STSP seeks a pair of pickup and delivery tours in two distinct networks,
where the two tours are related by LIFO contraints. We address here the problem
approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are
APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard
approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12

Toulouse, Sophie

English

arXiv

STSP seeks a pair of pickup and delivery tours in two distinct networks, where the two tours are related by LIFO contraints. We address here the problem approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12

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Approximability of the Multiple Stack TSP

Sophie Toulouse

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Electronic Notes in Discrete Mathematics

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STSP seeks a pair of pickup and delivery tours in two distinct networks, where the two tours are related by LIFO contraints. We address here the problem approxima-bility. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12

HAL-Paris 13

Approximability of the Multiple Stack TSPSophie ToulouseTo cite this version:Sophie Toulouse. Approximability of the Multiple Stack TSP. 2009. <hal-00521125>HAL Id: hal-00521125https://hal.archives-ouvertes.fr/hal-00521125Submitted on 25 Sep 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Approximability of the Multiple Stack TSPSophie Toulouse 1LIPN (UMR CNRS 7030) - Institut Galile´e - Universite´ Paris 13,99 av. Jean-Baptiste Cle´ment, 93430 Villetaneuse, FranceAbstractSTSP seeks a pair of pickup and delivery tours in two distinct networks, where thetwo tours are related by LIFO contraints. We address here the problem approxima-bility. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX,and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximationfor respectively Max2STSP, Max2STSP12 and Min2STSP12.Keywords: Approximation algorithms, multiple stack TSP, TSP, LIFO constraints1 IntroductionThe multiple Stack Traveling Salesman Probem, STSP, has been recently in-troduced in [1] and is very well motivated by the initial request of a trans-portation company. Consider a company that owns a vehicle fleet and a depotin m distinct cities C1, . . . , Cm. For any pair (Cj, Cj′) of cities, the companyhandles orders from Cj suppliers to Cj′ customers, proceeding as follows: inCj, a single vehicle picks up all the orders that involve a Cj′ customer; duringthis tour, the commodities are packed within a single container that consistsof a given number k of rows; the container is then sent to Cj′, where a localvehicle delivers the commodities. The operator thus is faced with two levels of1 Email: sophie.toulouse@lipn.univ-paris13.frtransportation: the local one inside the cities, the long-haul one between thecities. STSP models the local level problem; namely, in the context we havejust described, every pair of cities defines an instance of STSP. The problemspecificity relies on the way the containers are managed: their rows behavelike stacks, and no repacking is allowed. The rows of the containers thus aresubject to LIFO “Last In First Out” constraints. Therefore, the delivery tourin Cj′ must handle at first orders that have been handled at last in Cj.Formally, an instance I of STSP considers a number k of stacks (thecontainer rows), together with two distinct networks IA = (GA, dA), IB =(GB, dB), where V (Gα) ≡ [n + 1] and dα : E(Gα) → N, α ∈ {A,B}. In bothnetworks, vertex 0 represents the depot, whereas vertex i ∈ [n] representsthe ith order (the ith supplier in IA, the ith customer in IB). A solutionconsists of a pickup tour ~TA on IA, a delivery tour ~TB on IB, and a packingP = {P 1, . . . , P k} of the commodities into the stacks, where a stack P β isdescribed as a sequence (i1, . . . , ipβ) of commodities (index 1 refers to the bot-tom of P β). We denote by <α, α ∈ {A,B} the complete order that ~T α induceson [n]: i <α i′ iff ~T α handles order i earlier than it handles i′. Similary, wedenote by <P the partial order that P induces on [n]: i <P i′ iff P stacks iat a lower position than i′ in a same stack P β. The LIFO constraints maythus be expressed in terms of the three orders <A, <B, <P . Namely, a triple(P, ~TA, ~TB) is consistent iff it satisfies:∀i 6= i′ ∈ [n], i <P i′ ⇒ (i <A i′) ∧ (i >B i′)(1)The value of a solution (P, ~TA, ~TB) is given by the sum dA(~TA) + dB(~TB)of the distance of its two tours. The ultimate goal naturally consists in findingthe feasible triple (P, ~TA, ~TB) of optimum value.In what follows, the stacks are assumed to have unlimited capacities. Themost often, input graphs are complete. For TSP (the Traveling SalesmanProblem), as well as for STSP, A(S)TSP, ∆(S)TSP and (S)TSPab respec-tively refer the asymmetric, the metric and the bivaluated cases. Finally,kSTSP considers that k is a universal constant. Since one manipulates bothmaximization and minimization goals, we use notations , ≻, opt, opt in-stead of ≥, >, max, min (resp., ≤, <, min, max) if the goal is to maximize(resp., to minimize). Finally, because one jointly considers pickup and deliverytours, we respectively denote by E−1 = {(i′, i) | (i, i′) ∈ E}, G−1 = (V,E−1),d−1 : E−1 → N : d−1(i, i′) = d(i′, i) and I−1 = (G−1, d−1) the reverse of anedge set E, a graph G = (V,E), a distance d : E → N, a network I = (G, d).Hence, ~T is a tour of value d(~T ) on I iff ~T−1 is a tour of value d(~T ) on I−1.2 PreliminariesThe problem trivially is NP− hard, from TSP. Indeed, for extremal val-ues of the number of stacks, STSP strictly is equivalent to TSP: if STSPassumes a single stack, then it reduces to TSP, considering the distanced : (i, i′) 7→ dA(i, i′) + dB(i′, i); if on the opposite STSP assumes n stacks,then it reduces to two independent TSP (packings that store one commodityper stack do not induce any LIFO constraint). The litterature on computa-tional aspects of STSP is rather thin: see [1,2] for metaheuristic approachs,[3] for an exact approach (that solves 2STSP via the computation of the kbest tours in IA, IB), [4] for an analysis of STSP complexity. We here makeuse of natural connections to TSP and observations made in [4] in order tolocate STSP within the approximation hierarchy.Besides its practical impacts, STSP presents a rather intricate combina-torial structure. One the one hand, one could wonder on what part of theproblem does structure / impact the most the solution / its complexity. Itappeared in [4] that both the packing and the tour subproblems are tractable:Theorem 2.1 ([4]) In arbitrary input graphs:(i) Deciding, given a pair (~TA, ~TB), whether it admits or not a consistentpacking P (and to design this packing if the answer is YES) is in P.(ii) When k is a universal constant: given a packing P, computing a pairof tours that are optimal for P is in P.On the other hand, STSP has strong connections to coloring and schedul-ing. Indeed, subproblem (i) reduces to graph coloring in C(~TA, ~TB) = {i 6=i′ ∈ [n] | i <A i′ ⇔ i <B i′}, whereas subproblem (ii) reduces to schedulingunder <P (or >P) precedence constraints.3 A first classification of STSPProperty 1 (i) There exists a standard approximation preserving reductionfrom ASTSP to ATSP that maps a ρ-approximate solution of ATSP toa ρ/2, ρ/2 + (a/2b), ρ/2 + (b/2a)-approximate solution of ASTSP forMax(S)TSP, Max(S)TSPab and Min(S)TSPab, respectively.(ii) Case of arbitrary graphs. Any approximate factor ρ for any restrictionof STSP also holds for the same restricton of TSP.Proof. i. Associate with the instance I = (IA, IB) of ASTSP the pair (IA, IB)(i)i1j?i1j′i1j?i1j′i2hi2h′(ii)i1j i1j′i2hi2h′?(iii)i1j i1j+1i2h+1i2h?i1j i1j+1i2h+1i2h?Fig. 1. The three configurations that contradict consistency.of instances of ATSP. Compute ~T α, α ∈ {A,B} an approximate tour on Iα.Pick the tour ~TA or ~TB that performs the best among dA(~TA) + (dB)−1(~TA),(dA)−1(~TB) + dA(~TB). ii. Associate with the instance I = (G, d) of TSP theinstance I ′ = (I, I−1) of STSP. 2Proposition 3.1 (Immediate from Property 1)(i) Max(A)STSP, Max∆(A)STSP, Max(A)STSP01,Min(A)STSP12 are standard-approximable within the following factor:MaxSTSP MaxASTSP Max∆STSP Max∆ASTSP61/162− o(1), [8] 1/3, [9] 7/16− o(1), [7] 31/80, [10]MaxSTSP01 MaxASTSP01 MinSTSP12 MinASTSP123/7, [12] 3/8, [11] 11/7, [12] 13/8, [11](ii) Max∆STSP∈ MaxSNP-hard, [5]; MaxSTSP12∈ APX-hard, [6]; Min-STSP is not 2p(n+1) standard-approximable for any polynomial p, or P = NP4 A matching based heuristic for the symmetric 2STSPIn what follows, we say that an edge set N is consistent with a packing P iffthere exists a completion N ′ of N into a tour T s.t. T is feasible with respectto P. We first provide a characterization of consistency for 2STSP.Proposition 4.1 (Due to lack of space, the proof is omitted.) Given apacking P, Jβ = {0, . . . , pβ+1} denotes the index set on P β, β ∈ {1, 2}. Arti-ficial indexes 0, pβ+1 are introduced in order to represent the depot vertex, i.e.,iβ0 = iβpβ+1= 0. Let N be a collection of pairwise disjoint elementary chains on[n+1], then N is consistent with P iff it satifies the three following properties(configurations that contradict consistency are illustrated in Figure 1):(i) No jump: ∀β ∈ {1, 2}, ∀j < j′ ∈ Jβ,({iβj , iβj′} ∈ N)∨(∃h ≤ h′ ∈ J3−β, {iβj , i3−βh , . . . , i3−βh′ , iβj′} ⊆ N)⇒ j′ = j + 1(ii) No crossing edges: ∀j 6= j′ ∈ J1, ∀h 6= h′ ∈ J2, {{i1j , i2h}, {i1j′, i2h′}} ⊆ N⇒ (j < j′ ⇔ h < h′)(iii) No way back: ∀j ∈ J1, ∀h ∈ J2, {{i1j , i1j+1}, {i2h, i2h+1}} ⊆ N⇒({i1j , i2h})/∈ N ∧({i1j+1, i2h+1})/∈ NP 1 :P 2 :c11= 0c12c13c21 c22c23c24c31c32c41 c42 c43c44c45c46MATA, TB\MαMBe∗B AAB 0Fig. 2. Possible completions of the approximate packing P: case n even.j, j′ 6= 1, j ≡ j′ + 1[2]0c12 c13 c14 c15 c16c17c18c19e∗j, j′ 6= 1, j ≡ j′[2]0c12 c13 c14 c15 c16c19c18c170e∗j′ = 10c12 c13 c14c15c16c17c18c190e∗MAMBFig. 3. The approximate packing when n is even and p = 1.The algorithm relies on a pair (MA,MB) of optimum weight matchingson IA, IB. Given such a pair, together with some additional edge e∗, it aimsat computing a packing P s.t. Nα, α ∈ {A,B}, is consistent with P, whereNα coincides with Mα (resp., Mα ∪ {e∗}) when n is odd (resp., even). Let Hdenote the multigraph ([n+1],MA∪MB), and C1, . . . , Cp denote its connectedcomponents. We assume w.l.o.g., that the depot vertex (i.e., node 0) coincideswith vertex c11 in C1. If n is odd (iff n+1 is even), then every component Ch isan elementary cycle {ch1 , . . . , chqh, ch1} of even length qh ≥ 2. Otherwise, Ch ={ch1 , . . . , chqh, ch1}\{{chℓ , chℓ+1}} is an elementary chain of even length qh − 1 ≥ 0for a lonely index h ∈ [p]. We define the additional edge e∗ as follows:∀p, e∗ = arg opt{ optα=A,B{dα(f)} | f ∈ F (p)}, where:(2)F (1) ={({0} ∪ {c13, . . . , c1ℓ})×({0} ∪ {c1ℓ+1, . . . , c1n})}\{{0, 0}}(3)F (p) = ∪p(h<h′)=1V (Ch)× V (Ch′)(4)The heuristic is described in Algorithm 1: it first computes Mα, α ∈ {A,B}(Step (i)) and e∗ (case n even, Step (ii)); it then builds the approximatepacking P by considering the components the one after each other, startingwith C1 (Step (iv)); it finally computes the best tours T α(P), α ∈ {A,B},with respect to the P (Step (v)). Note that before it operates the packing,the algorithm may have in Step (iii) to perform some reindexation of the Ch,depending on e∗ (notably, if p ≥ 2, then the two connected components thatare incident to e∗ must be stacked consecutively). The even case is illustratedin Figures 2 (p ≥ 2) and 3 (p = 1).Claim 4.2 (i) Algorithm 1 has polynomial-time complexity.(ii) Nα, α ∈ {A,B}, is consistent with the approximate packing P.(iii) The solutions returned by the algorithm are 3/2, 3/4 and 1/2 standard ap-proximate for Min2STSP12, Max2STSP12 and Max2STSP, respectively;these ratios are (asymptotically) tight.Algorithm 1. An appproximate algorithm for 2STSP.(i) Compute Mα an optimum weight matching on Iα, α ∈ {A,B}(ii) Determine the connected components of H in such a way that c11 = 0If n is even Then compute e∗ = arg opt{optα=A,B{dα(f)} | f ∈ F (p)}(iii) If n is even and p ≥ 2 Then consider the components in such a way that:(a) If V (e∗) ∩ V (C1) ∈ {∅, {0}} Then ∃h ∈ [p]\{1} s.t. e∗ = {ch⌈qh/2⌉, ch+11 }Else e∗ = {c1j , c21} // h is taken modulo [p], j ∈ {2, . . . , q1} depends on c11(b) If q2 6= 2 ∧ e∗ = {c1j , c21} ∧ ∃α, {{0, c12}, {c21, c2q2}} ⊆Mα Then reverse C2(iv) If n is odd or p ≥ 2:j1 ← (If n ≡ 0[2] ∧ e∗ = {c1j , c21} Then j: ⌈(q1 − 1)/2⌉ + 1 otherwise)j2 ← (If n ≡ 0[2] ∧ e∗ = {c12, c21} ∧ q2 = 2 Then q2: ⌈q2/2⌉ otherwise)P 1 ← (c12, . . . , c1j1 , c21, . . . , c2j2 , c31, . . . , c3⌈q3/2⌉ , . . . , cp1, . . . , cp⌈qp/2⌉)P 2 ← (c1q1 , . . . , c1j1+1, c2q2 , . . . , c2j2+1, c3q3 , . . . , c3⌈q3/2⌉+1 , . . . , cpqp , . . . , cp⌈qp/2⌉+1)Else (thus p = 1, e∗ = {c1j , c1j′}, j ∈ {1}∪{3, . . . , ℓ}, j′ ∈ {1}∪{ℓ+1, . . . , n+1}):If j, j′ 6= 1 and j ≡ j′ + 1[2] Then P 1 ← (c12, . . . , c1ℓ ), P2 ← (c1n+1, . . . , c1ℓ+1)Else If j, j′ 6= 1 and j ≡ j′[2] Then P 1 ← (c12, . . . , c1ℓ ), P2 ← (c1ℓ+1, . . . , c1n+1)Else If j′ = 1 Then P 1 ← (c12, . . . , c1j ), P2 ← (c1n+1, . . . , c1j+1)Else If j = 1 Then P 1 ← (c12, . . . , c1j′−1), P2 ← (c1n+1, . . . , c1j′)(v) Compute Tα(P), α ∈ {A,B}, the best feasible tour for P = (P 1, P 2) on IαCh⌈qh/2⌉ − 1⌈qh/2⌉⌈qh/2⌉+ 1e∗Ch+11 2qh+1Ch⌈qh/2⌉ − 1⌈qh/2⌉⌈qh/2⌉ + 1e∗Ch+11 2qh+1C102q1e∗C21 2q2MAMBFig. 4. Case n even and p ≥ 2: edge e∗ induces “no jump” in Mα.Proof. Claim (i). Step (v) only requires a O((n+1)2)-time, [4]; for the othersteps, the fact is either famous, or trivial. Claim (ii). Nα, α ∈ {A,B} triv-ially satisfies Properties (ii) and (iii) (resp., (i)) of Propositon 4.1, in any case(resp., when n is odd or p = 1). Thus assume n ≡ 0[2] and p ≥ 2. Steps (iii-a)and (iv) of the algorithm indicate: ∃h ∈ [p], ∃j ∈ [qh] s.t. e∗ = {chj , ch+11 } ande∗ ∈ P 1 × (P 1 ∪ {0}). Mα possibly links chj (resp., ch+11 ) to vertices (whenthey exist) chj−1, chj+1 (resp., ch+11 , ch+1qh+1), with chj−1 ∈ P1 (unless e = {c12, c21}),chj+1 ∈ P2, ch+12 ∈ P1 and ch+1qh+1∈ P 2 (unless e∗ = {c12, c21} and q2 = 2). More-over, Steps (iii-b) and (iv) ensure that P satisfies: ¬({0, c12} ∈Mα∧{c21, c2q2} ∈Mα) ∧ (c21, c2q2) ∈ P1 × P 2). Hence, the chain {chj−1, chj , ch+11 , ch+1qh+1} (resp.,{chj+1, chj , ch+11 , ch+12 }) may never jump chj+1 (resp., ch+1qh+1) in P 2 (see Figure 4for some illustration). Claim (iii). Let (P∗, TA,∗, TB,∗) (resp., (P, TA, TB))denote an optimum (resp., the approximate) solution, we denote by OPTresp., APX) its value; we observe:(i) ∀α in {A,B}, ∃ N ′α s.t. T ′α = Nα ∪N ′α is a tour, and T ′α is consistentwith P. By optimality of T α with respect to P, we thus have:dα(T α)  dα(Mα) + dα(N ′α) + 1n≡0[2]dα(e∗), α ∈ {A,B}.(ii) If n is odd, then any tour on [n+1] is the union of two perfect matchingson [n+ 1]. By optimality of Mα, we thus have: 2dα(Mα)  dα(T α,∗).(iii) If n is even, then ∀ T tour on [n+ 1], ∀e ∈ T , T\{e} is the union of twoalmost perfect perfect matchings on [n+1]. Furthemore, by connectivityof T α,∗, ∃eα ∈ T α,∗ ∩ F (p), α ∈ {A,B}. By optimality of Mα, we thushave: 2dα(Mα)  dα(T α,∗)− dα(eα).From (i), (ii), (iii), we deduce:2APX  OPT + 2∑αdα(N ′α) + 1n≡0[2](2∑αdα(e∗)−∑αdα(eα))(5)What enables to conclude for Max2STSP, when n is odd. When n is even,just observe that e∗ is optimal on F (p), whereas eα ∈ F (p), α ∈ {A,B}:∑αdα(e∗)  optα{dα(e∗)}  optα{dα(eα)}  1/2∑αdα(eα)(6)For the bivalued case, any edge set S satisfies |S| ≤ dα(S) ≤ 2|S|, α ∈{A,B}. When n is odd, thus consider |T α,∗| = n+ 1 = |N ′α| and conclude:1/2OPT ≤ 2(n+ 1) ≤ 2∑αdα(N ′α) ≤ 4(n+ 1) ≤ 2OPT(7)When n is even, |N ′α| = n. Let Q denote the quantity 2(dA(e∗)+dB(e∗))−(dA(eA) + dB(eB)): if the goal is to maximize, assume that Q < 2; thendA(e∗) + dB(e∗) = 2 and dA(eA) + dB(eB) ≥ 3, what contradicts the fact thate∗ is optimal. Similary, Q may not reach values 5, 6 when minimizing.Tight instances In,a,b are defined on Kn+1, n ≥ 1, as: dα(u, v) = a if (v =u+2∧u ≡ 1[4]) or (v = u+2∧u ≡ 0[4]∧α = A) or (v = u+2∧u ≡ 2[4]∧α = B)or (v = u+1∧u ≡ 1[2]∧α = A) or (v = u+1∧u ≡ 0[2]∧α = B), dα(u, v) = botherwise. We consider subfamilies In,a,b s.t. (a, b) = (1, 0) (resp., (2, 1), (1, 2))for Max2STSP (resp., Max2STSP12, Min2STSP12) and n = 8q−1 (resp., 8q)for the odd (resp., even) case. 2Theorem 4.3 (Immediate form Claim 4.2) Min2STSP12, Max2STSP12and Max2STSP respectively are 3/2, 3/4 and 1/2 standard approximable.5 ConclusionTwo main questions arise from the proposed analysis: Min∆STP approxima-bility, STSP differential approximability.References[1] Petersen H.L., Madsen O.B.G.: The double travelling salesman problem withmultiple stacks - Formulation and heuristic solution approaches. EJOR 198(1):139-147 (2009)[2] Felipe A., Ortun˜o M., Tirado G.: New neighborhood structures for the DoubleTraveling Salesman Problem with Multiple Stacks. TOP 17(1):190-213 (2009).[3] R. Lusbyz, J. Larsenz, M. Ehrgott, D. Ryan: An Exact Method for the DoubleTSP with Multiple Stacks. Report 2.2009 DTU Management Engineering (2009).[4] S. Toulouse, R. Wolfler Calvo: On the complexity of the multiple stack TSP,kSTSP. TAMC 2009, LNCS 5532:360-369 (2009).[5] A. Barvinok, D. S. Johnson, G. J. Woeginger, R. Woodroofe: The MaximumTraveling Salesman Problem Under Polyhedral Norms. IPCO VI, LNCS1412:195-201 (1998).[6] C.H. Papadimitriou, M. Yannakakis: The traveling salesman problem withdistances one and two. Mathematics of Operations Research 18:1-11 (1993).[7] Z.-Z. Chen, T. Nagoya: Improved approximation algorithms for metric MaxTSP.J. of Combinatorial Optimization 13(4):321-336 (2007).[8] Z.-Z. Chen, Y. Okamoto, L. Wang: Improved deterministic approximationalgorithms for Max TSP. Inform. Proc. Letters 95(2):333-342 (2005).[9] H. Kaplan, N. Shafrir, M. Lewenstein, M. Sviridenko: Approximation Algorithmsfor Asymmetric TSP by Decomposing Directed Regular Multigraphs. J. of theACM 52(4):602-626 (2005).[10] M. Bla¨ser, L. Shankar Ram, M. Sviridenko: Improved approximation algorithmsfor metric maximum ATSP and maximum 3-cycle cover problems. OperationsResearch Letters 37(3):176-180 (2009).[11] M. Bla¨ser: A 3/4-Approximation Algorithm for Maximum ATSP with WeightsZero and One. APPROX-RANDOM 2004 LNCS 3122:61-71 (2004).[12] P. Berman; M. Karpinski: 8/7-approximation algorithm for (1,2)-tsp. SODA2006, ACM 641-648 (2006).

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