objective function,there is no natural notion of an optimal solution. Instead, we haveE-mail address: b.manthey@utwente.nl.0167-6377/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2011.10.012to content ourselveswith trade-off solutions. Therefore, the notionof Pareto curves has been introduced (cf. [5]): a Pareto curve is aset of solutions that are optimal trade-offs between the differentobjective functions.Let us describe thismore formally. An instance of k-Max-ℓ-Min-ATSP (or k-Max-ℓ-Min-STSP) is a directed (undirected) completegraph G = (V , E) with edge weights w1, . . . , wk : E → Q+and d1, . . . , dℓ : E → Q+, where each di satisfies the triangleinequality. The functions w1, . . . , wk should be maximized whiled1, . . . , dℓ should be minimized. We call w1, . . . , wk the maxobjectives and d1, . . . , dℓ the min objectives. For convenience, letw = (w1, . . . , wk) and d = (d1, . . . , dℓ). Inequalities of vectors aremeant component-wise. A Hamiltonian cycle H dominates anotherHamiltonian cycle H ′ if w(H) ≥ w(H ′) and d(H) ≤ d(H ′) and atleast one of these inequalities is strict. This means that H is strictlypreferable to H ′. A Pareto curve contains all solutions that are notdominated by another solution. For other optimization problems,this is defined analogously. Unfortunately, since Pareto curves forthe TSP cannot be computed efficiently (because, first, they canbe of exponential size and, second, we inherit the approximationhardness of the TSP with a single objective function), we have tobe satisfied with approximate Pareto curves.A set P of Hamiltonian cycles is called an (αmax, αmin)-approximate Pareto curve for the instance (G, w, d) if the followingholds: for every Hamiltonian cycle H̃ of G, there exists a Hamil-tonian cycle H ∈ P of G with w(H) ≥ αmaxw(H̃) and d(H) ≤αmind(H̃). We have αmax ≤ 1, αmin ≥ 1, and a (1, 1)-approximatePareto curve is a Pareto curve.An algorithm is called an (αmax, αmin)-approximation algorithmif, given G, w, and d, it computes an (αmax, αmin)-approximatePareto curve. It is called a randomized (αmax, αmin)-approximationif its success probability is at least 1/2. This success probability canbe amplified by executing the algorithm several times.B. Manthey / Operations Research Letters 40 (2012) 36–38 37Table 1Known approximation ratios for TSP and the results of this paper.Variant Single-criterion Multi-criteria CombinedMax-STSP 7/9 [13] 2/3 [7](2/3, 3+ ε)Min-STSP 3/2 [2] 2+ ε [12]Max-ATSP 2/3 [8] 1/2 [7](1/2, log2 n+ε)Min-ATSP Olog nlog log n[1] log2 n+ ε [10]A fully polynomial time approximation scheme (FPTAS) for amulti-criteria optimization problem computes (1 − ε, 1 + ε)-approximate Pareto curves in time polynomial in the size of theinstance and 1/ε for all ε > 0. Multi-criteria matching (computinga perfect of minimum or maximum weight or computing amaximum-weight matching that is not necessarily perfect) admitsa randomized FPTAS [14], i. e., the algorithm succeeds in computinga (1− ε, 1+ ε)-approximate Pareto curve with a probability of atleast 1/2. A cycle cover of a graph is a set of vertex-disjoint cyclessuch that every vertex is part of exactly one cycle. The randomizedFPTAS for matching yields also a randomized FPTAS for the multi-criteria cycle cover problem [12]. A path cover of a graph is a set ofvertex-disjoint paths such that every vertex is part of exactly onepath.Known results. ℓ-Min-STSP admits a (2 + ε)-approximationalgorithm [12]. For the special case of ℓ = 2, there even is a 2approximation [6]. ℓ-Min-ATSP can be approximated with a factorof log2 n + ε [9]. k-Max-ATSP and k-Max-STSP have first beenstudied by Bläser et al. [4], who have achieved approximationratios of 1k+1−ε and1k−ε, respectively. This has subsequently beenimproved to 1/2 and 2/3, respectively [7]. Table 1 summarizesthe known results and compares them to the approximation ratiosobtained for single-criterion TSP.As far as we are aware, nothing is known yet about theapproximability of multi-criteria TSP with both min and maxobjectives. In fact, with only a few exceptions [3,14], little is knownabout the approximability of any multi-criteria optimizationproblem with both min and max objectives.New results. We present a generic algorithm for computingapproximate Pareto curves for k-Max-ℓ-Min-ATSP and k-Max-ℓ-Min-STSP. Our algorithm achieves approximation ratios of(1/2, log2 n+ ε) and (2/3, 3+ ε), respectively. (See also Table 1.)The running-time of our algorithms is polynomial for any fixedℓ, k, and ε. Note that an exponential dependence on k and ℓ isunavoidable, as the size of even approximate Pareto curves canincrease exponentially with the number of objective functions.The main idea of our algorithms is to first find a collection ofpaths that have sufficient weight for the max objectives. This isbasically done using a (slightlymodified) approximation algorithmfor multi-criteria Max-TSP, while we have to take into accountthat these paths are not too heavy with respect to d. Afterthat, we connect these paths to get Hamiltonian cycles using anapproximation algorithm formulti-criteriaMin-TSP. In this secondstep, we only pay attention to the min objectives; we already haveenough weight for the max objectives, and adding further edgesdoes not decrease the weight. (A preliminary version of this paperhas been presented at the 7th Int. Workshop on ApproximationandOnlineAlgorithms [11]. The present paper improves the resultspresented there and simplifies the proofs.)2. Approximation algorithm for min and maxNowwe present and analyze our framework for approximationalgorithms for multi-criteria TSP that will yield the resultsmentioned above. The framework will be for both STSP and ATSP.For our framework to work, we need the concept of a canonicalalgorithm. An algorithm is called an (α, ε)-canonical algorithm ifPTSP ← TSP-Approx(G, w, d, ε)input: (un)directed complete graph G = (V , E), w : E → Qk+,d : E → Qℓ+, ε > 0output: (α, rn/2 + 1 + ε)-approximate Pareto curve withprobability at least 1/21: Ppaths ← GRW-Approx(G, w, d, ε/2)2: for all P ∈ Ppaths do3: let V ′ be the start-points of paths of P4: PTSP′← MinTSP-Approx(V ′, d, ε/2)5: for all H ′ ∈ PTSP′ do6: construct a tour H from P ∪ H ′ as in the proof ofTheorem 17: add H to PTSPAlgorithm1:Approximation algorithm for k-Max-ℓ-Min-ATSP andk-Max-ℓ-Min-STSP.it has the following property: on input (G = (V , E), d, w), itcomputes a set of path covers such that for each Hamiltonian cycleH̃ of G it outputs a path cover P of Gwith the following properties.First, by adding an edge from the end-point of each path in P to itscorresponding start-point, one obtains a cycle cover C ⊇ P suchthat d(C) ≤ (1 + ε) · d(H̃). Second, w(P) ≥ αw(H̃). Third, (V , P)consist of at most n/2 paths.Let us now describe how the algorithm of Glaßer et al. [7] fork-Max-STSP and k-Max-ATSP can be turned into an (2/3, ε) and(1/2, ε)-canonical algorithm, respectively, for any ε > 0 for thesymmetric and asymmetric variant. First, instead of computing1− 1n-approximate Pareto curves of cycle covers with respectto w (Glaßer et al. compute edge-fixed cycle covers, but this doesnot make a difference: we can fix some edges by removing themand the corresponding vertices from the graph), we compute1− 1n , 1+ ε-approximate Pareto curves of cycle covers withrespect to both w and d (using Papadimitriou and Yannakakis’randomized FPTAS [14]). We proceed with the original algorithm,ignoring d. From every such cycle cover C , the algorithm constructsa path cover P by removing exactly one edge from each cycle. IfC is a cycle cover with w(C) ≥1− 1n· w(H̃) for some tour H̃and C contains the right collection of fixed edges, then w(P) ≥23 · w(H̃) (in the symmetric case) or w(P) ≥12 · w(H̃) (in theasymmetric case). (In order to find such a cycle cover C , we haveto fix the heavy edges of the cycle cover in advance. Since thereare always only a few heavy edges, their algorithm iterates overall possibilities while maintaining polynomial running-time. Thus,we are guaranteed to find the right collection.) Furthermore, aswe compute1− 1n , 1+ ε-approximate Pareto curves of cyclecovers, we can achieve d(C) ≤ (1 + ε) · d(H̃). The graph (V , P)consists of at most n/2 paths. Finally, we remove the last line ofthe algorithm [7], where the paths are connected to a Hamiltoniancycle in an arbitrary way and simply output the path cover. Wedenote the canonical algorithm obtained from the algorithm byGlaßer et al. by GRW-Approx.Theorem 1. Assume that there is an (α, ε)-canonical randomizedpolynomial-time algorithm for k-Max-ℓ-Min-ATSP or k-Max-ℓ-Min-STSP for any ε > 0. Assume further that ℓ-Min-ATSP or ℓ-Min-STSP can be approximated with an approximation ratio of rn (for nvertices) by a randomized polynomial-time algorithm.Then TSP-Approx (Algorithm 1) is a randomized polynomial-time(α, rn/2+1+ ε)-approximation algorithm for k-Max-ℓ-Min-ATSP ork-Max-ℓ-Min-STSP.Proof. Let us analyze the approximation ratio first. To do this,we assume that all randomized computations are successful. Afterthat, we analyze success probability and running-time. Let H̃ be anarbitrary Hamiltonian cycle. Let GRW-Approx be (α, ε)-canonical.38 B. Manthey / Operations Research Letters 40 (2012) 36–38We have to show that there exists a Hamiltonian cycle H ∈ PTSPwith w(H) ≥ α · w(H̃) and d(H) ≤ (rn/2 + 1+ ε) · d(H̃).The set Ppaths contains a path cover P ⊆ E with d(P) ≤ (1+ ε) ·d(H̃) and w(P) ≥ αw(H̃). Even stronger, by assumption, there is acycle cover C ⊇ P with d(C) ≤ (1+ ε) · d(H̃).By construction, all paths of the path cover P consist of at leastone edge, i.e., (V , P) does not contain isolated vertices. Thus, wehave |V ′| ≤ |V |/2. This implies that PTSP′ (in the correspondingiteration) contains a tour H ′ with d(H ′) ≤ rn/2d(H̃). We obtainthe Hamiltonian cycle H from P and H ′ as follows: assume thatP contains a path from u to v and H ′ contains an edge from u tox. Then we add the path from u to v plus the edge (v, x) to H .We do this for all paths of P . The triangle inequality guaranteesd(v, x) ≤ d(v, u)+ d(u, x). Note that the edge (v, u) is part of thecorresponding cycle (that contains u and v) that is missing in P butis contained in C . This together yieldsd(H) ≤ d(H ′)+ d(C) ≤ (rn/2 + 1+ ε) · d(H̃).Furthermore, by construction, we have H ⊇ P . Thus, we havew(H) ≥ w(P) ≥ αw(H̃).The error probabilities of the randomized computations inlines 1 and 4 can be chosen small enough such that they sumup to at most 1/2. This yields that the probability that one ofthe computations fails is at most 1/2. The running-time followsbecause the running-times of MinATSP-Approx and GRW-Approxare polynomial if k and ℓ are fixed. Immediate consequences of the theorem above are the follow-ing two corollaries.Corollary 2. For every fixed k, ℓ ∈ N and every ε > 0, there is a(2/3, 3 + ε)-approximation algorithm for k-Max-ℓ-Min-STSP. Itsrunning-time is polynomial in the input size and 1/ε.Proof. By using rn = 2+ε andα = 2/3, we obtain a (2/3, 3+2ε)-approximation from Theorem 1 for every ε > 0, which proves theassertion. Corollary 3. For every fixed k, ℓ ∈ N and every ε > 0, there is a(1/2, log2 n + ε)-approximation algorithm for k-Max-ℓ-Min-ATSP.Its running-time is polynomial in the input size and 1/ε.Proof. We use rn = log2 n + ε and α = 1/2. Then we obtain anapproximation ratio of (1/2, log2(n/2)+ε+1+ε) = (1/2, log2 n+2ε) for every ε > 0. References[1] Arash Asadpour, Michel X. Goemans, Aleksander Madry, Shayan OveisGharan, Amin Saberi, An O(log n/ log log n)-approximation algorithm for theasymmetric traveling salesman problem, in: Proc. of the 21st Ann. ACM-SIAMSymp. on Discrete Algorithms, SODA, SIAM, 2010, pp. 379–389.[2] Giorgio Ausiello, Pierluigi Crescenzi, Giorgio Gambosi, Viggo Kann, AlbertoMarchetti, Marco Protasi, Complexity and Approximation: CombinatorialOptimization Problems and Their Approximability Properties, Springer,1999.[3] André Berger, Vincenzo Bonifaci, Fabrizio Grandoni, Guido Schäfer, Budgetedmatching and budgeted matroid intersection via the gasoline puzzle,Mathematical Programming 128 (1–2) (2011) 355–372.[4] Markus Bläser, BodoManthey, Oliver Putz, Approximatingmulti-criteriaMax-TSP, in: Proc. of the 16th Ann. European Symp. on Algorithms, ESA, in: LectureNotes in Computer Science, vol. 5193, Springer, 2008, pp. 185–197.[5] Matthias Ehrgott, Multicriteria Optimization, Springer, 2005.[6] Christian Glaßer, Christian Reitwießner, Maximilian Witek, Improved andgeneralized approximations for two-objective traveling salesman, Report 09-076, Revision 1, Electronic Colloquium on Computational Complexity (ECCC),2010.[7] Christian Glaßer, Christian Reitwießner, Maximilian Witek, Applications ofdiscrepancy theory inmultiobjective approximation, in: Proc. of the 31st Conf.on Foundations of Software Technology and Theoretical Computer Science,FSTTCS, 2011 (in press).[8] Haim Kaplan, Moshe Lewenstein, Nira Shafrir, Maxim I. Sviridenko, Approx-imation algorithms for asymmetric TSP by decomposing directed regularmultigraphs, Journal of the ACM 52 (4) (2005) 602–626.[9] Bodo Manthey, On approximating multi-criteria TSP, ACM Transactions onAlgorithms (in press).[10] Bodo Manthey, On approximating multi-criteria TSP, in: Proc. of the 26th Int.Symp. on Theoretical Aspects of Computer Science, STACS, 2009, pp. 637–648.[11] BodoManthey, Multi-criteria TSP: min andmax combined, in: Proc. of the 7thWorkshop on Approximation and Online Algorithms, WAOA 2009, in: LectureNotes in Computer Science, vol. 5893, Springer, 2010, pp. 205–216.[12] Bodo Manthey, L. Shankar Ram, Approximation algorithms for multi-criteriatraveling salesman problems, Algorithmica 53 (1) (2009) 69–88.[13] Katarzyna Paluch, Marcin Mucha, Aleksander Madry, A 7/9 approximationalgorithm for the maximum traveling salesman problem, in: Proc. of the 12thInt. Workshop on Approximation Algorithms for Combinatorial OptimizationProblems, APPROX, in: Lecture Notes in Computer Science, vol. 5687, Springer,2009, pp. 298–311.[14] Christos H. Papadimitriou, Mihalis Yannakakis, On the approximability oftrade-offs and optimal access of web sources, in: Proc. of the 41st Ann. IEEESymp. on Foundations of Computer Science, FOCS, IEEE Computer Society,2000, pp. 86–92.

Multi-criteria TSP:Min and max combined

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