A 3/2-Approximation for the Metric Many-visits Path TSP
AA { -Approximation for the Metric Many-visits Path TSP ∗ Krist´of B´erczi † Matthias Mnich ‡ Roland Vincze § Abstract
In the
Many-visits Path TSP , we are given a set of n cities along with their pairwisedistances (or cost) c p uv q , and moreover each city v comes with an associated positive integerrequest r p v q . The goal is to find a minimum-cost path, starting at city s and ending at city t ,that visits each city v exactly r p v q times.We present a { -approximation algorithm for the metric Many-visits Path TSP , thatruns in time polynomial in n and poly-logarithmic in the requests r p v q . Our algorithm canbe seen as a far-reaching generalization of the { -approximation algorithm for Path TSP byZenklusen (SODA 2019), which answered a long-standing open problem by providing an efficientalgorithm which matches the approximation guarantee of Christofides’ algorithm from 1976 formetric TSP.One of the key components of our approach is a polynomial-time algorithm to compute a con-nected, degree bounded multigraph of minimum cost. We tackle this problem by generalizing afundamental result of Kir´aly, Lau and Singh (Combinatorica, 2012) on the
Minimum BoundedDegree Matroid Basis problem, and devise such an algorithm for general polymatroids, evenallowing element multiplicities.Our result directly yields a { -approximation to the metric Many-visits TSP , as well as a { -approximation for the problem of scheduling classes of jobs with sequence-dependent setuptimes on a single machine so as to minimize the makespan. Keywords:
Traveling salesman problem, degree constraints, generalized polymatroids. ∗ Supported by DAAD with funds of the Bundesministerium f¨ur Bildung und Forschung (BMBF) and by DFGproject MN 59/4-1. † MTA-ELTE Egerv´ary Research Group, Department of Operations Research, E¨otv¨os Lor´and University, Budapest,Hungary. [email protected] . ‡ TU Hamburg, Hamburg, Germany. [email protected] . § TU Hamburg, Hamburg, Germany. [email protected] . a r X i v : . [ c s . D M ] J u l Introduction
The traveling salesman problem (TSP) is one of the cornerstones of combinatorial optimization.Given a set V of n cities with non-negative costs c p uv q for each cities u and v , the objective is tofind a minimum cost closed walk visiting each city. TSP is well-known to be NP -hard even in thecase of metric costs, i.e. when the cost function c satisfies the triangle inequality. For metric costs,the best known approximation ratio that can be obtained in polynomial time is { , discoveredindependently by Christofides [9] and Serdyukov [50].In the traveling salesman path problem, or Path TSP , two distinguished vertices s and t aregiven, and the goal is to find a minimum cost walk from s to t visiting each city. Approximating themetric Path TSP has a long history, from the first { -approximation by Hoogeveen [27], throughsubsequent improvements [2,22,48,49,57] to the recent breakthroughs. The latest results eventuallyclosed the gap between the metric TSP and the metric Path TSP : Traub and Vygen [53] provideda p { ` ε q -approximation for any ε ą
0, Zenklusen [58] provided a { -approximation and finally thethree authors showed a reduction from the Path TSP to the TSP [55].We consider a far-reaching generalization of the metric
Path TSP , the metric
Many-visitsPath TSP , where in addition to the costs c on the edges, a requirement r p v q is given for eachcity v . The aim is to find a minimum cost walk from s to t that visits each city v exactly r p v q times. The cycle version of this problem, where s “ t , is known as Many-visits TSP and was firstconsidered in 1966 by Rothkopf [45]. Psaraftis [44] proposed a dynamic programming approach thatsolves the problem in time p r { n q n for r “ ř v P V r p v q . Later, Cosmadakis and Papadimitriou [10]gave the first algorithm for Many-visits TSP with logarithmic dependence on r , though thespace and time requirements of their algorithm were still superexponential in n . Recently, Bergeret al. [5] simultaneously improved the run time to 2 O p n q ¨ log r and reduced the space complexity topolynomial. (The algorithm by Berger et al. [5] can be slightly modified to solve the path version aswell.) Lately, Kowalik et al. [37] made further fine-grained time complexity improvements. To thebest of our knowledge, no constant-factor approximation algorithms for the metric Many-visitsTSP or metric Many-visits Path TSP are currently known.Besides being of scientific interest in itself, the
Many-visits Path TSP can be used for mod-eling various problems. The aircraft sequencing problem or aircraft landing problem is one of themost referred applications in the literature [3, 6, 40, 44], where the goal is to find a schedule ofdeparting and/or landing airplanes that minimizes an objective function and satisfies certain con-straints. The aircraft are categorized into a small number of classes, and for each pair of classes anon-negative lower bound is given denoting the minimum amount of time needed to pass betweenthe take off/landing of two planes from the given classes. The problem can be embedded in the Many-visits Path TSP model by considering the classes to be cities and the separation timesto be costs between them, while the number of airplanes in a class corresponds to the number ofvisits of a city.As another illustrious example, the
Many-visits Path TSP is equivalent to the high-multipli-city job scheduling problem 1 | HM, s ij , p j | ř C j , where each class j of jobs has a processing time p j and there is a setup time s ij between processing two jobs of different classes. There is only ahandful of constant-factor approximation algorithms for scheduling problems with setup times [1], At the Hausdorff Workshop on Combinatorial Optimization in 2018, Rico Zenklusen brought up the topic ofapproximation algorithms for the metric version of
Many-visits TSP in the context of iterative relaxation techniques;he suggested an approach to obtain a 1.5-approximation, which is unpublished.
Many-visits Path TSP would further extend the list of such results.A different kind of application comes from geometric approximation. Recently, Kozma andM¨omke provided an EPTAS for the
Maximum Scatter TSP [38]. Their approach involved group-ing certain input points together and thus reducing the input size. The reduced problem is exactlythe
Many-visits TSP . The same problem arises as a subproblem in the fixed-parameter algorithmfor the
Hamiltonian Cycle problem on graphs with bounded neighborhood diversity [39].Our work relies on a polymatroidal optimization problem with degree constraints. An illustriousexample of such problem is the
Minimum Bounded Degree Spanning Tree problem, wherethe goal is to find a minimum cost spanning tree in a graph with lower and upper bounds on thedegree of each vertex. Checking feasibility of a degree-bounded spanning tree contains the NP -hard Hamiltonian Path problem, and several algorithms were given that were balancing between thecost of the spanning tree and the violation of the degree bounds [7, 8, 19, 21, 35, 36]. Based on aniterative rounding approach [30] combined with a relaxation step, Singh and Lau [51] provided apolynomial-time algorithm that finds a spanning tree of cost at most the optimum value violatingeach degree bound by at most 1. Kir´aly et al. [33] later showed that similar results can be obtainedfor the more general
Minimum Bounded Degree Matroid Basis Problem . Our results
In this paper we provide the first efficient constant-factor approximation algorithm for the metric
Many-visits Path TSP . Formally, a graph G “ p V, E q is given with a positive integer r p v q foreach v P V , and a non-negative cost c p uv q for every pair of vertices u, v ; finally, a departure city s and an arrival city t are specified. We seek a minimum cost s - t -walk that visits each city v exactly r p v q times, where leaving city s as well as arriving to city t counts as one visit.The cost function c : E Ñ R ě is assumed to be metric. Besides the triangle inequality c p uw q ď c p uv q ` c p vw q for every triplet u, v, w this implies that the cost of a self-loop c p vv q atvertex v is at most the cost of leaving city v to any other city u and returning, that is: c p vv q ď ¨ min u P V ´ v c p uv q for all v P V .
The assumption of metric costs is necessary, as the TSP, and therefore the
Many-visits TSP ,does not admit any non-trivial approximation for unrestricted cost functions assuming that P ‰ NP (see e.g. Theorem 6.13 in the book of Garey and Johnson [20]).We start with a simple approximation idea, that leads to a constant factor approximation instrongly polynomial time: Theorem 1.
There is a polynomial-time { -approximation for the metric Many-visits PathTSP , that runs in time polynomial in n and log r . The approximation factor { in Theorem 1 still leaves a gap to the best-known factor { forthe metric Path TSP , which is due to Zenklusen [58]. His recent { -approximation for the metric Path TSP uses a Christofides-Serdyukov-like construction that combines a spanning tree and amatching, with the key difference that it calculates a constrained spanning tree in order to boundthe costs of the tree and the matching by { times the optimal value.3ur main algorithmic result matches this approximation ratio for the metric Many-visitsPath TSP . Theorem 2.
There is a polynomial-time { -approximation for the metric Many-visits PathTSP . The algorithm runs in time polynomial in n and log r . As a direct consequence of Theorem 2, we obtain the following:
Corollary 3.
There is a { -approximation for the metric Many-visits TSP that runs in timepolynomial in n and log r . Our approach follows the main steps of Zenklusen’s work [58]. However, the presence of requests r p v q makes the problem significantly more difficult and several new ideas are needed to design analgorithm which returns a tour with the correct number of visits and still runs in polynomial time.For instance, whereas the backbone of both Christofides and Zenklusen’s algorithm is a spanningtree (with certain properties), the possibly exponentially large number of (parallel) edges in amany-visits TSP solution requires us to work with a structure that is more general than spanningtrees. We therefore consider the problem of finding a minimum cost connected multigraph withlower bounds ρ on the degree of vertices, and lower and upper bounds L and U , respectively,on the number of occurrences of the edges. We call this task the Minimum Bounded DegreeConnected Multigraph with Edge Bounds problem, and show the following:
Theorem 4.
There is an algorithm for the
Minimum Bounded Degree Connected Multi-graph with Edge Bounds problem that, in time polynomial in n and log ř v ρ p v q , returns aconnected multigraph T with ρ p V q { edges, where each vertex v has degree at least ρ p v q ´ and thecost of T is at most the cost of min t c T x | x P P CG p ρ, L, U qu , where (1) P CG p ρ, L, U q : “ $’’&’’% x P R E ě ˇˇˇˇˇˇˇˇ supp p x q is connected x p E q “ ř v ρ p v q { x p δ p v qq ě ρ p v q @ v P VL p vw q ď x p vw q ď U p vw q @ v, w P V ,//.//- . Note that an optimal solution x ˚ to the Minimum Bounded Degree Connected Multi-graph with Edge Bounds problem is a minimum cost integral vector from the polytope P CG .We use the result of Theorem 4 to obtain a multigraph that serves a key role in our approximationalgorithm for the metric Many-visits Path TSP ; the values ρ , L and U depend on the instanceand the details are given in Section 4.The Minimum Bounded Degree Connected Multigraph with Edge Bounds problemshows a lot of similarities to the
Minimum Bounded Degree Spanning Tree problem. However,neither the result of Singh and Lau [51] nor the more general approach by Kir´aly et al. [33] appliesto our setting, due to the presence of parallel edges and self-loops in a multigraph.One of our key contributions is therefore an extension of the result of Kir´aly et al. [33] togeneralized polymatroids, which might be of independent combinatorial interest. Formally, the
Bounded Degree g-polymatroid Element with Multiplicities problem takes as input ag-polymatroid Q p p, b q defined by a paramodular pair p, b : 2 S Ñ R , a cost function c : S Ñ R ,a hypergraph H “ p S, E q with lower and upper bounds f, g : E Ñ Z ě and multiplicity vectors m ε : S Ñ Z ě for ε P E satisfying m ε p s q “ s P S ´ ε . The objective is to find a minimum-costintegral element x of Q p p, b q such that f p ε q ď ř s P ε m ε p s q x p s q ď g p ε q for each ε P E . We give apolynomial-time algorithm for finding a solution of cost at most the optimum value with boundson the violations of the degree prescriptions. 4 heorem 5. There is an algorithm for the
Bounded Degree g-polymatroid Element withMultiplicities problem which returns an integral element x of Q p p, b q of cost at most the optimumvalue such that f p ε q ´ ` ď ř s P ε m ε p s q x p s q ď g p ε q ` ´ for each ε P E , where ∆ “ max s P S t ř ε P E : s P ε m ε p s qu . The run time of the algorithm is polynomial in n and log ř ε p f p ε q ` g p ε qq . When only lower bounds (or only upper bounds) are present, we call the problem
Lower(Upper) Bounded Degree g-polymatroid Element with Multiplicities . Similarly toKir´aly et al. [33], we obtain an improved bound on the degree violations when only lower or upperbounds are present: Theorem 6.
There is an algorithm for
Lower Bounded Degree g-polymatroid Elementwith Multiplicities which returns an integral element x of Q p p, b q of cost at most the optimumvalue such that f p ε q ´ ∆ ` ď ř s P ε m ε p s q x p s q for each ε P E . An analogous result holds for UpperBounded Degree g-polymatroid Element , where ř s P ε m ε p s q x p s q ď g p ε q ` ∆ ´ . The runtime of these algorithms is polynomial in n and log ř ε f p ε q or log ř ε g p ε q , respectively. Basic notation.
Throughout the paper, we let G “ p V, E q be a finite, undirected complete graphon n vertices, whose edge set E also contains a self-loop at every vertex v P V . For a subset F Ď E of edges, the set of vertices covered by F is denoted by V p F q . The number of connected components of the graph p V p F q , F q is denoted by comp p F q . For a subset X Ď V of vertices, the set of edgesspanned by X is denoted by E p X q . Given a multiset F of edges (that is, F might contain severalcopies of the same edge), the multiset of edges leaving the vertex set C Ď V p F q is denoted by δ F p C q . Similarly, denote the multiset of regular edges (i.e. excluding self-loops) in F incident to avertex v P V is denoted by δ F p v q . Denote the multiset of all edges (i.e. including self-loops) in F incident to a vertex v P V by δ F p v q , then the degree of v in F is denoted by deg F p v q : “ | δ F p v q| ,where every copy of the self-loop at v in F is counted twice. We will omit the subscript when F contains all the edges of G , that is, F “ E . For a vector x P R E , we denote the sum of the x -valueson the edges incident to v by x p δ p v qq . Note that the x -value of the self-loop at v is counted twicein x p δ p v qq . Let us denote the set of edges between two disjoint vertex sets A and B by δ p A, B q .Given two graphs or multigraphs H , H on the same vertex set, H ` H denotes the multigraphobtained by taking the union of the edge sets of H and H .Given a vector x P R S and a set Z Ď S , we use x p Z q “ ř s P Z x p s q . The lower integer part of x is denoted by t x u , so t x u p s q “ t x p s q u for every s P S . This notation extends to sets, so by t x u p Z q wemean ř s P Z t x u p s q . The support of x is denoted by supp p x q , that is, supp p x q “ t s P S | x p s q ‰ u .The difference of set B from set A is denoted by A ´ B “ t s P A | s R B u . We denote a single-element set t s u by s , and with a slight abuse of notation, we write A ´ s to indicate A ´ t s u . Letus denote the symmetric difference of two sets A and B by A (cid:52) B : “ p A ´ B q Y p B ´ A q and the characteristic vector of a set A by χ A .For a collection T of subsets of S , we call L Ď T an independent laminar system if for any pair X, Y P L : (i) they do not properly intersect, i.e. either X Ď Y , Y Ď X or X X Y “ H , and (ii)the characteristic vectors χ Z of the sets Z P L are independent over the real numbers. A maximal independent laminar system L with respect to T is an independent laminar system in T such that for The results in Theorem 5, Theorem 6 and Corollary 3 appeared in an unpublished work [4] by a superset of theauthors. In order to make the paper self-contained, we include all the details and proofs in this paper as well. Y P T ´ L the system L Yt Y u is not independent laminar. In other words, if we include any set Y from T ´ L , it will intersect at least one set Y from L , or χ Y can be given as a linear combination of t χ Z | Z P L u . Given a laminar system L and a set X Ď S , the set of maximal members of L lyinginside X is denoted by L max p X q , that is, L max p X q “ t Y P L | Y Ĺ X, E Y P L s.t. Y Ĺ Y Ĺ X u . Many-visits Path TSP.
Recall that in the
Many-visits Path TSP , we seek for a minimumcost s - t -walk P such that P visits each vertex v P V exactly r p v q times. Let r p V q “ ř v P V r p v q . Thesequence of the edges of P has length r p V q ´
1, which is exponential in the size of the input, as thevalues r p v q are stored using log r p V q space. For this reason, instead of explicitly listing the edgesin a walk (or tour) we always consider compact representations of the solution and the multigraphsthat arise in our algorithms. That is, rather than storing an p r p V q ´ q -long sequence of edges, forevery edge e we store its multiplicity z p e q in the solution. As there are at most n different edgesin the solution each having multiplicity at most max v P V r p v q , the space needed to store a feasiblesolution is O p n log r p V qq . Therefore, a vector z P Z E ě represents a feasible tour if supp p z q is aconnected subgraph of G and deg z p v q “ ¨ r p v q holds for all v P V ´ t s, t u and deg z p v q “ ¨ r p v q ´ v P t s, t u . (Note that each self-loop vv contributes 2 in the value deg p v q “ | δ p v q| .)Denote by P ‹ c,r,s,t an optimal solution for an instance p G, c, r, s, t q of the Many-visits PathTSP . Let us denote by P ‹ c, ,s,t an optimal solution for the single-visit counterpart of the problem,i.e. when r p v q “ v P V . Relaxing the connectivity requirement for solutions of the Many-visits Path TSP yields Hitchcock’s transportation problem [26], where supply and demandvertices t a v u v P V and t b v u v P V are given. The supplies for v P V ´ s are then defined by r p v q , thesupply of s by r p s q ´
1; the demand of each vertex v P V ´ t by r p v q and the demand of t by r p t q ´
1. Finally, by setting the transportation costs between a u and b v as c p uv q , the objectiveis to fulfill the supply and demand requirements by transporting goods from vertices t a v u v P V tovertices in t b v u v P V , while keeping the total cost minimal. The transportation problem is solvablein polynomial time using a minimum cost flow algorithm [13] and we denote an optimal solutionby TP ‹ c,r,s,t , where s and t denote the special vertices with decreased supply and demand value,respectively. Lemma 7.
Let TP ‹ c,r,s,t be an optimal solution to the Hitchcock transportation problem, where supply p v q` demand p v q is odd for v P t s, t u and it is even otherwise. Then TP ‹ c,r,s,t can be decomposedinto cycles and exactly one s - t -path.Proof. Any solution X to the transportation problem is essentially a multigraph that has an evendegree for vertices v P V ´ t s, t u , and an odd degree for v P t s, t u . Hence, because of a parityargument, there has to be an s - t -path U in X , possibly covering other vertices W Ă V ´ t s, t u .Vertices w P W have an even degree in U . Therefore, deleting the edges of U from X , all vertices v P V will have an even degree in the modified multigraph X . Thus X can be decomposed into aunion of (not necessarily distinct) cycles, and the lemma follows.The decomposition provided by the lemma is called a path-cycle representation. Such a repre-sentation can be stored as a path P and a collection C of pairs p C, µ C q , where each C is a simpleclosed walk (cycle) and µ C is the corresponding integer denoting the number of copies of C . Belowwe show that one can always calculate a path-cycle decomposition in polynomial time, and such adecomposition takes polynomial space. 6 emma 8. Let P c,r,s,t be a many-visits TSP path with endpoints s, t , and TP c,r,s,t be a transportationproblem solution with special vertices s, t . There is a path-cycle representation of P c,r,s,t and TP c,r,s,t ,both of which take space polynomial in n and log r p V q , and can be computed in time polynomialin n and log r p V q .Proof. We first show the proof for a many-visits TSP path P c,r,s,t . Let us first add an edge ts to P c,r,s,t , and denote the resulting multigraph by T . Observe that T is a many-visits TSP tour withthe same number of visits, since it is connected and the degree of every vertex v in T is 2 ¨ r p v q . Wecan now use the procedure ConvertToSequence by Grigoriev and van de Klundert [23], whichtakes the edge multiplicities of T , denoted by t x uv u u,v P V as input, and outputs a collection C ofpairs p C, µ C q . Here, C is a simple closed walk, and µ C is the corresponding integer denoting thenumber of copies of the walk C in P . Lastly, choose an arbitrary cycle C , such that ts P C ,and transform one copy of C into a path as follows. Let C : “ C , and remove the edge ts from C , resulting in an s - t -path P . Update µ C : “ µ C ´
1. Now p P , C q is a compact path-cyclerepresentation of P c,r,s,t .In every iteration, the procedure ConvertToSequence looks for a cycle C and removes each ofits occurrences from t x uv u u,v P V . The procedure stops when t x uv u u,v P V represents a graph withoutedges. This demonstrates that the input need not represent a connected graph in the first place,as the edge removals possibly make it disconnected during the process. Note that the only struc-tural difference between TP c,r,s,t and P c,r,s,t is that the underlying multigraph of TP c,r,s,t might bedisconnected. This means that the procedure ConvertToSequence can be applied to obtain acompact path-cycle representation of TP c,r,s,t the same way as in the case of P c,r,s,t .Finally, the number of cycles C in C can be bounded by O p n q (as removing all occurrencesof a cycle C sets at least one variable x uv to zero), and the algorithm has a time complexity of O p n q [23]. The edge insertion and deletion, and other graph operations during the process, canalso be implemented efficiently. This concludes the proof.From now on, we assume that the path-cycle decompositions appearing in this paper are storedin space polynomial in n and log r p V q .Let p P , C q be a compact path-cycle representation of a many-visits TSP path P c,r,s,t . One canobtain the explicit order of the vertices from p P , C q the following way: traverse the s - t -path P , andwhenever a vertex u is reached for the first time, traverse µ C copies of every cycle C containing u .Note that while the size of C is polynomial in n , the size of the explicit order of the vertices isexponential, hence the approaches presented in this paper consider symbolic rather than literaltraversals of many-visits TSP paths and tours. { -Approximation for Metric Many-visits Path TSP In this section we give a simple { -approximation algorithm for the metric Many-visits PathTSP that runs in polynomial time. The algorithm is as follows:
Theorem 1.
There is a polynomial-time { -approximation for the metric Many-visits PathTSP , that runs in time polynomial in n and log r .Proof. The algorithm is presented as Algorithm 1. Since P αc, ,s,t is connected, and P contains allthe edges of P αc, ,s,t , P is also connected. Let p P , C q be the compact path-cycle decomposition of TP ‹ c,r,s,t . The graph P thus consists of P αc, ,s,t and the cycles of C . The edges of P αc, ,s,t contribute a7 lgorithm 1 A polynomial-time p α ` q -approximation for metric Many-visits Path TSP . Input:
A complete undirected graph G “ p V, E q , costs c : E Ñ R ě satisfying the triangle inequality,requests r : V Ñ Z ě , distinct vertices s, t P V . Output: An s - t -path that visits each v P V exactly r p v q times. Calculate an α -approximate solution P αc, ,s,t for the single-visit metric Path TSP instance p G, c, , s, t q . Calculate an optimal solution TP ‹ c,r,s,t for the corresponding transportation problem, together with acompact path-cycle decomposition p P , C q , where C is a collection of pairs p C, µ C q . Let P be the union of P αc, ,s,t and µ C copies of every cycle C P C . Do shortcuts in P and obtain a solution P , such that P visits every city v exactly r p v q times (that is,deg P p v q “ ¨ r p v q for every vertex v P V ´ t s, t u , and deg P p v q “ ¨ r p v q ´ return P . degree of 1 in case of s and t , and 2 for v P V ´ t s, t u ; the cycles of C contribute degrees of 2 ¨ r p v q ´ v P t s, t u , and degrees of 2 ¨ r p v q or 2 ¨ r p v q ´ v P V ´ t s, t u . Let us denote the latter setby W , matching the notation in the proof of Lemma 7. The total degree of v in P is:2 ¨ r p v q ´ v P t s, t u , ¨ r p v q for v P W, and2 ¨ r p v q ` V ´ p W Y t s, t uq . As a direct consequence of the degrees and connectivity, P is an open walk that starts in s , visitsevery vertex v P V either r p v q or r p v q ` t . Since the edge costs are metric, wecan use shortcuts at the vertices w P V ´ p W Y t s, t uq to reduce their degrees by 2. We describethe procedure below. Shortcutting.
At Step 3, p P αc, ,s,t , C q denotes the compact path-cycle representation of P . Letus construct an auxiliary multigraph A on the vertex set V by taking the edges of P αc, ,s,t and eachcycle C from C exactly once. Note that parallel edges appear in A if and only if an edge appearsin multiple distinct cycles, or in the path P αc, ,s,t and at least one cycle C . Due to the construction, s and t have odd degree, while every other vertex has an even degree in A , which means that thereexist an Eulerian trail in A . Moreover, there are O p n q cycles [23], hence the total number of edgesin A is O p n q . Consequently, using Hierholzer’s algorithm, we can compute an Eulerian trail η in A in O p n q time [15, 25]. The trail η covers the edges of each cycle C once. Now an implicit orderof the vertices in the many-visits TSP path P is the following. Traverse the vertices of the Euleriantrail η in order. Every time a vertex u appears the first time, traverse all cycles C that containthe vertex µ C times. Denote this trail by η . It is easy to see that the sequence η is a sequence ofvertices that uses the edges of P αc, ,s,t once and the edges of each cycle C exactly µ C times, meaningthis is a feasible sequence of the vertices in the path P . Moreover, the order itself takes polynomialspace, as it is enough to store indices of O p n q vertices and O p n q cycles.Denote the surplus of visits of a vertex w P W by γ p w q : “ deg P p w q { ´ r p w q . In Step 4, we removethe last γ p w q occurrences of every vertex w P W from P by doing shortcuts: if an occurrence of w ispreceded by u and superseded by v in P , replace the edges uw and wv by uv in the sequence. Thiscan be done by traversing the compact representation of η backwards, and removing the vertex w from the last γ p w q cycles C p w q r p w q´ γ p w q` , . . . , C p w q r p w q . As ř w γ p w q can be bounded by O p n q , thisoperation makes O p n q new cycles, keeping the space required by the new sequence of vertices andcycles polynomial. Moreover, since the edge costs are metric, making shortcuts the way described8bove cannot increase the total cost of the edges in P . Finally, using a similar argument as inthe algorithm of Christofides, the shortcutting does not make the trail disconnected. The resultinggraph is therefore an s - t -walk P that visits every vertex v exactly r p v q times, that is, a feasiblesolution for the instance p G, c, r, s, t q .Note that by construction, P is such that the surplus of visits γ p w q equals to either 0 or 1.However, the same shortcutting procedure is used in Algorithm 2 later in the paper, where γ p w q can take higher values as well. Costs and complexity.
The cost of the path P constructed by Algorithm 1 equals to c p P q ď c p P αc, ,s,t q ` c p TP ‹ c,r,s,t q . Since c p TP ‹ c,r,s,t q is an optimal solution to a relaxation of the Many-visitsPath TSP , its cost is a lower bound to the cost of the corresponding optimal solution, P ‹ c,r,s,t .Since the cost of P αc, ,s,t is at most α times the cost of an optimal single-visit TSP path P ‹ c, ,s,t ,and c p P ‹ c, ,s,t q ď c p P ‹ c,r,s,t q holds for any r , Algorithm 1 provides an p α ` q -approximation for the Many-visits Path TSP . Using Zenklusen’s recent polynomial-time { -approximation algorithmon the single-visit metric Path TSP [58] in Step 1 yields the approximation guarantee of { statedin the theorem.The transportation problem in Step 2 can be solved in O p n log n q operations using the approachof Orlin [43] or its extension due to Kleinschmidt and Schannath [34]. Step 3 can also be performedin polynomial time [23], and the number of closed walks can be bounded by O p n q . Moreover, thetotal surplus of degrees in P is at most n ´
2, therefore the number of operations performed duringshortcutting in Step 4 is also bounded by O p n q . This proves that the algorithm has a polynomialtime complexity. Remark 1.
The TSP, as well as the
Path TSP can also be formulated for directed graphs, wherethe costs c are asymmetric. (Note that c still has to satisfy the triangle inequality, which impliesthe following bound for the self-loops: c p vv q ď max u ‰ v t c p vu q ` c p uv qu .) In a recent breakthrough,Svensson et al. [52] gave the first constant-factor approximation for the metric ATSP . In subsequentwork, Traub and Vygen [54] improved the constant factor to ` ε for any ε ą . Moreover, Feigeand Singh [14] proved that an α -approximation for the metric ATSP yields a p α ` ε q -approximationfor the metric Path-ATSP , for any ε ą . By combining these results with a suitable modificationof Algorithm 1, we can obtain a p ` ε q -approximation for the metric Many-visits ATSP , anda p ` ε q -approximation for any ε ą for the metric Many-visits Path-ATSP in polynomialtime. { -Approximation for the Metric Many-visits Path TSP In this section we show how to obtain a { -approximation for the metric Many-visits Path TSP .Our approach follows the general strategy of Zenklusen [58], but we need to make several crucialmodifications for the many-visits setting with exponentially large requests. This means that insteadof calculating a constrained spanning tree, we use the result in Theorem 4 to obtain a connected One can obtain a { -approximation for the metric Many-visits TSP by simply running Algorithm 1 for everypair p u, v q P V ˆ V and setting s “ u and t “ v , then choosing a solution whose cost together with the cost of theedge uv is minimal. However, Algorithm 1 can be simplified while maintaining the same approximation guarantee.This approach appeared in the unpublished manuscript [4] by a superset of the authors and has a simpler proof, asthe algorithm does not involve making shortcuts. P with a sufficiently large number of edges. Then compute a matching M so that allthe degrees in P ` M have the correct parity, and the cost of P ` M is at most { times the optimalcost. In order to ensure this bound, we have to enforce certain restrictions on P , similarly to thecomputation of the spanning tree in [58]. However, as we will show, the many-visits setting leadsto further challenges.For technical reasons, from now on we assume that the two endpoints s and t are different. Let usstart by defining the Held-Karp relaxation of the Many-visits Path TSP as min t c T x | x P P MVHK u ,where P MVHK denotes the following polytope:(2) P MVHK : “ $’’’’’&’’’’’% x P R E ě ˇˇˇˇˇˇˇˇˇˇˇ x p δ p C qq ě @ C Ă V, C ‰ H , | C X t s, t u| P t , u x p δ p C qq ě @ C Ď V, | C X t s, t u| “ x p δ p v qq “ ¨ r p v q @ v P V ´ t s, t u x p δ p s qq “ ¨ r p s q ´ x p δ p t qq “ ¨ r p t q ´ ,/////./////- The Q -join polytope (where Q Ď V is of even cardinality) is defined as follows:(3) P Ò Q -join : “ (cid:32) x P R E ě ˇˇ x p δ p C qq ě @ Q -cut C Ă V ( , where a Q -cut is a set C Ď V with | C X Q | odd.In the following we assume arbitrary but fixed parameters c, r and denote the optimal many-visits TSP path by P ‹ “ P ‹ c,r,s,t . Given a solution y of the linear program min t c T x | x P P MVHK u ,the vector y { is not necessarily in P Ò Q -join for every even set Q Ď V . Indeed, y only needs to havea load of 1 on s - t -cuts, therefore y { may violate some of the constraints of P Ò Q -join . This meansthat calculating a minimum cost perfect matching on an arbitrary even set Q Ď V might lead toa matching M with higher cost than c p P ‹ q { . Therefore, simply taking a solution P provided byTheorem 4 and a minimum cost matching M on the vertices with degrees having incorrect parity,then applying shortcuts would not lead to a { -approximation.To circumvent this problem, we would like to have a control over the vertices of P that takepart in the perfect matching phase of the algorithm. Similarly to Zenklusen [58], we calculate apoint q that is feasible for the Held-Karp relaxation of the Many-visits Path TSP , and that isonly needed for the analysis of the algorithm. Let odd p P q denote the vertices v with an odd degreein P . We need P and q to meet the following requirements:(R1) c p P q ď c p P ‹ q ,(R2) c p q q ď c p P ‹ q , and(R3) q { P P Ò Q P -join , where Q P : “ odd p P q (cid:52) t s, t u ,where c p q q stands for the cost of the vector q with respect to c , that is, c p q q “ ř e P E c p e q q p e q .Adding a shortest Q P -join J to the multigraph P results in a multigraph P where every vertex v P V ´ t s, t u has an even degree at least 2 ¨ r p v q , and every v P t s, t u has an odd degree at least2 ¨ r p v q ´
1. Due to (R3), the cost of the shortest Q P -join J satisfies c p J q ď c p q q { . Therefore, usingWolsey’s analysis for Christofides’ algorithm, the solution P obtained by taking the edges of P and applying shortcuts has cost at most { ¨ c p P ‹ q .Let x ˚ be an optimal solution to the Held-Karp relaxation of the Many-visits Path TSP :(4) min t c T x | x P P MVHK u .
10n order to obtain P and q that satisfy the conditions (R1)-(R3) above, we will calculateanother solution y P P MVHK with c p y q ď c p P ‹ q , and set q to be the midpoint between x ˚ and y , thatis, q “ x ˚ { ` y { . The construction of P also depends on y , the details are given in Algorithm 2 andthe reasoning in the proof of Theorem 2. Being a convex combination of two points in P MVHK , q isin P MVHK as well. We would like to ensure the existence of a multigraph P such that q { P P Ò Q P -join ,therefore we need to construct y accordingly.Let Q Ď V be a set of even cardinality. Recall the definition of P Ò Q -join at Equation (3), whichrequires that the load on Q -odd cuts is at least 1. Since q is in P MVHK , q p δ p C qq { ě s - t -cuts, i.e. any cuts C Ă V, C ‰ H with | C X t s, t u| P t , u . However, for s - t -cuts, the property y P P MVHK only implies y p δ p C qq ě
1. If in addition x ˚ p δ p C qq ě q p δ p C qq { ě P . If, however, x ˚ p δ p C qq ă P Ò Q P -join that correspond to s - t -cuts, where the x ˚ -load is strictly less than 3. Let us denote these cuts by B p x ˚ q , that is,(5) B p x ˚ q : “ t C Ď V | s P C, t R C, x ˚ p δ p C qq ă u . For a family B Ď t C Ď V | s P C, t R C u of s - t -cuts, we say that a point y P P MVHK is B -good , iffor every B P B we have(i) either y p δ p B qq ě y p δ p B qq “
1, and y is integral on the edges δ p B q .Therefore, if y P P MVHK is B p x ˚ q -good, then q “ x ˚ { ` y { satisfies q p δ p C qq { ě Q P -cut C .We will refer to a cut B satisfying condition (i) as a type (i) cut , and if it satisfies condition (ii)we will refer to it as a type (ii) cut . Note that condition (ii) translates to having a single edge f P δ p B q with y p f q “ y p e q “ e from δ p B q . The notion of B -goodness wasintroduced by Zenklusen for the elements of the polytope P HK in relation to metric Path TSP . Lemma 9.
The characteristic vector χ U of any many-visits s - t path U is B -good for any family B of s - t -cuts.Proof. The lemma easily follows from the fact that a many-visits s - t path U crosses any s - t -cut anodd number of times.We present our algorithm for the metric Many-visits Path TSP as Algorithm 2.In Step 4 of the algorithm, we use Theorem 4 to obtain a multigraph with additional propertiesbesides the degree requirements. In the single-visit counterpart of the problem, one can showthat even though x ˚ p δ p B qq ă y p δ p B qq “ B , the corresponding point q { “ x ˚ { ` y { is still in P Ò Q P -join . However, due to the possible parallel edges in P , the parityargument given by Zenklusen [58] does not hold, therefore we need to treat this case separately.For this reason we make the following distinction. Let E y denote the set of edges that correspondto type (ii) cuts in y , that is(6) E y : “ t e P E | D B P B : supp p y q X δ p B q “ e u . We let U ‹ p e q : “ e P E y , U ‹ p e q : “ `8 for the rest of the edges of supp p y q , and U ‹ p e q : “ e P E ´ supp p y q . Finally, we set L ‹ p e q : “ e P E . According to the claim11 lgorithm 2 A { -approximation algorithm for the metric Many-visits Path TSP
Input:
A complete undirected graph G “ p V, E q , costs c : E Ñ R ě satisfying the triangle inequality,requests r : V Ñ Z ě , distinct vertices s, t P V . Output: An s - t -path that visits each v P V exactly r p v q times. Calculate an optimal solution x ˚ to the Held-Karp relaxation of the Many-visits Path TSP , i.e. x ˚ : “ argmin t c T x | x P P MVHK u . Determine a B p x ˚ q -good solution y P P MVHK minimizing c T y . Let B Ă ¨ ¨ ¨ Ă B k denote the type (ii) cuts with respect to y . Compute a connected multigraph P on p V, supp p y qq such that a: each vertex v P V ´ t s, t u has degree at least 2 ¨ r p v q ´ b: each vertex v P t s, t u has degree at least 2 ¨ r p v q ´
2, and c: P contains no parallel edges leaving B i for i “ , . . . , k . Compute a minimum-cost matching M with respect to c on the vertices odd p P q (cid:52) t s, t u . Let P denote the many-visits path P ` M . Do shortcuts in P and obtain an s - t -walk P that visits each city v exactly r p v q times. return P . of Theorem 4, we can compute a multigraph P satisfying the conditions in Steps 4.a to 4.c, suchthat the cost of P is at most min t c T x | x P P CG p ρ ‹ , L ‹ , U ‹ qu , where the polytope P CG p ρ ‹ , L ‹ , U ‹ q depends on the instance p G, c, r, s, t q and can be written in the following form:(7) P CG p ρ ‹ , L ‹ , U ‹ q : “ $’’’’’’’’’&’’’’’’’’’% x P R E ě ˇˇˇˇˇˇˇˇˇˇˇˇˇˇˇ supp p x q is connected x p E q “ r p V q x p δ p v qq ě ¨ r p v q @ v P V ´ t s, t u x p δ p v qq ě ¨ r p v q ´ @ v P t s, t u ď x p e q ď @ e P E y ď x p e q ď `8 @ e P supp p y q ´ E y x p e q “ @ e P E ´ supp p y q ,/////////./////////- It is not difficult to see that y P P CG p ρ, L, U q , and thus(8) c p P q ď min ! c T x ˇˇˇ x P P CG p ρ ‹ , L ‹ , U ‹ q ) ď c T y , therefore c p P q ď c T y holds; this is one of the reasons behind restricting P to supp p y q . Moreover,according to Lemma 9, the inequality c T y ď c p P ‹ q holds, hence the bound c p P q ď c p P ‹ q follows.Now we have all the ingredients to prove our main theorem. Theorem 2.
There is a polynomial-time { -approximation for the metric Many-visits PathTSP . The algorithm runs in time polynomial in n and log r .Proof. Recall that Q P “ odd p P q (cid:52) t s, t u . First prove that q “ x ˚ { ` y { implies that q { is in P Ò Q P -join .For that we need to show that we calculated the solution y in a way that q satisfies q p δ p C qq { ě C Ă V for which | C X odd p P q (cid:52) t s, t u| is odd.Clearly, q P P MVHK , as q is the midpoint of two points from P MVHK . Therefore, for any Q P -cut C Ď V that is a not an s - t -cut, we have q p δ p C qq { ě Q P -cut C Ď V that is an s - t -cut and is not included in B p x ˚ q , we have x ˚ p δ p C qq ě
3, and so(9) 12 q p δ p C qq “ ` x ˚ p δ p C qq ` y p δ p C qq ˘ ě , y P P MVHK , and thus y p δ p C qq ě Q P -cuts C Ď V that are in B p x ˚ q . Since y is B p x ˚ q -good by construction,either y p δ p C qq ě
3, or y p δ p C qq “ y being integral on the edges δ p C q . If y p δ p C qq ě
3, then q p δ p C qq { ě x ˚ p δ p C qq ě q . If y p δ p C qq “ y is integral onthe edges δ p C q , it holds that y p e q “ δ p C q except for one f P δ p C q where y p f q “ P . Since supp p P q Ď supp p y q , and theload on an edge e P E y is at most 1 in P , the only edge of P with a positive load on δ p C q is f , andthat load is at most 1. Moreover, every cut has at least 1 load in P , which means | P X δ p C q| “ s - t -cut C Ď V with | δ P p C q| odd cannot be a Q P -cut because of the following:(10) | C X odd p P q| ” ÿ v P C | δ P p v q| p mod 2 q“ ¨ |t uv P P | u, v P C u| ` | δ P p C q| . Equation (10) implies that | C X odd p P q| is odd, and hence | C X Q P | “ | C X p odd p P q (cid:52) t s, t uq| is even because C is an s - t -cut. By the above, any cut of type (ii) partitions the vertices ofodd p P q (cid:52) t s, t u into two subsets of even cardinality. This means that no cut constraint of P Ò Q P -join requires a load of 1 for q { on C , and so q { P P Ò Q P -join holds.The cost of the matching M can therefore be bounded as follows:(11) c p M q ď c ´ q ¯ “ c T x ˚ ` c T y ď c p P ‹ q , since c T x ˚ ď c p P ‹ q . Thus, the multigraph obtained from P ` M has cost at most { ¨ c p P ‹ q , asclaimed. Shortcuts and complexity.
According to Theorem 4, every vertex v P V ´ t s, t u has degree atleast 2 ¨ r p v q ´
1, while vertices s and t have degrees at least 2 ¨ r p s q ´ ¨ r p t q ´ P . The matching M provides 1 additional degree for vertices with the wrongparity, therefore P will have an even degree at least r p v q for all v P V ´ t s, t u and an odd degreeat least r p v q ´ v P t s, t u . This means that P corresponds to a many-visits s - t -path that visitseach vertex v at least r p v q times, but possibly more. In Step 7 we proceed with taking shortcutsthe way described in Algorithm 1, so that P is a feasible solution to the Many-visits Path TSP instance p G, c, r, s, t q .Now we turn to the complexity analysis. The constraints of the Held-Karp relaxation (Equa-tion (4)) of the Many-visits Path TSP can be tested in time polynomial in n and log r p V q , hencecalculating x ˚ takes a poly p n, log r p V qq time as well [47, § n and log r p V q . According to Lemma 14, Step 2 also has polynomial time complexity,and By Theorem 4, Step 4 takes polynomial time and calculating a matching in Step 5 can be doneefficiently as well. Finally, since the number of edges in P is r p V q and the matching M contributesat most n { edges, we remove at most n { edges from P to obtain our solution P . This meansthat the number of operations performed in Step 7 can be bounded by O p n q . The claimed timecomplexity follows. For a cut C with y p δ p C qq “ y being integral on δ p C q , the term | T X δ p C q| in the proof of Theorem 2.1 ofZenklusen [58] corresponds to the term | δ P p C q| in Equation (10). Since the spanning tree T computed on supp p y q inthe algorithm of [58] cannot contain parallel edges, | T X δ p C q| has a value of 1 without enforcing an upper bound onthe edge e P δ T p C q for y p e q “ orollary 3. There is a { -approximation for the metric Many-visits TSP that runs in timepolynomial in n and log r .Proof. Let G “ p V, E q be a graph and p G, c, r q denote a metric Many-visits TSP instance. Choosean arbitrary vertex v P V , and construct a metric Many-visits Path TSP instance p ˆ G, ˆ c, ˆ r, s v , t v q as follows. Let ˆ G be an undirected graph on the vertex set ˆ V : “ V ´ v Y t s v , t v u and edge setˆ E : “ ˆ V ˆ ˆ V . We define ˆ c p s v t v q as the cost of a self loop at v , c p vv q , and ˆ c p s v u q “ ˆ c p t v u q : “ c p vu q for every vertex u P V ´ v . Moreover, the self-loops at v s and v t have cost c p vv q as well. It is easyto check that ˆ c satisfies the triangle inequality. Finally, set ˆ r p s v q : “ r p v q and ˆ r p t v q : “ Many-visits TSP instance p G, c, r q and the corresponding Many-visits Path TSP instance p ˆ G, ˆ c, ˆ r, s v , t v q can be reduced to each other. First let T be a solution to p G, c, r q . Choose an edge vu from T such that v ‰ u , and let P denote the many-visits s v - t v -pathobtained from T by deleting v , replacing each occurrence of any edge vw P T with a copy of s v w if w P V ´ t u, v u or with a copy of the loop on s v if w “ v , and replacing all but one occurrence ofthe edge vu P T with a copy of s v u , while one copy of vu is substituted by t v u . In other words, s v ‘inherits’ all copies of all edges and self-loops incident to v , except one copy of uv , and t v inheritsone copy of uv . This means that deg P p s v q “ ¨ r p v q ´ P p t v q “
1. Note that each edge of T is replaced by an edge of the same cost, and every vertex w P V ´ v has the same degree in T and P , hence deg P p w q “ ¨ r p w q . Therefore, P is a feasible solution to p ˆ G, ˆ c, ˆ r, s v , t v q of the samecost as T .Now consider a multigraph P that is a solution to p ˆ G, ˆ c, ˆ r, s v , t v q . Identify the vertices s v and t v , denote the new vertex by v , and introduce an edge vv for every copy of the edge s v t v in P .Let us denote the resulting multigraph by T . Since c p uv q “ ˆ c p s v u q “ ˆ c p t v u q for all u P V ´ v and c p vv q “ ˆ c p s v s v q “ ˆ c p t v t v q “ ˆ c p s v t v q , replacing s v and t v by v the way described above doesnot change the cost of the multigraph. Moreover, the degree of v in T is deg T p v q “ deg P p s v q ` deg P p t v q “ ¨ r p v q ´ ` “ ¨ r p v q . The degrees of vertices w P V ´ v remain unchanged, thus T is a feasible solution to p G, c, r q of the same cost as P .We therefore showed that for every solution of p G, c, r q there exists a solution of p ˆ G, ˆ c, ˆ r, s v , t v q with the same cost, and vice versa. Let now p G, c, r q be a metric Many-visits TSP instance.Pick an arbitrary vertex v P V , and consider the corresponding metric Many-visits Path TSP instance p ˆ G, ˆ c, ˆ r, s v , t v q , and obtain a { -approximation P using Algorithm 2. Identify s v and t v into v again, and substitute each copy of the edge s v t v in P by a copy of the self-loop vv . By theabove, the resulting multigraph T gives a { -approximation to the instance p G, c, r q . Remark 2.
Alternatively, one can directly obtain a { -approximation for the metric Many-visitsTSP by performing Step 4.a, Step 5 and Step 7 of Algorithm 2. More precisely, calculate a connectedmultigraph T with degrees at least ¨ r p v q ´ and cost at most the optimum using the result ofTheorem 4, then calculate a matching on the odd degree vertices and apply shortcuts. This procedurewas described by a superset of the authors [4]. Before we show how to calculate a B p x ˚ q -good point y P P MVHK , let us show that the number ofcuts in B is polynomial in n , and that the set B can be computed efficiently: Lemma 10.
Let q P P MVHK . Then the family B p q q of s - t -cuts of q -value strictly less than 3 satisfies | B p q q| ď n and can be computed in O p mn q time, where n : “ | V | and m : “ supp p q q .Proof. Let us define an auxiliary graph H “ p V, E q whose edge set E consists of the edges insupp p q q and an additional st edge. Let q H “ q ` χ st . Clearly, for non- s - t -cuts we have q H p δ H p C qq “ p δ p C qq , while for s - t -cuts we have q H p δ H p C qq “ q p δ p C qq ` ě st . Therefore, the family B p q q can be written as B p q q “ t C Ă V | s P C, t R C, q H p δ H p C qq ă u . The minimum cut has a load of at least 2, and due to Karger [32] the number of cuts with a load lessthan k times the minimum cut is at most O p n k q . Moreover, using an algorithm by Nagamochi etal. [42], we can enumerate the cuts of size at most k times the minimum cut in time O p m n ` n k m q .These results prove that the number of cuts in | B p q q| is O p n q , and that they can be enumeratedin time O p mn q . The dynamic program
Given a family B of s - t -cuts, our goal is to determine a minimum cost B -good point y P P MVHK . Weuse the dynamic programming approach introduced by Traub and Vygen [54] and improved uponby Zenklusen [58]. More precisely, the goal of the dynamic program is to determine which cuts in B are of type (i), and which ones are of type (ii). Our approach is constructive as the dynamicprogram also determines a point y that is B p x ˚ q -good.Consider a B p x ˚ q -good point y . Let B , . . . , B k denote the type (ii) s - t cuts in B with respectto y , that is, y p v i u i q “ v i u i P δ p B i q and y p e q “ e P δ p B i q ´ v i u i . It is notdifficult to see that these cuts necessarily form a chain (see e.g. [58]), thus we set the indices suchthat B Ĺ ¨ ¨ ¨ Ĺ B k . The endpoints of v i u i are named such that v i P B i , u i R B i . Furthermore, wedefine B : “ H , B k ` : “ V , u : “ s and v k ` : “ t for notational convenience. Note that u i and v i ` might coincide for some i “ , . . . , k ` B “ t s u and B k “ V ´ t t u , because the constraints of P MVHK enforce a degree of 1 on vertices s and t . In the many-visits setting, however, this is not necessarily true, as the instance possiblyrequires more than one visit for s or t .Assume for a moment that we knew the cuts B , . . . , B k and the edges v i u i , and we are lookingfor a B p x ˚ q -good point y P P MVHK such that among all cuts in B the cuts B , . . . , B k are preciselythose where (a) y is integral, and (b) y p δ p B i qq “ i “ , . . . , k . Then the B -good points y P P MVHK that satisfy these constraints (a) and (b) have the following properties for all i “ , . . . , k :(P1) y p v i u i q “ y p e q “ e P δ p B i q ´ v i u i ,(P2) the restriction of y to the vertex set B i ` ´ B i is a solution to the Held-Karp relaxation forthe Many-visits Path TSP with endpoints u i and v i ` , with the additional property that y p δ p B qq ě B P B such that B i Y u i Ď B Ď B i ` ´ v i ` .The dynamic program thus aims to find cuts B , . . . , B k while exploiting the properties (P1)and (P2) above. Formally, it is defined to find a shortest path on an auxiliary directed graph. Letus define the auxiliary directed graph H “ p N, A q with node set N , arc set A , and length function d : A Ñ R ě . The node set N is defined by N “ N ` Y N ´ , where N ` “ tp B, u q P B ˆ V | u R B u Y tpH , s qu , and N ´ “ tp B, v q P B ˆ V | v P B u Y tp V, t qu .15he arc set A is given by A “ A HK Y A E , where A HK “ (cid:32)` p B ` , u q , p B ´ , v q ˘ P N ` ˆ N ´ ˇˇ B ` Ď B ´ , u, v P B ´ ´ B ` ( , and A E “ (cid:32)` p B ´ , v q , p B ` , u q ˘ P N ´ ˆ N ` ˇˇ B ´ “ B ` ( .Finally, the lengths d : A Ñ R ě are defined as follows: d p a q “ c p vu q if a “ pp B, v q , p B, u qq P A E , OPT p LP p a qq if a P A HK , where OPT p LP p a qq denotes the optimum value ofmin c T x subject to x P P MVHK p B ´ ´ B ` , u, v q (LP p a q ) x p δ p B qq ě B P B s.t. B ` Ď B Ď B ´ ,u P B, v R B ,where a “ pp B ` , u q , p B ´ , v qq .For y -values across the cuts B P B so that B R t B , . . . , B k u , we require that y p δ p B qq ě Many-visits Path TSP instances between cuts B i and B i ` for every i “ , . . . , k . More precisely, such an instance isdefined on the subgraph of G induced on the vertex set B i ` ´ B i with distinguished vertices u i and v i ` , with the additional property that it has a y -load of at least 3 on each cut B P B with B i Ă B Ă B i ` , as shown in (LP p a q ). In case u ‰ v , the polytope P MVHK p W, u, v q is defined asfollows:(12) P MVHK p W, u, v q : “ $’’’’’’’’’&’’’’’’’’’% x P R E ě ˇˇˇˇˇˇˇˇˇˇˇˇˇˇˇ x p δ p C qq ě @ C Ă W, C ‰ H , | C X t u, v u| P t , u x p δ p C qq ě @ C Ă W, | C X t u, v u| “ x p δ p w qq “ ¨ r p w q @ w P W ´ t u, v u x p δ p u qq “ ¨ r p u q ´ x p δ p v qq “ ¨ r p v q ´ x p e q “ @ e P E ´ E r W s ,/////////./////////- Note that unlike in the single-visit case [58], we allow u being equal to s or v being equal to t inEquation (12), and the corresponding polytopes P MVHK p B , s, v q and P MVHK p V ´ B k , u k , t q are feasible.Let us now cover the case when for some index i P t , . . . , k u , vertices u i and v i ` coincide. In the single-visit
Path TSP , the solution is defined to be the all-zero vector if u i “ v i ` is theonly vertex in B i ` ´ B i , and there exists no solution otherwise. However, since we allow for avertex to be visited more than once (i.e. have a degree more than 2) in a solution to the Held-Karprelaxation for the Many-visits Path TSP , we use a different extension in our approach. Wedefine the corresponding subproblem as the Held-Karp relaxation for the
Many-visits TSP . Firstassume that u i ‰ s and u i ‰ t . Since y p v i u i q “ y p u i u i ` q “ u i in the Many-visits TSP subproblem is two less that in P MVHK , namely r p u i q ´ Note that the corresponding arc in H will have the form ` p B ` , w q , p B ´ , w q ˘ P A HK . u i “ s (or u i “ t ), then due to y p s u q “ y p v k t q “
1) the degree requirement for u i inthe subproblem is one less that in P MVHK , which also equals to r p u i q ´
2. Note that if u i “ v i ` there is no cut B P B with u i P B and v i ` R B , thus the linear program LP p a q has the formmin t c T x | x P P MVHK p W, u, u qu , where:(13) P MVHK p W, u, u q : “ $’’&’’% x P R E ě ˇˇˇˇˇˇˇˇ x p δ p C qq ě @ C Ă W, C ‰ H ,x p δ p w qq “ ¨ r p w q @ w P W ´ ux p δ p u qq “ ¨ r p u q ´ x p e q “ @ e P E ´ E r W s ,//.//- . If the requirement for u is r p u q “ | W | ą
1, the polytope P MVHK p W, u, u q is empty, andthus the linear program LP p a q has no solution. In this case the cost of the arc a is defined tobe infinity. Note however that if r p u q “ W “ t u u , the corresponding linear program has anon-zero solution, namely a vector that has value r p u q ´ uu ,and 0 otherwise.To find a B -good point with minimum cost c T y , we compute a shortest pH , s q – p V, t q pathwith respect to d in H ; due to Lemmas 9 and 11 this path has finite length. Let pH , s q , p B , v q , p B , u q , p B , v q , . . . , p B k , u k q , p V, t q be the nodes on this shortest path, and similarly as before,define B : “ H , u : “ s and B k ` : “ V, v k ` : “ t . By construction of H , we have B Ă B Ă ¨ ¨ ¨ Ă B k ` . Let x i P R E be an optimal solution to LP p a q for a “ pp B i , u i q , p B i ` , v i ` qq . Set(14) y : “ k ÿ i “ x i ` k ÿ i “ χ v i u i . By the definition of the lengths d in H , c T y necessarily equals the length (cid:96) ˚ of a shortest pH , s q – p V, t q path in H with respect to d . We now show that y computed in Equation (14) is indeed a B p x ˚ q -good point of minimum cost. Lemma 11.
The length (cid:96) ˚ of a shortest pH , s q – p V, t q path in H with respect to d satisfies (cid:96) ˚ ď min t c T z | z P P MVHK , z is B -good u .Proof. Let B z Ď B be the family of cuts B P B such that z p f q “ f P δ p B q ,and z p e q “ e P δ p B q ´ f . These are the sets in B that are type (ii) cuts withrespect to z , and also B z forms a chain: B Ă ¨ ¨ ¨ Ă B k holds, where B i P B z for i “ , . . . , k . Thecuts t B , . . . , B k u defines a partition of V into sets B : “ B , B : “ B ´ B , . . . , B k ´ : “ B k ´ B k ´ , B k : “ V ´ B k . For i P t , . . . , k u , let v i u i be the unique edge in δ p B i q where z p v i u i q “
1, so that v i P B i and u i R B i .Consider the path along nodes p B , u q , p B , v q , p B , u q , . . . , p B k ` , v k ` q . It suffices to showthat the length (cid:96) of the path is at most c T z . For each i P t , . . . , k u , the vector z i P R E is definedto be the restriction of z to E r B i ` ´ B i s . Assume for a moment that z i is a feasible solution ofLP p a q with a “ pp B i , u i q , p B i ` , v i ` qq . Then the total length (cid:96) is equal to ř ki “ c T x i ` ř ki “ c p v i u i q by definition, which is at most ř ki “ c T z i ` ř ki “ c p v i u i q “ c T z . Since (cid:96) ˚ is minimum among allpossible (cid:96) ’s, we get (cid:96) ˚ ď (cid:96) ď c T z .Since z is B -good, and z i p δ p B qq “ z p δ p B qq for any cut B with B i Ĺ B Ĺ B i ` , that means z i p δ p B qq “ z p δ p B qq ě
3. It remains to show that z i P P MVHK p B i , u i , v i ` q or z i P P MVHK p B i , u i , u i q follows for i “ , . . . , k . 17 istinct endpoints. Let us start with the case when u i ‰ v i ` . By definition, z i p e q “ z p e q if bothendpoints of e are in B i and z i p e q “ w P B i such that w R t u i , v i ` u , z i p δ p w qq “ z p δ p w qq “ ¨ r p w q . First assume that u i ‰ s and v i ` ‰ t . Both of the endpoints u i and v i ` have a total z -value of 1 on the edge set δ p B i q due to z p v i u i q “ z p v i ` u i ` q “ z i -value on the edges incident to the endpoints is z i p δ p u i qq “ z p δ p u i qq ´ “ ¨ r p u i q ´ z i p δ p v i ` qq “ z p δ p v i ` qq ´ “ ¨ r p v i ` q ´
1. If u i “ s or v i ` “ t , their z i -values equal totheir z -values, that is 2 ¨ r p s q ´ ¨ r p t q ´
1, respectively. The degree constraints are thereforesatisfied.Finally, we have to show that for a cut C Ď B i , z i p δ p C qq ě C is a u i - v i ` -cut,and z i p δ p C qq ě C does not separate u i and v i ` . For the single-visit variant, the proof goesby showing that z i is in the spanning tree polytope of G r B i s , using the fact that z is in thespanning tree polytope of G and the degree constraints of v P B i [58]. However, these terms do notimmediately generalize to the many-visits setting, so we show that the connectivity of z i followsfrom the properties of z . Non- u i - v i ` -cuts: First let us consider the cuts that do not separate u i and v i ` , and provethat the total z i -value across these cuts is at least 2. We may assume that C Ă B i does not containneither u i nor v i ` throughout this paragraph. In case u i , v i ` P C , we can take B i ´ C and weare done, as z i p δ p C qq “ z i p δ p B i ´ C qq “ z p δ p C qq ě
2, yielding z i p δ p C qq ě
2. Assume first that u i ‰ s and v i ` ‰ t , and let C Ă B i . Then z i p δ p C qq ě C is a non- s - t -cut andthus z p δ p C qq ě
2. Now let u i be equal to s , and let C Ă B i be a cut that does not contain either s or v . Likewise, z p δ p C qq ě C is a non- s - t -cut and thus z p δ p C qq ě
2. The argument for v i ` “ t goes the same way. u i - v i ` -cuts: Here we have to prove that if C is a u i - v i ` -cut, then z i p δ p C qq ě u i ‰ s and v i ` ‰ t . Without the loss of generality, we may assume that u i P C .Since C is a non- s - t -cut, z p δ p C qq ě
2. If we account for z p v i u i q “
1, then z i p δ p C qq ě u i R C and v i ` P C . Assume now that u i “ s and u i P C ;then C is an s - t -cut, so z p δ p C qq ě
1. Moreover, v i ` R C , so v i ` u i ` R δ p C q , therefore z i p δ p C qq “ z p δ p C qq ě
1. If u i “ s and v i ` P C , then z i p δ p C qq ě z p δ p C qq ´ z p v i ` u i ` q ě
1, since C is anon- s - t -cut and thus z p δ p C qq ě
2. The case v i ` “ t can be proved similarly. Same endpoints.
Now we cover the case when u i “ v i ` . We need to prove that z i P P MVHK p B i , u i , u i q ,as defined in Equation (13). The argument about the degrees is analogous to the case above, thefact that z i p δ p w qq “ ¨ r p w q directly follows for vertices w P B i ´ u i . Moreover, if u i R t s, t u , theendpoint u i has a z -load of 2 on δ p B i q because of z p v i u i q “ z p v i ` u i ` q “ z p u i u i ` q “ z i p δ p u i qq “ z p δ p u i qq ´ “ ¨ r p u i q ´
2, as desired. If u i P t s, t u , then z p δ p u i qq “ ¨ r p u i q ´ z p s u q “ z p v k t q “ u i “ s or u i “ t , respectively. In both cases this means z i p δ p u i qq “ z p δ p u i qq ´ “ ¨ r p u i q ´ C Ă B i be a cut. We can assume that u i R C , otherwisewe take B i ´ C , and we are done. If u i R C , then z i p δ p C qq “ z p δ p C qq ě C is a non- s - t -cut.The proof is complete. Lemma 12. y P P MVHK .Proof.
To prove this claim, we use the properties of x i . We again distinguish two cases, based onwhether the endpoints of subproblem LP p a q are the same or different.18 egree constraints. First, consider the indices i with u i ‰ v i ` . By definition, the vector x i satisfies x i P P MVHK p B i ` ´ B i , u i , v i ` q , meaning that it is a solution to the Held-Karp relaxationfor Many-visits Path TSP in the induced subgraph G r B i ` ´ B i s with endpoints u i and v i ` .Recall that y “ ř ki “ x i ` ř ki “ χ v i u i by definition. For a v P B i ` ´ B i , the value y p δ p v qq is equalto 1. x i p δ p v qq if v “ u i “ s or v “ v i ` “ t ,2. x i p δ p v qq ` “ ¨ r p v q if v R t s, t u and v “ u i or v “ v i ` for some i , due to the edge v i u i or v i ` u i ` , respectively; and3. x i p δ p v qq “ ¨ r p v q otherwise.By the above, y p δ p s qq “ ¨ r p s q ´ y p δ p t qq “ ¨ r p t q ´
1, and y p δ p v qq “ ¨ r p t q for v R t s, t u ,therefore the degree constraints are satisfied for all v P V .If u i “ v i ` , the value y p δ p v qq is equal to1. x i p δ p v qq ` v “ s or v “ t , because of the edge vu or v k v , respectively,2. x i p δ p v qq ` “ ¨ r p v q if v R t s, t u and v “ u i “ v i ` for some i , due to the edge v i u i and v i ` u i ` ; and3. x i p δ p v q “ ¨ r p v q otherwise.Again, we get y p δ p s qq “ ¨ r p s q ´ y p δ p t qq “ ¨ r p t q ´
1, and y p δ p v qq “ ¨ r p t q for v R t s, t u ,therefore the degree constraints are satisfied for all v P V . Cut constraints.
It remains to show that y satisfies the cut constraints. As in the proof ofLemma 11, instead of building on a spanning subgraph polytope, we directly prove that the cutconstraints hold. As before, the cuts t B , . . . , B k u define a partition of V into sets B : “ B , B : “ B ´ B , . . . , B k ´ : “ B k ´ B k ´ , B k : “ V ´ B k .Let us first consider the value of y on s - t -cuts. For B i P t B , . . . , B k u the y -load on δ p B i q equalsto 1 due to the edge v i u i , therefore it satisfies the constraint y p δ p B i qq ě
1. If C is a s - t -cut suchthat C R t B , . . . , B k u , then there is at least one index i P t , . . . , k u , such that both B i X C and B i ´ C are not empty. In other words, there is at least one vertex from B i on both sides of the cut C . If u i ‰ v i ` , x i satisfies the constraints of LP for a “ pp B i , u i q , p B i ` , v i ` qq , we have x i P P MVHK p B i , u i , v i ` q . That means x i has a load of at least 1 on edges leaving every proper subset of B i , including B i X C , and x i p δ p B i X C qq ě y p δ p B i X C qq ě
1, which yields y p δ p C qq ě u i “ v i ` , then x i P P MVHK p B i , u i , u i q , which means that y p δ p B i ´ C, B i X C qq ě
2, so y p δ p C qq ě C is a non- s - t -cut, we distinguish the following three cases. Note that in neither of the casesis B Ď C or B k Ď C a possibility, as that would make C an s - t -cut. If C Ĺ B i for some i , and C is a u i - v i ` -cut so that u i P C (or v i ` P C ), then y p δ p C qq has at least 1 load from the fact that x i P P MVHK p B i , u i , v i ` q , and 1 load from the edge v i ´ u i (or v i u i ` ). If C is not a u i - v i ` -cut, and u i , v i ` are in C , then y p δ p C qq ě v i u i and v i ` u i ` ; while if u i , v i ` are not in C then y p δ p C qq ě x i P P MVHK p B i , u i , v i ` q .Note that if u i “ v i ` , C can only be a non- u i - v i ` -cut. In that case x i p δ p C qq ě x i P P MVHK p B i , u i , u i q , and thus y p δ p C qq ě f C “ Ť i P I B i for some I Ă t , . . . , k u , let us define i min : “ min t i | B i Ă C u and i max : “ max t i | B i Ă C u . Then y p δ p C qq ě y p v i min ´ u i min q ` y p v i max u i max ` q “ Else there exists a set B i such that C X B i ‰ H and C Ę B i and B i Ę C hold , then letus define i min and i max as follows: i min : “ min t i | C X B i ‰ H , C Ę B i , B i Ę C u ,i max : “ max t i | C X B i ‰ H , C Ę B i , B i Ę C u . Suppose that i min ‰ i max . Then, δ p C q has at least 1 y -load on δ p B i min X C, B i min ´ C q , as well asat least 1 y -load between on δ p B i max X C, B i max ´ C q , thus y p δ p C qq ě
2. In case of i min “ i max ,there must exist another index i ‰ i min such that B i Ă C (as otherwise we are back in one of theprevious two cases), in which case there is at least one edge e in δ p C X B i q such that y p e q “ v i u i or v i ` u i ` or both); in total y p δ p C qq ě Lemma 13. y is B -good.Proof. The proof follows the lines of the corresponding proof of Lemma 3.3 of Zenklusen [58]: therethe claim can be deduced from cut constraints of P HK , while in our case it follows from those ofthe polytope P MVHK . Nevertheless, we include the full proof here for the sake of completeness.For i P t , . . . , k u , we have by construction of y that y p v i u i q “ y p e q “ e P δ p B i q . This means that all cuts B i satisfy (ii) of the definition of B -goodness, i.e. the y -valueis 1 and y is integral. Let us show that for any other cut B P B ´ t B , . . . , B k u , the y -load satisfies(i) of the definition.First suppose that t B , . . . , B k uY B is not a chain, in this case there is some index j P t , . . . , k u ,such that neither B Ď B j nor B j Ď B is true. Hence y p δ p B qq ` “ y p δ p B qq ` y p δ p B j qqě y p δ p B ´ B j qq ` y p δ p B j ´ B qqě . The first line follows from y p δ p B j qq “
1, this was shown at the beginning of the proof. The firstinequality holds by the cut functions C Ñ y p δ p C qq being symmetric and submodular. Since B and B j are s - t -cuts, B ´ B j and B j ´ B are non- s - t -cuts, and the y -load of both of these cuts is at least2, hence the second inequality follows.Suppose that t B , . . . , B k u Y B is a chain, then there is an index j such that B j Ĺ B Ĺ B j ` .If u j P B and v j ` R B , then x j p δ p B qq ě p a q , where a “ pp B j , u j q , p B j ` , v j ` qq . Since y ě x j holds for all j component-wise, y p δ p B qq ě u j R B and v j ` P B , then both the edges v j u j and v j ` u j ` are in δ p B q ;moreover x j p δ p B qq ě B is a u j - v j ` -cut, therefore y p δ p B qq ě x j p δ p B qq ` y p v j u j q ` y p v j ` u j ` q ě . Finally, if B is not an u j - v j ` -cut, x j p δ p B qq ě x j P P MVHK p B j ` ´ B j , u j , v j ` q . Moreover,either u i v i or u i ` v i ` is an edge in δ p B q , depending on whether u i and v j ` are in B or not; bothof the possibilities imply y p δ p B qq ě
3. 20 emma 14.
Let B Ď t C Ď V | s P C, t R C u . One can determine in time polynomial in | B | andthe input size of p G, s, t, c q a B -good point y P P MVHK of minimum cost.Proof.
The number of nodes and arcs in H are polynomial in | B | . Calculating a shortest path on H takes time polynomial in | H | . The feasibility of the linear programs (LP p a q ) can be checked intime poly p n, log r, | B |q , therefore an optimal solution can also be found, using the ellipsoid method,in time polynomial in n , log r and | B | (see the discussion in § Remark 3.
It is worth considering how Algorithm 2 proceeds when applied to the single-visitTSP, that is, when r p v q “ for each v P V . The output of Algorithm 3 in Step 4 is then aconnected multigraph with r p V q ´ “ n ´ edges. This means that each vertex v has degree at least ¨ r p v q ´ “ , which boils down to a connected graph with n ´ edges, therefore a spanning treeon G with the additional properties (R1)-(R3). Thus Algorithm 2 performs the same operations,as the algorithm of Zenklusen [58] for the Path TSP . In what follows, we make use of some basic notions and theorems of the theory of generalizedpolymatroids. For background, see for example the paper of Frank and Tardos [18] or Chapter 14in the book by Frank [17].Given a ground set S , a set function b : 2 S Ñ Z is submodular if b p X q ` b p Y q ě b p X X Y q ` b p X Y Y q holds for every pair of subsets X, Y Ď S . A set function p : 2 S Ñ Z is supermodular if ´ p issubmodular. As a generalization of matroid rank functions, Edmonds introduced the notion ofpolymatroids [12]. A set function b is a polymatroid function if b pHq “ b is non-decreasing, and b is submodular.We define P p b q : “ t x P R S ě | x p Y q ď b p Y q for every Y Ď S u . The set of integral elements of P p b q is called a polymatroidal set . Similarly, the base polyma-troid B p b q is defined by B p b q : “ t x P R S | x p Y q ď b p Y q for every Y Ď S, x p S q “ b p S qu . Note that a base polymatroid is just a facet of the polymatroid P p b q . In both cases, b is called the border function of the polyhedron. Although non-negativity of x is not assumed in the definitionof B p b q , this follows by the monotonicity of b and the definition of B p b q : x p s q “ x p S q ´ x p S ´ s q ě b p S q ´ b p S ´ s q ě s P S . The set of integral elements of B p b q is called a basepolymatroidal set . Edmonds [12] showed that the vertices of a polymatroid or a base polymatroidare integral, thus P p b q is the convex hull of the corresponding polymatroidal set, while B p b q is theconvex hull of the corresponding base polymatroidal set. For this reason, we will call the sets ofintegral elements of P p b q and B p b q simply a polymatroid and a base polymatroid.21assin [24] introduced polyhedra bounded simultaneously by a non-negative, monotone non-decreasing submodular function b over a ground set S from above and by a non-negative, monotonenon-decreasing supermodular function p over S from below, satisfying the so-called cross-inequality linking the two functions: b p X q ´ p p Y q ě b p X ´ Y q ´ p p Y ´ X q for every pair of subsets X, Y Ď S .
We say that a pair p p, b q of set functions over the same ground set S is a paramodular pair if p pHq “ b pHq “ p is supermodular, b is submodular, and they satisfy the cross-inequality.The slightly more general concept of generalized polymatroids was introduced by Frank [16]. A generalized polymatroid , or g-polymatroid is a polyhedron of the form Q p p, b q : “ (cid:32) x P R S | p p Y q ď x p Y q ď b p Y q for every Y Ď S ( , where p p, b q is a paramodular pair. Here, p p, b q is called the border pair of the polyhedron. It isknown (see e.g. [17]) that a g-polymatroid defined by an integral paramodular pair is a non-emptyintegral polyhedron.A special g-polymatroid is a box β p L, U q “ t x P R S | L ď x ď U u where L : S Ñ Z Y t´8u , U : S Ñ Z Y t8u with L ď U . Another illustrious example is base polymatroids. Indeed, givena polymatroid function b with finite b p S q , its complementary set function p is defined for X Ď S by p p X q : “ b p S q ´ b p S ´ X q . It is not difficult to check that p p, b q is a paramodular pair and that B p b q “ Q p p, b q . Theorem 14.3.9 (Frank [17]).
The intersection Q of a g-polymatroid Q “ Q p p, b q and a box β “ β p L, U q is a g-polymatroid. If L p Y q ď b p Y q and p p Y q ď U p Y q hold for every Y Ď S , then Q is non-empty, and its unique border pair p p , b q is given by p p Z q “ max t p p Z q ´ U p Z ´ Z q ` L p Z ´ Z q | Z Ď S u ,b p Z q “ min t b p Z q ´ L p Z ´ Z q ` U p Z ´ Z q | Z Ď S u . (15)Given a g-polymatroid Q p p, b q and Z Ă S , by deleting Z Ď S from Q p p, b q we obtain a g-polymatroid Q p p, b qz Z defined on set S ´ Z by the restrictions of p and b to S ´ Z , that is, Q p p, b qz Z : “ t x P R S ´ Z | p p Y q ď x p Y q ď b p Y q for every Y Ď S ´ Z u . In other words, Q p p, b qz Z is the projection of Q p p, b q to the coordinates in S ´ Z .Extending the notion of contraction from matroids to g-polymatroids is not immediate. A setcan be naturally identified with its characteristic vector, that is, in the case of matroids contractionis basically an operation defined on 0 ´ I is an independent of M { Z if and only if F Y I is independent in M for any maximal independent set F of Z .With this property in mind, we define the g-polymatroid obtained by the contraction of anintegral vector z P Q p p, b q to be the polymatroid Q p p , b q : “ Q p p, b q{ z on the same ground set S with the border functions p p X q : “ p p X q ´ z p X q b p X q : “ b p X q ´ z p X q . p is obtained as the difference of a supermodular and a modular function, implyingthat it is supermodular. Similarly, b is submodular. Moreover, p pHq “ b pHq “
0, and b p X q ´ p p Y q “ b p X q ´ z p X q ´ p p Y q ` z p Y qě b p X ´ Y q ` p p Y ´ X q ´ z p X ´ Y q ` z p Y ´ X q“ b p X ´ Y q ´ p p Y ´ X q , hence p p , b q is indeed a paramodular pair. The main reason for defining the contraction of anelement z P Q p p, b q is shown by the following lemma. Lemma 15.
Let Q p p , b q be the polymatroid obtained by contracting z P Q p p, b q . Then x ` z P Q p p, b q for every x P Q p p , b q .Proof. Let x P Q p p , b q . By definition, this implies p p Y q ď x p Y q ď b p Y q for Y Ď S . Thus p p Y q “ p p Y q ` z p Y q ď x p Y q ` z p Y q ď b p Y q ` z p Y q “ b p Y q , concluding the proof. The aim of this section is to prove Theorem 5 and Theorem 6. Theorem 5 extends the resultof Kir´aly et al. [33] from matroids to g-polymatroids. However, adapting their algorithm is notimmediate due to the following major differences. A crucial step of their approach is to relax theproblem by deleting a constraint corresponding to a hyperedge ε with small g p ε q value. This stepis feasible when the solution is a 0 ´ Bounded Degree g-poly-matroid Element Problem :minimize ÿ s P S c p s q x p s q subject to p p Z q ď x p Z q ď b p Z q @ Z Ď S (LP) f p ε q ď ÿ s P ε m ε p s q x p s q ď g p ε q @ ε P E Although the program has an exponential number of constraints, it can be separated in poly-nomial time using submodular minimization [28, 41, 46]. Algorithm 3 generalizes the approach byKir´aly et al. [33]. We iteratively solve the linear program, delete elements which get a zero value inthe solution, update the solution values and perform a contraction on the polymatroid, or removeconstraints arising from the hypergraph. In the first round, the bounds on the coordinates solelydepend on p and b , while in the subsequent rounds the whole problem is restricted to the unit cube. Theorem 5.
There is an algorithm for the
Bounded Degree g-polymatroid Element withMultiplicities problem which returns an integral element x of Q p p, b q of cost at most the optimum lgorithm 3 Approximation algorithm for the
Bounded Degree g-polymatroid Elementwith Multiplicities problem.
Input:
A g-polymatroid Q p p, b q on ground set S , cost function c : S Ñ R , a hypergraph H “ p S, E q ,lower and upper bounds f, g : E Ñ Z ě , multiplicities m ε : S Ñ Z ě for ε P E satisfying m ε p s q “ s P S ´ ε . Output: z P Q p p, b q of cost at most OPT LP , violating the hyperedge constraints by at most 2∆ ´ Initialize z p s q Ð s P S . while S ‰ H do Compute a basic optimal solution x for (LP).(Note: starting from the second iteration, 0 ď x ď a: Delete any element s with x p s q “
0. Update each hyperedge ε Ð ε ´ s and m ε p s q Ð Q p p, b q Ð Q p p, b qz s by deletion. b: For all s P S update z p s q Ð z p s q ` t x u p s q .Apply polymatroid contraction Q p p, b q Ð Q p p, b q{ t x u , that is, redefine p p Y q : “ p p Y q ´ t x u p Y q and b p Y q : “ b p Y q ´ t x u p Y q for every Y Ď S .Update f p ε q Ð f p ε q ´ ÿ s P ε m ε p s q t x u p s q and g p ε q Ð g p ε q ´ ÿ s P ε m ε p s q t x u p s q for each ε P E . c: If m ε p ε q ď ´
1, let E Ð E ´ ε . d: if it is the first iteration then Take the intersection of Q p p, b q and the unit cube r , s S , that is, p p Y q : “ max t p p Y q´| Y ´ Y | | Y Ď S u and b p Y q : “ min t b p Y q ` | Y ´ Y | | Y Ď S u for every Y Ď S . return z value such that f p ε q ´ ` ď ř s P ε m ε p s q x p s q ď g p ε q ` ´ for each ε P E , where ∆ “ max s P S t ř ε P E : s P ε m ε p s qu . The run time of the algorithm is polynomial in n and log ř ε p f p ε q ` g p ε qq .Proof. Our algorithm is presented as Algorithm 1.
Correctness.
First we show that if the algorithm terminates then the returned solution z satisfiesthe requirements of the theorem. In a single iteration, the g-polymatroid Q p p, b q is updated to p Q p p, b qz D q{ t x u , where D “ t s : x p s q “ u is the set of deleted elements. In the first iteration, theg-polymatroid thus obtained is further intersected with the unit cube. By Lemma 15, the vector x ´ t x u restricted to S ´ D remains a feasible solution for the modified linear program in the nextiteration. Note that this vector is contained in the unit cube as its coordinates are between 0 and 1.This remains true when a lower degree constraint is removed in Step 3.c as well, therefore the costof z plus the cost of an optimal LP solution does not increase throughout the procedure. Hence thecost of the output z is at most the cost of the initial LP solution, which is at most the optimum.By Lemma 15, the vector x ´ t x u ` z is contained in the original g-polymatroid, although it mightviolate some of the lower and upper bounds on the hyperdeges. We only remove the constraintscorresponding to the lower and upper bounds for a hyperedge ε when m ε p ε q ď ´
1. As theg-polymatroid is restricted to the unit cube after the first iteration, these constraints are violatedby at most 2∆ ´
1, as the total value of ř s P ε m ε p s q z p s q can change by a value between 0 and 2∆ ´ Termination.
Suppose, for sake of contradiction, that the algorithm does not terminate. Thenthere is some iteration after which none of the simplifications in Steps 3.a to 3.c can be performed.24his implies that for the current basic LP solution x it holds 0 ă x p s q ă s P S and m ε p ε q ě
2∆ for each ε P E . We say that a set Y is p-tight (or b-tight ) if x p Y q “ p p Y q (or x p Y q “ b p Y q ), and let T p “ t Y Ď S : x p Y q “ p p Y qu and T b “ t Y Ď S : x p Y q “ b p Y qu denote thecollections of p -tight and b -tight sets with respect to solution x .Let L be a maximal independent laminar system in T p Y T b . Claim 16. span pt χ Z | Z P L uq “ span pt χ Z | Z P T p Y T b uq Proof of Claim 16.
The proof uses an uncrossing argument. Let us suppose indirectly that thereis a set R from T p Y T b for which χ R R span pt χ Z | Z P L uq . Choose this set R so that it isincomparable to as few sets of L as possible. Without loss of generality, we may assume that R P T p . Now choose a set T P L that is incomparable to R . Note that such a set necessarily existsas the laminar system is maximal. We distinguish two cases. Case 1. T P T p . Because of the supermodularity of p , we have x p R q ` x p T q “ p p R q ` p p T q ď p p R Y T q ` p p R X T q ď x p R Y T q ` x p R X T q“ x p R q ` x p T q , hence equality holds throughout. That is, R Y T and R X T are in T p as well. In addition, since χ R ` χ T “ χ R Y T ` χ R X T and χ R is not in span pt χ Z | Z P L uq , either χ R Y T or χ R X T is not containedin span pt χ Z | Z P L uq . However, both R Y T and R X T are incomparable with fewer sets of L than R , which is a contradiction. Case 2. T P T b . Because of the cross-inequality, we have x p T q ´ x p R q “ b p T q ´ p p R q ě b p T z R q ´ p p R z T q ě x p T z R q ´ x p R z T q“ x p T q ´ x p R q , implying T z R P T b and R z T P T p . Since χ R ` χ T “ χ R z T ` χ R z T ` χ R Y T and χ R is not inspan pt χ Z | Z P L uq , one of the vectors χ R z T , χ R z T and χ R Y T is not contained in span pt χ Z | Z P L uq .However, any of these three sets is incomparable with fewer sets of L than R , which is a contradic-tion.The case when R P T b is analogous to the above. This completes the proof of the Claim. ♦ We say that a hyperedge ε P E is tight if f p ε q “ ř s P ε m ε p s q x p s q or g p ε q “ ř s P ε m ε p s q x p s q .As x is a basic solution, there is a set E Ď E of tight hyperedges such that t m ε | ε P E u Yt χ Z | Z P L u are linearly independent vectors with | E | ` | L | “ | S | .We derive a contradiction using a token-counting argument. We assign 2∆ tokens to eachelement s P S , accounting for a total of 2∆ | S | tokens. The tokens are then redistributed in sucha way that each hyperedge in E and each set in L collects at least 2∆ tokens, while at least oneextra token remains. This implies that 2∆ | S | ą | E | ` | L | , leading to a contradiction.We redistribute the tokens as follows. Each element s gives ∆ tokens to the smallest mem-ber in L it is contained in, and m ε p s q tokens to each hyperedge ε P E it is contained in. As ř ε P E : s P ε m ε p s q ď ∆ holds for every element s P S , thus we redistribute at most 2∆ tokens perelement and so the redistribution step is valid. Now consider any set U P L . Recall that L max p U q consists of the maximal members of L lying inside U . Then U ´ Ť W P L max p U q W ‰ H , as otherwise χ U “ ř W P L max p U q χ W , contradicting the independence of L . For every set Z in L , x p Z q is aninteger, meaning that x p U ´ Ť W P L max p U q W q is an integer. But also 0 ă x p s q ă s P S ,25hich means that U ´ Ť W P L max p U q W contains at least 2 elements. Therefore, each set U in L receives at least 2∆ tokens, as required. By assumption, m ε p ε q ě
2∆ for every hyperedge ε P E ,which means that each hyperedge in E receives at least 2∆ tokens, as required.If ř ε P E : s P ε m ε p s q ď ∆ holds for any s P S or L max p S q is not a partition of S , then an extratoken exists. Otherwise, ř ε P E m ε “ ∆ ¨ χ S “ ∆ ¨ ř W P L max p S q χ W , contradicting the independenceof t m ε | ε P E u Y t χ Z | Z P L u . Time complexity.
Solving an LP, as well as removing a hyperedge in Step 3.a or removing anelement from a hyperedge in Step 3.c can be done in polynomial time. In Steps 3.b and 3.d, wecalculate the value of the current functions p and b for a set Y only when it is needed during theellipsoid method. We keep track of the vectors t x u that arise during contraction steps (there is onlya polynomial number of them), and every time a query for p or b happens, it takes into accountevery contraction and removal that occurred until that point.Step 3.a can be repeated at most | S | times, while Step 3.c can be repeated at most | E | times.Starting from the second iteration, we are working in the unit cube. That is, when Step 3.b addsthe integer part of a variable x p s q to z p s q and reduces the problem, then the given variable will be 0in the next iteration and so element s is deleted. This means that the total number of iterations ofStep 3.b is at most O p| S |q .Now we consider case when only lower or only upper bounds are given. Theorem 6.
There is an algorithm for
Lower Bounded Degree g-polymatroid Elementwith Multiplicities which returns an integral element x of Q p p, b q of cost at most the optimumvalue such that f p ε q ´ ∆ ` ď ř s P ε m ε p s q x p s q for each ε P E . An analogous result holds for UpperBounded Degree g-polymatroid Element , where ř s P ε m ε p s q x p s q ď g p ε q ` ∆ ´ . The runtime of these algorithms is polynomial in n and log ř ε f p ε q or log ř ε g p ε q , respectively.Proof. The proof is similar to the proof of Theorem 5, the main difference appears in the countingargument. When only lower bounds are present, the condition in Step 3.c changes: we delete ahyperedge ε if f p ε q ď ∆ ´
1. Suppose, for the sake of contradiction, that the algorithm does notterminate. Then there is an iteration after which none of the simplifications in Steps 3.a to 3.c canbe performed. This implies that in the current basic solution 0 ă x p s q ă s P S and f p ε q ě ∆ for each ε P E . We choose a subset E Ď E and a maximal independent laminar system L of tight sets the same way as in the proof of Theorem 5. Recall that | E | ` | L | “ | S | .Let Z , . . . , Z k denote the members of the laminar system L . As L is an independent system, Z i ´ Ť W P L max p Z i q W ‰ H . Since x p s q ă s P S , x p Z i ´ ď W P L max p Z i q W q ă | Z i ´ ď W P L max p Z i q W | . As we have integers on both sides of this inequality, we get | Z i ´ ď W P L max p Z i q W | ´ x p Z i ´ ď W P L max p Z i q W q ě i “ , . . . , k . ř s P ε m ε p s q x p s q ě f p ε q ě ∆ for all hyperedges; therefore, | E | ` | L | ď ÿ ε P E ř s P ε m ε p s q x p s q ∆ ` k ÿ i “ »– | Z i ´ ď W P L max p Z i q W | ´ x p Z i ´ ď W P L max p Z i q W q fifl “ ÿ s P S x p s q ∆ ÿ ε P E s P ε m ε p s q ` ÿ W P L max p S q | W | ´ ÿ W P L max p S q x p W q ď | S | . In the last line, the first term is at most x p S q since ř ε P E : s P ε m ε p s q ď ∆ holds for each element s P S .From x p S q ´ ř W P L max p S q x p W q ď | S | ´ ř W P L max p S q | W | the upper bound of | S | follows. As | S | “| L | ` | E | , we have equality throughout. This implies that ÿ ε P E m ε “ ∆ ¨ χ S “ ∆ ¨ ÿ W P L max p S q χ W , contradicting linear independence.If only upper bounds are present, we remove a hyperedge ε in Step 3.c when g p ε q` ∆ ´ ě m ε p ε q .Suppose, for the sake of contradiction, that the algorithm does not terminate. Then there is aniteration after which none of the simplifications in Steps 3.a to 3.c can be performed. This impliesthat in the current basic solution 0 ă x p s q ă s P S and m ε p ε q ´ g p ε q ě ∆ for each ε P E . Again, we choose a subset E Ď E and a maximal independent laminar system L of tightsets the same way as in the proof of Theorem 5.Let Z , . . . , Z k denote the members of the laminar system L . As L is an independent system, Z i ´ Ť W P L max p Z i q W ‰ H and so x p Z i ´ ď W P L max p Z i q W q ě . By ř s P ε m ε p s q x p s q ď g p ε q , we get ř s P ε m ε p s q ´ ř s P ε m ε p s q x p s q ě m ε p ε q ´ g p ε q ě ∆. Thus, | E | ` | L | ď ÿ ε P E ř s P ε m ε p s q ´ ř s P ε m ε p s q x p s q ∆ ` k ÿ i “ x p Z i ´ ď W P L max p Z i q W q“ ÿ s P S ´ x p s q ∆ ÿ ε P E s P ε m ε p s q ` ÿ W P L max p S q x p W qď ÿ s P S ´ x p s q ∆ ÿ ε P E s P ε m ε p s q ` x p S q ď | S | . In the last line, the first term is at most | S | ´ x p S q since ř ε P E : s P ε m ε p s q ď ∆ holds for every element s P S . Therefore, the upper bound of | S | follows. As | S | “ | L | ` | E | , we have equality throughout.This implies that ř ε P E m ε “ ∆ ¨ χ S “ ∆ ¨ ř W P L max p S q χ W , contradicting linear independence. Remark 4.
Note that Theorems 5 and 6 only provide a solution if there exists a (fractional) solutionto the underlying linear program in (LP). Consequently, Theorem 4 only provides a solution if thepolytope P CG in Equation (1) is not empty.
27e have seen in Section 5.1 that base polymatroids are special cases of g-polymatroids. Thisimplies that the results of Theorem 6 immediately apply to polymatroids. In the
Lower BoundedDegree Polymatroid Basis with Multiplicities problem, we are given a base polymatroid B p b q “ p S, b q with a cost function c : S Ñ R , and a hypergraph H “ p S, E q on the same groundset. The input contains lower bounds f : E Ñ Z ě and multiplicity vectors m ε : ε Ñ Z ě forevery hyperedge ε P E . The objective is to find a minimum-cost element x P B p b q such that f p ε q ď ř s P ε m ε p s q x p s q holds for each ε P E . Corollary 17.
There is a polynomial-time algorithm for the
Lower Bounded Degree Poly-matroid Basis with Multiplicities problem which returns an integral element x of B p b q of costat most the optimum value such that f p ε q ´ ∆ ` ď ř s P ε m ε p s q x p s q for each ε P E . In this section we show that Algorithm 3 can be applied in order to obtain an approximation tothe
Minimum Bounded Degree Connected Multigraph with Edge Bounds problem, asdescribed in Theorem 4.
Theorem 4.
There is an algorithm for the
Minimum Bounded Degree Connected Multi-graph with Edge Bounds problem that, in time polynomial in n and log ř v ρ p v q , returns aconnected multigraph T with ρ p V q { edges, where each vertex v has degree at least ρ p v q ´ and thecost of T is at most the cost of min t c T x | x P P CG p ρ, L, U qu , where (1) P CG p ρ, L, U q : “ $’’&’’% x P R E ě ˇˇˇˇˇˇˇˇ supp p x q is connected x p E q “ ρ p V q { x p δ p v qq ě ρ p v q @ v P VL p vw q ď x p vw q ď U p vw q @ v, w P V ,//.//- . Let us take a
Minimum Bounded Degree Connected Multigraph with Edge Bounds problem instance p G, c, ρ, L, U q on a graph G p V, E q , where c , ρ , L , U are non-negative and ρ p V q “ ř v P V ρ p v q is even. Note that we do not require c to satisfy the triangle inequality. We startwith defining the specific input variables passed over Algorithm 3. Then, we show that given thespecified input, the algorithm yields an approximate solution to the Minimum Bounded DegreeConnected Multigraph with Edge Bounds problem. From now on we use ˆ ρ “ ρ p V q { ´| V |` S as the edge set E of our original graph G . In the hypergraph H “ p S, E q , the elements of S thus correspond to the edges of G . Moreover, there is a hyperedge ε for every vertex in V , defined the following way: E : “ t δ p v q | v P V u . The multiplicity of an element s in a hyperedge ε is 1, that is, m ε p s q : “ s corresponds to a regular edge e P E , and m ε p s q : “ s corresponds to a self-loop. We set the lower bound f for a hyperedge ε according to the degreerequirement of the corresponding vertex v , that is f p ε q : “ ρ p v q .We now define the second input of Algorithm 3, a g-polymatroid Q p S, p, b q . This is done intwo steps, by first defining an auxiliary polymatroid Q p S, p , b q , then taking the intersection of theg-polymatroid Q with a box. We define the border function p as the zero vector on S , and b p Z q as follows: Due to the handshaking lemma, the sum of degrees in a graph is even, therefore ρ p V q being even is necessary. emma 18. Let b denote the following function defined on sets Z Ď S : (16) b p Z q “ | V p Z q| ´ comp p Z q ` ˆ ρ, if Z ‰ H , , otherwise .Then b is a polymatroid function.Proof. By definition, b pHq “ b is monotone increasing. It remains to show that b issubmodular. Let X, Y Ď S . The submodular inequality clearly holds if one of X and Y isempty. If none of X and Y is empty then the submodular inequality follows from the fact that | V p Z q| ´ comp p Z q is the rank function of the graphical matroid.Consider the g-polymatroid B p p , b q determined by the border functions defined in Equa-tion (16). Let us define the set B “ t x P Z E ě : x p E q “ ρ p V q { , supp p x q is connected u . Lemma 19. B “ B p p , b q X Z E ě .Proof. Take an integral element x P B p p , b q and let C Ď E be an arbitrary cut between V and V for some partition V Z V of V . Then x p C q “ x p E q ´ x ` E p V q Y E p V q ˘ ě | V | ´ ` ˆ ρ ´ p| V | ` | V | ´ comp ` E p V q Y E p V q ˘ ` ˆ ρ qě , thus supp p x q is connected. As x p E q “ | V | ´ ` ˆ ρ “ ρ p V q { , we obtain x P B , showing that B p p , b q Ď B .To see the other direction, take an element x P B . As supp p x q is connected, x p E ´ F q ě comp p F q ` | V | ´ | V p F q| ´ F Ď E . That is, x p F q “ x p E q ´ x p E ´ F qď r p V q ´ p| V ´ V p F q| ` comp p F q ´ q“ | V p F q| ´ comp p F q ` ˆ ρ, thus x p F q ď b p F q . As x p E q “ r p V q “ | V |´ ` ˆ ρ , we obtain x P B p p , b q , showing B Ď B p p , b q .So far we proved that the integral points of Q p S, p , b q correspond to a connected multigraphon V that has ρ p V q { edges. Let β p L, U q be the box defined by β p L, U q : “ (cid:32) x P R S ě | L p s q ď x p s q ď U p s q @ s P S ( . Let us define the polymatroid Q “ p S, p, b q as the intersection of the polymatroid Q and thebox β , where the border functions p, b are defined as in Equation (15). We now prove that taking H “ p S, E q and Q p S, p, b q as input, the output of Algorithm 3 corresponds to a multigraph withthe properties stated in Theorem 4. Proof of Theorem 4.
Consider the linear program (LP) that is defined in the iterative roundingmethod for the g-Polymatroid Element with Multiplicities problem. The constraints re-garding the bounds on the hyperedges imply ρ p v q ď x p δ p v qq for every v P V : note that m ε p s q “ uv to the value x p δ p v qq . This, together with Lemma 19 and the fact that x is contained in the box β p L, U q , impliesthat Algorithm 3 returns an integral solution z such that the cost of z is at most the minimum costelement of P CG .According to Theorem 6, the integral solution z violates the bounds f on the hyperedges byat most ∆ ´
1, where ∆ : “ max s P S t ř ε P E : s P ε m ε p s qu . But we defined m ε p s q to be equal to 2if s corrensponds to a self-loop in G and 1 otherwise, meaning that the solution z violates thebounds on the hyperedges f and thus the bounds on the vertices ρ by at most 1. The solution z is also connected, with a total number of edges ρ p V q { and z satisfies the edge bounds L, U ; due toTheorem 6. Therefore, the solution z corresponds to a multigraph, that admits the properties inthe claim of Theorem 4. The transportation problem in Algorithm 1 can be solved in strongly polynomial time [34, 43].Computing compact path-cycle representations uses the algorithm of Grigoriev and van de Klun-dert [23]; which, along with computing the Eulerian trail and making shortcuts, can be done instrongly polynomial time. Moreover, the algorithm uses the { -approximation for the Path TSP by Zenklusen [58] as a black-box, which can also be implemented in strongly polynomial time.Making shortcuts in Algorithm 1 can be performed in strongly polynomial time as well, thus wecan find a { -approximation for the metric Many-visits Path TSP in strongly polynomial time.Algorithm 2 involves solving three types of LPs. According to § x can be tested in polynomial time, then the ellipsoidmethod can find a solution in strongly polynomial time.In Step 1, we calculate an optimal solution to P MVHK as defined in Equation (2), while a numberof linear programs of form LP p a q arise throughout the dynamic program in Step 2. The feasibilityof the cut constraints can be checked in strongly polynomial time, by solving a minimum cutproblem. The number of degree constraints in both types of these LPs and the number of constraints x p δ p B qq ě p a q ) is polynomial in n . Finally, in (LP), the number of constraints involvinghyperedges is polynomial in n , and one can check the feasibility of the constraints involving theborder functions using submodular minimization [29]. This means all of the linear programs arisingin Algorithm 2 can be solved in strongly polynomial time.According to Lemma 10, the number of cuts in B is polynomial in n , hence Steps 1, 2 and 4can be performed in strongly polynomial time. This is true for computing a matching in Step 5,as well as all the remaining graph operations, using the same arguments as in case of Algorithm 1.Therefore we provide a { -approximation for the metric Many-visits Path TSP in stronglypolynomial time.
In this paper we gave an approximation algorithm for a far-reaching generalization of the metric
Path TSP , the metric
Many-visits Path TSP where each city v has a (potentially exponentiallylarge) requirement r p v q ě
1. Our algorithm yields a { -approximation for the metric Many-visitsPath TSP in time polynomial in the number n of cities and the logarithm of the r p v q ’s. It therefore30eneralizes the recent fundamental result by Zenklusen [58], who obtained a { -approximation forthe metric Path TSP , finishing a long history of research.At the heart of our algorithm is the first polynomial-time approximation algorithm for theminimum-cost degree bounded g-polymatroid element with multiplicities problem. That algorithmyields a solution of cost at most the optimum, which violates the lower bounds only by a constantfactor depending on the weighted maximum element frequency ∆.Finally, we show a simple approach, that gives a { -approximation for the metric Many-visitsTSP in strongly polynomial time, and an O p q -approximation for the metric Many-visits ATSP in polynomial time.
Acknowledgements.
The authors are grateful to Rico Zenklusen for discussions on techniques to obtaina { -approximation for the metric version of the Many-visits TSP , and to Tam´as Kir´aly and Gyula Pap fortheir suggestions. Krist´of B´erczi was supported by the J´anos Bolyai Research Fellowship of the HungarianAcademy of Sciences and by the ´UNKP-19-4 New National Excellence Program of the Ministry for Innovationand Technology. Projects no. NKFI-128673 and no. ED 18-1-2019-0030 (Application-specific highly reliableIT solutions) have been implemented with the support provided from the National Research, Developmentand Innovation Fund of Hungary, financed under the FK 18 and the Thematic Excellence Programme fundingschemes, respectively.
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