The Salesman's Improved Tours for Fundamental Classes
TThe Salesman’s Improved Tours for FundamentalClasses
Sylvia Boyd and Andr´as Seb˝o Abstract.
Finding the exact integrality gap α for the LP relaxation of the met-ric Travelling Salesman Problem (TSP) has been an open problem for over thirtyyears, with little progress made. It is known that 4 / ≤ α ≤ /
2, and a famousconjecture states α = /
3. It has also been conjectured that there exist half-integer basic solutions of the linear program for which the highest integralitygap is reached.For this problem, essentially two “fundamental” classes of instances have beenproposed. This fundamental property means that in order to show that the inte-grality gap is at most ρ for all instances of the metric TSP, it is sufficient to showit only for the instances in the fundamental class. However, despite the impor-tance and the simplicity of such classes, no apparent effort has been deployed forimproving the integrality gap bounds for them. In this paper we take a natural firststep in this endeavour, and consider the 1 / / / / Given the complete graph K n = ( V n , E n ) on n nodes with non-negative edge costs c ∈ R E n , the Traveling Salesman Problem (henceforth TSP) is to find a Hamiltonian cycleof minimum cost in K n . When the costs are metric , i.e. satisfy the triangle inequality c i j + c jk ≥ c ik for all i , j , k ∈ V n , the problem is called the metric TSP. If the metric isdefined by the shortest (cardinality) paths of a graph, then it is called a graph metric ;the TSP specialized to graph metrics is the graph TSP .For G = ( V , E ) , x ∈ R E and F ⊆ E , x ( F ) : = ∑ e ∈ F x e ; for U ⊆ V , δ ( U ) : = δ G ( U ) : = { uv ∈ E : u ∈ U , v ∈ V \ U } ; E [ U ] : = { uv ∈ E : u ∈ U , v ∈ U } . The scalar product ofvectors a and x of the same dimension will simply be denoted by ax . A path is the edgeset of a connected subgraph with two nodes of degree 1 and all other nodes of degree 2,and a cycle is the edge set of a connected subgraph with all node degrees equal to 2. SEECS, University of Ottawa, Ottawa, Canada. Partially supported by the National Sciencesand Engineering Research Council of Canada. This work was done during visits in LaboratoireG-SCOP, Grenoble; support from the CNRS and Grenoble-INP is gratefully acknowledged. CNRS, Laboratoire G-SCOP, Univ. Grenoble Alpes, Supported by LabEx PERSYVAL-Lab(ANR 11-LABX-0025), ´equipe-action GALOIS. a r X i v : . [ c s . D M ] O c t A natural linear programming relaxation for the TSP is the following subtour LP :minimize cx (1)subject to: x ( δ ( v )) = v ∈ V n , (2) x ( δ ( S )) ≥ (cid:54) = S (cid:40) V n , (3)0 ≤ x e ≤ e ∈ E n . (4)For a given cost function c ∈ R E n , we use LP ( c ) to denote the optimal solution valuefor the subtour LP and OPT ( c ) to denote the optimal solution value for the TSP. Thepolytope associated with the subtour LP, called the subtour elimination polytope anddenoted by S n , is the set of all vectors x satisfying the constraints of the subtour LP, i.e. S n = { x ∈ R E n : x satisfies ( ) , ( ) , ( ) } .The metric TSP is known to be NP-hard. One approach taken for finding reason-ably good solutions is to look for a ρ -approximation algorithm for the problem, i.e. apolynomial-time algorithm that always computes a solution of value at most ρ times theoptimum. Currently the best such algorithm known for the metric TSP is the algorithmdue to Christofides [9] for which ρ = . Although it is widely believed that a better ap-proximation algorithm is possible, no one has been able to improve upon Christofidesalgorithm in four decades. For arbitrary nonnegative costs not constrained by the trian-gle inequality there does not exist a ρ -approximation algorithm for any ρ ∈ R unless P = NP , since such an algorithm would be able to decide if a given graph is Hamilto-nian.For an approximation guarantee of a minimization problem one needs lower boundsfor the optimum, often provided by linear programming. For the metric TSP with costfunction c , a commonly used lower bound is LP ( c ) . Then finding a solution, i.e. aHamiltonian cycle, of objective value at most ρ LP ( c ) in polynomial time implies atthe same time a ρ -approximation algorithm, and establishes that the integrality gapOPT ( c ) / LP ( c ) is at most ρ for any input. (The input consists of the nodes and themetric function c on pairs of nodes; again, without the metric assumption this ratiois unbounded already for the graph TSP by putting infinite costs on non-edges of thedefining graph.) Since up until now the bounds on the integrality gap have been provedvia polynomial algorithms constructing the Hamiltonian cycles, the approximation ratiofor metric costs is conjectured not to be larger than the integrality gap.It is known that the integrality gap for the subtour LP for any metric c is at most ([10], [25], [26]), however no example for which the integrality gap is greater than is known. In fact, a famous conjecture, often referred to as the Conjecture , states thefollowing:
Conjecture 1.
The integrality gap for the subtour LP with metric c is at most .Well-known examples have an integrality gap asymptotically equal to . In almost thirtyyears, there have been no improvements made to the upper bound of or lower boundof for the integrality gap for the subtour LP with metric c .Define a tour to be the edge set of a spanning Eulerian (i.e. connected with all de-grees even) multisubgraph of K n . If none of the multiplicities can be decreased, then all multiplicities are at most two; however, there are some technical advantages to allowinghigher multiplicities.For any multiset J ⊆ E n , the incidence vector of J , denoted by χ J , is the vector in R E n for which χ Je is equal to the number of copies of edge e in J for all e ∈ E n . We use T n to denote the convex hull of incidence vectors of tours of K n , and for costs c ∈ R E n weuse OPT T n ( c ) to denote the cost of a minimum cost tour. Note that T n is an unboundedpolyhedron, as T n + R E n + = T n : each edge may have arbitrarily large multiplicity.For any ρ ∈ R , ρ S n denotes { y ∈ R E n : y = ρ x , x ∈ S n } . The definition of the in-tegrality gap can be reformulated in terms of a containment relation between the twopolyhedra ρ S n and T n (Theorem 1) that does not depend on the objective function. Wenot only use this reformulation here, but also develop a specific way of exploiting it, andfor our arguments this is the very tool that works. Showing for some constant ρ ∈ R that ρ x ∈ T n for each x ∈ S n , i.e. that ρ x is a convex combination of incidence vectorsof tours, gives an upper bound of ρ on the integrality gap for the subtour LP: it impliesthat for each x ∈ S n and any cost function c ∈ R E n such that cx = LP ( c ) , at least oneof the tours in the convex combination has cost at most ρ ( cx ) = ρ LP ( c ) . If the costsare metric, this tour can be shortcut to a TSP solution of cost at most ρ LP ( c ) , givinga ratio of OPT ( c ) / LP ( c ) ≤ ρ . A shortcut means to fix an Eulerian tour and replace asequence of nodes a , b , c ) by a , c , whenever b has already been visited by the tour. Theessential part “(i) implies (ii)” of the following theorem, due to Goemans [14] (also see[8]), asserts that the converse is also true: if ρ is at least the integrality gap then ρ S n isa subset of T n . Theorem 1. [14][8] Let K n = ( V n , E n ) be the complete graph on n nodes and let ρ ∈ R , ρ ≥ . The following statements are equivalent:(i) For any metric cost function c : E n −→ R + , OPT ( c ) ≤ ρ LP ( c ) .(ii) For any x ∈ S n , ρ x ∈ T n .(iii) For any vertex x of S n , ρ x ∈ T n . By Theorem 1, Conjecture 1 can be equivalently reformulated as follows:
Conjecture 2.
The polytope S n is contained in the polyhedron T n , that is, S n ⊆ T n .Given a vector x ∈ S n , the support graph G x = ( V n , E x ) of x is defined with E x = { e ∈ E n : x e > } . We call a point x ∈ S n -integer if x e ∈ { , , } for all e ∈ E n . Forsuch a vector we call the edges e ∈ E n -edges if x e = and 1 -edges if x e =
1. Notethat the 1-edges of -integer points form a set of disjoint paths that we call 1 -paths of x , and the -edges form a set of edge-disjoint cycles we call the -cycles of x .For Conjecture 1, it seems that -integer vertices play an important role (see [1],[7],[20]).In fact it has been conjectured by Schalekamp, Williamson and van Zuylen [20] that asubclass of these -integer vertices are the ones that give the largest gap. Here we statea weaker version of their conjecture: Conjecture 3.
The integrality gap for the subtour LP is reached on -integer vertices.Very little progress has been made on the above conjectures, even though they havebeen around for a long time and have been well-studied. For the special case of graph TSP an upper bound of is known for the integrality gap [23]. Conjecture 2 has beenverified for the so-called triangle vertices x ∈ S n for which the values are -integer,and the -edges form triangles in the support graph [4]. The lower bound of for theintegrality gap is provided by triangle vertices with just two triangles.A concept first introduced by Carr and Ravi [7] (for the 2-edge-connected subgraphproblem) is that of a fundamental class , which is a class of points F in the subtour elim-ination polytopes S n , n ≥ ρ x is a convexcombination of incidence vectors of tours for all vertices x ∈ F implies the same holdsfor all vertices of polytopes S n , and thus implies that the integrality gap for the subtourLP is at most ρ .Two main classes of such vertices have been introduced, one by Carr and Vempala[8], the other by Boyd and Carr [4]. In this paper we will focus on the latter one, thatis, we define a Boyd-Carr point [4] to be a point x ∈ S n that satisfies the followingconditions:(i) The support graph G x of x is cubic.(ii) In G x , there is exactly one 1-edge incident to each node.(iii) The fractional edges of G x form disjoint 4-cycles.A Carr-Vempala point [8] is one that satisfies (i), (ii) and instead of (iii), the frac-tional edges form a Hamiltonian cycle. We use fundamental point as a common namefor points that are either Boyd-Carr or Carr-Vempala points. It has been proved thatthe Boyd-Carr points [4] and Carr-Vempala points [8] each form fundamental classes.The support graph of a fundamental point will be called a fundamental graph . In otherwords, a fundamental graph is a cubic graph where there exists a perfect matchingwhose deletion leads to a graph whose components are 4-cycles, or a Hamiltonian cy-cle. Note that the 3-edge-connected instances of each of the two classes of fundamentalpoints also form fundamental classes: for Boyd-Carr points this can be checked fromthe construction itself [4]; for Carr-Vempala points this is obvious, since the Hamilto-nian cycle of edges { e ∈ E n : x ( e ) < } has at least two edges in every cut, but two suchedges alone do not suffice for constraints (3) of the subtour LP.Despite their significance and simplicity, no effort has been deployed to exploringnew integrality gap bounds for these classes, and no improvement on the general upper bound on the integrality gap has been made for them, not even for special cases.A natural first step in this endeavour is to try to improve the general bounds for thespecial case of -integer Boyd-Carr or Carr-Vempala points.In this paper we improve the upper bound for the integrality gap from to for -integer Boyd-Carr points, and also provide a -approximation algorithm for metricTSP for any cost function for which the subtour LP is optimized by a Boyd-Carr point.In fact we generalize these results to a superclass of these points. Replacing the 1-edgesin Boyd-Carr points by 1-paths of arbitrary length between their two endpoints, we getall the square points , that is, -integer points of S n for which the -edges form disjoint4-cycles, called squares of the support graph. We also show that there exists a subclassof square points that provide instances where the integrality gap is at least . Thussquare points provide new tight examples for the lower bound of the conjectures. In the endeavour to find improved upper bounds on the integrality gap we exam-ine the structure of the support graphs of Boyd-Carr points. We show that they are allHamiltonian, an important ingredient of our bounding of their integrality gap. The proofuses a theorem of Kotzig [19] on Eulerian trails with forbidden transitions. An
Euleriantrail in a graph is a closed walk containing each of its edges exactly once. Contrary totours, it is more than just an edge set, the order of the edges also plays a role. The con-nection of tours to Eulerian trails leads us to delta-matroids and to developing relatedalgorithms, which are discussed in Section 4.In Section 2.1 we show a first, basic application of these ideas, where some partsof the difficulties do not occur. We prove that all edges can be uniformly covered 6 / / x of the subtour elimination polytope as the convex combinationof incidence vectors of “rainbow” spanning trees in edge-coloured graphs. The idea ofusing spanning trees with special structures to get improved results has recently beenused successfully in [13] for graph TSP, and in [15] and [24] for a related problem,namely the metric s − t path TSP. However, note that we obtain and use our trees in acompletely different way.Our main results concerning the integrality ratio of -integer Boyd-Carr points andsquare points are proved in Section 3. We conclude that section by outlining a potentialstrategy for using the Carr-Vempala points of [8] for proving the Conjecture.Finally, in Section 4, we provide polynomial-time optimization algorithms for someof the existence theorems of previous sections, including a -approximation algorithmfor metric TSP for any cost function which is optimized by a square point for the subtourLP. The methods used relate to delta matroids, and their relevance is discussed.
In this section we discuss some useful and powerful tools that we need in the proof ofour main result in Section 3. We begin with some preliminaries.Given a graph G = ( V , E ) with a node in V labelled 1, a 1 -tree is a subset F of E such that | F ∩ δ ( ) | = F \ δ ( ) forms a spanning tree on V \{ } . The convex hullof the incidence vectors of 1-trees of G , which we will refer to as the 1 -tree polytope ofthe graph G , is given by the following [16, page 262]: { x ∈ R E : x ( δ ( )) = , x (( E [ U ])) ≤ | U | − (cid:54) = U ⊆ V \{ } , ≤ x e ≤ e ∈ E , x ( E ) = | V |} . (5)It is well-known that the 1-trees of a connected graph satisfy the basis axioms of amatroid (see [16, page 262-263]).Given G = ( V , E ) and T ⊆ V , | T | even, a T -join of G is a set J ⊆ E such that T isthe set of odd degree nodes of the graph ( V , J ) . A cut C = δ ( S ) for some S ⊂ V is called a T -cut if | S ∩ T | is odd. We say that a vector majorates another if it is coordinatewisegreater than or equal to it. The set of all vectors x that majorate some vector y in theconvex hull of incidence vectors of T -joins of G is given by the following [12]: { x ∈ R E : x ( C ) ≥ T -cut C, x e ≥ e ∈ E } . (6)This is the T -join polyhedron of the graph G .The following two results are well-known (see [10], [25], [26]), but we include theproofs as they introduce the kind of polyhedral arguments we will use: Lemma 1. [10] [25] [26] If x ∈ S n , then (i) it is a convex combination of incidence vec-tors of -trees of K n , and (ii) x / majorates a convex combination of incidence vectorsof T -joins of K n for every T ⊆ V n , | T | even.Proof. Constraints (2) and (3) of the subtour LP together imply that x ( E n ) = n and x ( E n [ S ]) ≤ | S | −
1, for all /0 (cid:54) = S (cid:40) V n . Thus x ∈ S n satisfies all of the constraints ofthe 1-tree polytope of K n and (i) of the lemma follows. To check (ii), note that for all T ⊆ V n , | T | even, x / T -join polyhedron of K n (in fact x ( C ) / ≥ C ), that is, it majorates a convex combination of incidencevectors of T -joins. (cid:117)(cid:116) Theorem 2. [10] [25] [26] If x ∈ S n , x ∈ T n .Proof. By (i) of Lemma 1, x is a convex combination of incidence vectors of 1-trees of K n . Let F be any 1-tree of such a convex combination, and T F be the set of odd degreenodes in the graph ( V n , F ) . Then by (ii) of Lemma 1, x / T F -joins. So χ F + x / x + x / x ∈ T n . (cid:117)(cid:116) The tools of the following two subsections are new for the TSP and appear to bevery useful.
Let G = ( V , E ) be a connected 4-regular multigraph. For any node v ∈ V , a bitransition (at v ) is a partition of δ ( v ) into two pairs of edges. Clearly every Eulerian trail of G uses exactly one bitransition at every node, meaning the two disjoint pairs of consecutiveedges of the trail at the node. There are three bitransitions at every node and the theorembelow, equivalent to a result of Kotzig [19], states that we can forbid one of these andstill have an allowed Eulerian trail. As we will show, this provides Hamiltonian cyclesin the support of square points, used in Section 3 to prove our main result.
Theorem 3. [19] Let G = ( V , E ) be a 4-regular connected multigraph with a forbid-den bitransition for every v ∈ V . Then G has an Eulerian trail not using the forbiddenbitransition of any node. A square graph is defined as a pair ( G , M ) where G = ( V , E ) is a cubic 2-edge-connectedgraph, and M is a perfect matching of G such that the edges E \ M form squares. Weassociate Boyd-Carr points to square graphs, where M is defined to be the set of 1-edges. Lemma 2.
A square graph ( G , M ) has a Hamiltonian cycle containing M.Proof. Let G = ( V , E ) . We assume G has at least two squares as the lemma is triviallytrue otherwise.Suppose first that G has a square u u , u u , u u , u u in E \ M , with a chord, say u u , in M , that is, exactly the two edges au and u b ( a , b ∈ V ) are leaving the set { u , u , u , u } ⊆ V . In this case we can delete u , u , u , u from the graph and add theedge ab to G to form a new square graph ( ˆ G , ˆ M ) with ˆ M = M − u u + ab . From aHamiltonian cycle of this reduced graph containing ˆ M we obtain a Hamiltonian cycleof G containing M by deleting ab , and adding au , u u , u u , u u , u b . We can thussuppose that G has no such square.Contracting all squares of G , we obtain a 4-regular connected multigraph G (cid:48) =( V (cid:48) , E (cid:48) ) whose edges are precisely M and whose nodes are precisely the squares of G \ M . To each contracted square C we associate the forbidden bitransition consistingof the pairs of edges of M incident with the diagonally opposite nodes of C in G , asshown in Figure 1. By Theorem 3, there is an Eulerian trail K of G (cid:48) that does not usethese forbidden bitransitions. The two pairs of consecutive edges in K at each node in G (cid:48) can then be completed by a perfect matching of the corresponding square in G , formingthe desired Hamiltonian cycle. (cid:117)(cid:116) vv u G ’ v v uu uuv vv
234 1 234 v G x Fig. 1: Shrinking a square in G to node u ; forbidden: { ( uv , uv ) , ( uv , uv ) } .The exhibited connection of Eulerian graphs with forbidden bitransitions sends usto a link on delta-matroids [3] with well-known optimization properties. We explorethis link in Section 4, where we provide a direct self-contained algorithmic proof (witha polynomial-time, greedy algorithm) for a weighted generalization of Lemma 2. Wecontent ourselves in this section by providing a simple, first application of Lemma 2which shows a basic idea we will use in the proof of our main result in Section 3,without the additional difficulty of the more refined application.Given a graph G = ( V , E ) and a value λ , the everywhere λ vector for G is thevector y ∈ R E | V | for which y e = λ for all edges e ∈ E and y e = in the complete graph K | V | . In Theorem 4 below we show that for any cubic 3-edge-connected graph with a Hamiltonian cycle, the everywhere 6 / T n . Sincefundamental graphs are Hamiltonian (by Lemma 2 for Boyd-Carr, and by definition forCarr-Vempala), the theorem applies to their 3-edge-connected instances. Both classesof such 3-edge-connected graphs are also fundamental classes as was noted earlier (seethe remark after the definition). Theorem 4.
If G = ( V , E ) is a cubic, -edge-connected Hamiltonian graph, then theeverywhere / vector for G is in T n .Proof. Let H be a Hamiltonian cycle of G, and let M : = E \ H be the perfect matchingcomplementary to H . Note that χ H is the incidence vector of a tour, and we will use itin the convex combination for the everywhere 6 / H very often in order to achieve our goal. To thisend we consider the point x ∈ R E | V | defined by x e = e ∈ M , x e = / e ∈ H and x e = x is in the subtour elimination polytope S | V | , thusby Theorem 2, x ∈ T n .Now take the convex combination of tours t : = χ H +
47 32 x . Then for edges e ∈ M we have t e = +
47 32 = . For edges e ∈ H we have t e = +
47 32 12 = , and x e = e not in G , finishing the proof. (cid:117)(cid:116) Replacing Hamiltonian cycles by other, relatively small tours or convex combina-tions of tours in the proof of Theorem 4, one may get similar results weakening theHamiltonicity condition. Such results are particularly interesting for general, cubic 3-edge-connected graphs. For such graphs the everywhere 1 vector is in T n , as noticedin [22], where it is then asked whether the everywhere vector belongs to T n . Theseare the values one gets from Theorem 2 and Conjecture 2 respectively, applied to theeverywhere vector, feasible for S n . Note that the analogous problem for the s − t pathTSP has been solved [24].The above possibility has been explored by Haddadan, Newman and Ravi [17],who proved that the everywhere 18 /
19 vector is in T n , getting this constant below 1 forthe first time, and with a polynomial-time algorithm. They replaced the Hamiltoniancycle in the proof of Theorem 4 by a convex combination of tours proved by Kaiserand ˇSkrekovski [18] and by Boyd, Iwata, Takazawa [5] algorithmically. Using this con-vex combination they also deduce better bounds with simple proofs for node-weightedgraphs. Let us do the same for node-weighted Hamiltonian graphs.In a node-weighted TSP every node v of a given graph G = ( V , E ) is given a weight f v ∈ R + , and the cost c uv of an edge uv ∈ K | V | is f u + f v if uv ∈ E , and the cost of ashortest path in G otherwise. Note that c is metric. Theorem 5.
Let G = ( V , E ) be a node-weighted cubic -edge-connected graph forwhich the everywhere λ vector y is in T n . Then OPT ( c ) ≤ λ LP ( c ) .Proof. Since G is cubic we have c ( E ) = ∑ ( f v : v ∈ V ) . Since y ∈ T n there is a tour J such that c ( J ) ≤ λ c ( E ) = λ ∑ ( f v : v ∈ V ) . (7) As observed in [17], it follows from constraints (2) of the subtour LP and the fact that theeverywhere vector on E ( G ) is feasible for S | V | that LP ( c ) = ∑ ( f v : v ∈ V ) . Togetherwith (7) and the fact that OPT ( c ) ≤ c ( J ) for metric costs completes the proof. (cid:117)(cid:116) Corollary 1.
If G = ( V , E ) is a node-weighted cubic, -edge-connected Hamiltoniangraph, or in particular a -edge-connected fundamental graph, then OPT ( c ) ≤ LP ( c ) . -trees We now use matroid intersection to prove that not only is x is in the convex hull ofincidence vectors of 1-trees, but we can also require that these 1-trees satisfy someadditional useful properties. We will use this in the proof of our main result in Section3. Given a graph G = ( V , E ) , let every edge of G be given a colour. We call a 1-tree F of G a rainbow -tree if every edge of F has a different colour. Rainbow trees are discussedby Broersma and Li in [6], where they note they are the common independent sets oftwo matroids, a fact combined here with a polyhedral argument to obtain the followingtheorem: Theorem 6.
Let x ∈ S n be -integer, and let P be any partition of the -edges intopairs. Then x is in the convex hull of incidence vectors of -trees that each containexactly one edge from each pair in P .Proof. Let G x = ( V n , E x ) be the support graph of x . Consider the partition matroid (cf.[21]) defined on E x by the partition P ∪ {{ e } : e ∈ E x , e is a 1-edge } . By Lemma 1, x is in the convex hull of incidence vectors of 1-trees in E x ; since x ( Q ) = Q of the defined partition matroid, it is also in the convex hull of its bases. Thus by [21,Corollary 41.12d], which is a corollary of Edmonds’ matroid intersection theorem [11], x is in the convex hull of incidence vectors of the common bases of the two matroids. (cid:117)(cid:116) -integer Points In this section we show that x ∈ T n for all square points x ∈ S n , and thus for all -integer Boyd-Carr points x as well. We also analyse the possibility of a similar proof forCarr-Vempala points that would have the advantage of also implying a ratio of for all -integer points in S n , as we will discuss at the end of this section. We begin by statingtwo properties we prove later to be sufficient to guarantee x ∈ T n for any -integervector x in S n :(A) The support graph G x of x has a Hamiltonian cycle H .(B) Vector x is a convex combination of incidence vectors of 1-trees of K n such thateach 1-tree satisfies the following condition: it contains an even number of edgesin every cut of G x consisting of four -edges in H . We will use χ H of (A) as part of the convex combination for x , which is globallygood, since H has only n edges, but the -edges of H have too high a value (equal to 1),contributing too much for the convex combination. To compensate for this, property (B)ensures that x is not only a convex combination of 1-trees, but these 1-trees are even forcertain edge cuts δ ( S ) , allowing us to use a value essentially less than the x = for edges in H for the corresponding T -join. The details of how to ensure feasibility for the T -join polyhedron will be given in the proof of Theorem 7.While condition (A) may look at first sight impossibly difficult to meet, Lemma 2shows that one can count on the bonus of the naturally arising properties: any squarepoint x satisfies property (A), and the additional property stated in this lemma togetherwith the “rainbow 1-tree decomposition” of Theorem 6 will also imply (B) for squarepoints. The reason we care about the somewhat technical property (B) instead of itsmore natural consequences is future research: in a new situation we may have to use themost general condition. Lemma 3.
Let x be any square point. Then x satisfies both (A) and (B).Proof.
If we replace the 1-paths in the support graph G x by single 1-edges, then byLemma 2 there is a Hamiltonian cycle for the new graph that contains all of the 1-edges, and thus G x also has a Hamiltonian cycle H that contains all of its 1-edges. Thuspoint x satisfies Property (A) by Lemma 2. Moreover, since H contains all the 1-edgesin G x , it follows that H contains a perfect matching from each square of G x .Define P to be the partition of the set of -edges of G x into pairs whose classes arethe perfect matchings of squares. Then by Theorem 6, x is in the convex hull of inci-dence vectors of 1-trees that contain exactly one edge from each pair P ∈ P . Property(B) follows, since every cut C that contains four -edges of H is partitioned by twoclasses P , P ∈ P . (Indeed, we saw at the end of the first paragraph of this proof that asquare met by H is met in a perfect matching.) Since P and P are met by exactly oneedge of each tree of the constructed convex combination, C is met by exactly two edgesof each tree. (cid:117)(cid:116) Next we prove that properties (A) and (B) are sufficient to guarantee that x ∈ T n for any -integer point of S n . Recall that properties (A) and (B) are more general thanwhat we need for square points: the condition of the theorem we prove does not requirethat the Hamiltonian cycle of property (A) contains the 1-edges of G x , as Lemma 2asserts for square points. We keep this generality of (A) and (B) to remain open toeventual posterior demands of future research. Theorem 7.
Let x ∈ S n be a -integer point satisfying properties (A) and (B). Then x ∈ T n .Proof. Let G x = ( V n , E x ) be the support graph of x , and let H be any Hamiltonian cycleof G x , which exists according to (A). Let the 1-trees in the convex combination forproperty (B) be F i , i = , , ..., k , and for any 1-tree F of G x let T F be the set of odddegree nodes in the graph ( V n , F ) . Consider the vector y ∈ R E n defined as y : = x − χ H . Claim . For any 1-tree F of G x which satisfies the condition of property (B), vector y isin the T F -join polyhedron of K n . To prove the claim, we show that y satisfies the constraints of the T F -join polyhedron(6) of K n . Clearly y e ≥ e ∈ E n . Let C be a T F -cut in K n . We must show y ( C ) ≥ | H ∩ C | (cid:54) = H ⊆ E x .Case 1: | H ∩ C | =
2. Since x ( C ) ≥ y ( C ) ≥ ( ) − ( ) =
1, as required.Case 2: | H ∩ C | =
4. Since x ∈ S n , we have x ( C ) ≥ x is -integer the -edges form edge-disjoint cycles, so x ( C ) is integer: x ( C ) ≥
3, since x ( C ) = C consists of the four edges of H ∩ C , and by (B) C is then not a T F -cut; so y ( C ) ≥ ( ) − ( ) ≥ | H ∩ C | ≥
6. Then for all e ∈ H ∩ C : y e = x e − χ He ≥ ( ) − = , so y ( C ) ≥
1, which completes the proof of the claim.According to the claim, for all i ∈ { , . . . , k } y majorates a convex combination of T F i -joins, and adding any one of these to χ F i , we get tours. The convex combination ofthese tours is majorated by χ F i + y , so χ F i + y ∈ T n for all i = , . . . , k , and therefore x + y ∈ T n . Thus z : = χ H + ( x + y ) = x is also in T n , which completes the proof. (cid:117)(cid:116) Our main result is an immediate corollary of this theorem:
Theorem 8.
Let x be a square point. Then x ∈ T n .Proof. By Theorem 7 it is enough to make sure that x satisfies properties (A) and (B),which is exactly the assertion of Lemma 3. (cid:117)(cid:116) Since -integer Boyd-Carr points are square points we have: Corollary 2.
If x is a -integer Boyd-Carr point, then x ∈ T n . Theorem 8 immediately implies the following optimization form of the above corol-lary:
Corollary 3.
Let c ∈ R E n be a metric cost function optimized by a square point x, i.e.cx = LP ( c ) . Then there exists a Hamiltonian cycle of cost at most LP ( c ) in K n .Proof. We have LP ( c ) = cx = c ( x ) ≥ OPT T n ( c ) , where the last inequality fol-lows from x being a convex combination of tours by Theorem 8. As OPT T n ( c ) ≥ OPT ( c ) for metric costs, the result follows. (cid:117)(cid:116) In the following section we show that the above existence theorems and their corol-laries can also be accompanied with polynomial algorithms. We finish this section byshowing that the integrality gap of square points is at least , providing new examplesshowing that Conjecture 1 cannot be improved.Define a subclass of square points we call k-donuts , k ∈ Z , k ≥
2: the support graph G x = ( V n , E x ) consists of k -squares arranged in a cyclic order, where the squares arejoined by 1-paths, each of length k . In Figure 2 the support graph of a 4-donut is shown.In the figure, dashed edges represent -edges and solid edges represent 1-edges. We define the cost of each edge in E x to be 1, except for the pair of -edges in eachsquare that are opposite to one another, with one edge on the inside of the donut, andone on the outside, which are defined to have cost k (see the figure, where only edgesof cost k are labelled, and all other edges in E x have cost 1). The costs of other edges of K n not in E x are defined by the metric closure (cost of shortest paths in G x ). For thesedefined costs c ( k ) , we have OPT ( c ( k ) ) = k − k + LP ( c ( k ) ) ≤ c ( k ) x = k + k ,thus lim k → ∞ OPT ( c ( k ) ) LP ( c ( k ) ) ≥ . Along with Theorem 8, this gives the following: Corollary 4.
The integrality gap for square points lies between and . k k kk k kkk Fig. 2: Graph G x for a k -donut x , k =4.To conclude this section we briefly discuss the structure of Carr-Vempala points.Note that for the Boyd-Carr points that have been our focus, the transformation usedfrom general vertices x ∈ S n to these Boyd-Carr points does not completely preservethe denominators. In particular, -integer vertices of S n get transformed into Boyd-Carrpoints x ∗ with x ∗ e values in { , , , , } . However, for the Carr-Vempala points, it isclear from their construction in [8] that general -integer vertices of S n lead to -integerCarr-Vempala vertices. In fact we have the following theorem which, if Conjecture 3 istrue, would provide a nice approach for proving Conjecture 2, since it is given for freethat Carr-Vempala vertices satisfy property (A): Theorem 9. If ρ x ∈ T n for each -integer Carr-Vempala point x ∈ S n , then ρ x ∈ T n for every -integer point x ∈ S n . In light of these results and conjectures it seems worthwhile to study fundamentalclasses further and the role of -integer points in them. In this section we show that some of the existence theorems stated in the previoussections can be accompanied by simple combinatorial algorithms that can be executed in polynomial time. The main result of this section is a polynomial-time algorithm forfinding the Hamiltonian cycle of Corollary 3 in a completely elementary way. We pointout that the bridge taking us to this result also leads to a polynomial-time algorithm forfinding a minimum cost Hamiltonian cycle containing the M edges in a square graph,generalizing Lemma 2.It turns out that the greedy algorithm is the main ingredient of our algorithms, anddelta-matroids are the structure behind this phenomenon. We give a short introductionto delta-matroids, their greedy algorithm and their connection to our results. We firstintroduce the algorithm directly on our combinatorial objects below. Greedy Algorithm (HAM) for Hamiltonian cycles in square graphsInput : A square graph ( G , M ) and cost vector c ∈ R E ( G ) . Output : A Hamiltonian cycle containing M .1. For each square C of G let w C be the absolute value of the difference of the cost ofthe two perfect matchings of C . Order the squares in non-increasing order of w C , from C to C t ; i : = i ≤ t do:We keep exactly one of the two perfect matchings of C i and delete both edges ofthe other perfect matching according to the following rule:If the graph remains connected after both of these choices, keep the perfectmatching of C i which has smaller cost (if the costs are equal, break ties arbi-trarily).If the graph remains connected after exactly one of these two possible choices,then make this choice.There is no other case according to the following claim: Claim:
The graph remains connected with at least one of the two choices.
Proof : Suppose that at iteration i , G − C i (edge-deletion) is not connected. Thenit has at least two of the four arising nodes of degree 1 (the only nodes of odddegree after contracting the remaining squares) in each component. It follows thatit has two components. If adding one of the two perfect matchings it is still notconnected, then all edges of C i are induced by one of the two components, so G isalso not connected, a contradiction. (cid:117)(cid:116)
3. Output the constructed graph which is a Hamiltonian cycle containing M (all degreesare two, and it is connected because of the claim). end We will see that this algorithm determines the Hamiltonian cycle of minimum costcontaining M (Theorem 12), and this is also not difficult to prove directly along the samelines as the optimality of the greedy algorithm for optimal spanning trees, as follows:Suppose for a contradiction that the algorithm finds H , while K is a Hamiltonian cycleof smaller cost containing M . Then there exists a square C i for which K uses a differentperfect matching than H and the cost of this perfect matching is strictly less than theone used by H . Let i be the smallest index for which this is true, and assume that overall minimum cost Hamiltonian cycles containing M , we chose K to be the one for whichthis i is as large as possible. By the algorithm, when we considered square C i , we know that removing the smaller cost perfect matching disconnected the graph (or it wouldhave been chosen). Thus there must exist another square C j crossing the cut formedby this disconnection for which K chooses a different perfect matching than H , and j < i . By choice of index i , we know that the perfect matching used in C j by K has costgreater than or equal to the one used by H . Now consider the new Hamiltonian cycle K (cid:48) obtained by taking K , and swapping the perfect matchings used by K in squares C i and C j . We have c ( K (cid:48) ) = c ( K ) + w C i − w C j and since w C j ≥ w C i by the index ordering,we must have K (cid:48) is another minimum cost Hamiltonian cycle. Again there must exista square C r for which K (cid:48) uses a different perfect matching than H and the cost of thisperfect matching is strictly less than the one used by H , but by construction of K (cid:48) wehave r > i . But this contradicts our choice of K .We now propose the following algorithm for finding a tour of relatively small costfor cost functions optimized at square points. Algorithm (TOUR) for tours in the case of a square optimum for the subtour LPInput : Costs c ∈ R E n and a square point x optimizing c on the subtour LP, i.e. c ( x ) = LP ( c ) . Output : A tour in G x .1. Determine the support graph G x , and call (HAM) for the square graph ( G , M ) thatresults from replacing each 1-path of G x by one single edge whose cost is the sum ofthe costs of the replaced edges, and defining M to be the set of these single edges. Let H be the Hamiltonian cycle of G x obtained by taking the output of (HAM) and replacingall edges of M by their respective 1-paths.2. As in Theorem 6 and Lemma 3, determine the partition P of the -edges of G x intopairs whose classes are the perfect matchings of the squares of G x , and find the spanningtree F ∗ of G x of minimum cost having exactly one edge in each P ∈ P in polynomialtime with Edmonds matroid intersection algorithm [11].3. Let T F ∗ be the set of odd degree nodes in the graph ( V n , F ∗ ) . Find a minimum cost T F ∗ -join in G x . Note this can be done in polynomial time (cf. [21]).4. Let J ∗ be the union of F ∗ and the T F ∗ -join from Step 3, and output the one of H or J ∗ having smaller cost.We can now complete Corollary 3 with an algorithmic postulate. Theorem 10.
Let c ∈ R E n be a metric cost function optimized by a square point x,cx = LP ( c ) . Then there exists a Hamiltonian cycle of cost at most LP ( c ) in K n and(TOUR) can be used to find such a cycle in polynomial time.Proof. We have to prove only the latter part of the last sentence concerning the algo-rithm, as the rest has already been included in Corollary 3.Let H be any Hamiltonian cycle of the support graph G x that contains all of the1-edges, provided algorithmically in polynomial time by (HAM), and let J ∗ be the tourof G x determined in polynomial time by (TOUR). Note that the 1-tree F ∗ from Step 2of (TOUR) satisfies the condition of property (B), and thus by the Claim in the proofof Theorem 7 we have y : = x − χ H is in the T F ∗ -join polyhedron (6) of K n . Thus the cost of the minimum cost T F ∗ -join found in Step 3 is at most c ( y ) . Similarly, byTheorem 6, c ( F ∗ ) ≤ c ( x ) , which gives c ( J ∗ ) ≤ c ( x ) + c ( y ) . Thusmin { c ( H ) , c ( J ∗ ) } ≤ c ( H )+ c ( J ∗ ) ≤ c ( H )+ c ( x )+ ( c ( x ) − c ( H )) = LP ( c ) is the cost of a tour, and shortcutting this tour we get a Hamtiltonian cycle of K n notlarger in cost. (cid:117)(cid:116) Note that this proof actually used less than what (HAM) produces: for TOUR it issufficient to find any
Hamiltonian cycle in G x containing its 1-edges, not necessarilythe optimal one! However, sharper or more general results may need the exact opti-mum here. This motivates us to sketch some details about delta-matroids that are in thebackground.Delta-matroids were introduced by Bouchet [2]. For the introduction and the basicsabout them we follow [3]: the family D (cid:54) = /0 of subsets of a finite set S , or the pair ( S , D ) is called a delta-matroid if the following symmetric exchange axiom (also calledthe 2-step axiom) is satisfied: For D , D ∈ D and j ∈ D ∆ D there exists k ∈ D ∆ D such that D ∆ { j , k } ∈ D . Note that k = j is possible, and we then naturally define { j , j } : = { j } .A delta-matroid ( S , D ) may have an exponential number of elements, too manyto be given explicitly. Fortunately, in order to work with delta-matroids we need lessthan giving all of its elements as input. The basic and simple greedy algorithm alreadynecessitates a solution of the following problem [3, (2.1)]: let ( S , D ) be a delta-matroid,then for given A , B ⊆ S decide whether there exists D ∈ D such that D ⊇ A , D ∩ B = /0.Let us call an oracle that solves this problem the extendability oracle for the given delta-matroid. This oracle can be executed in polynomial time for all the relevant applications,and we will have to check that this also holds for the delta matroid for which we needit. Think about the set A as a set of elements chosen to be in the solution, and B the setof elements chosen not to be in the solution. Roughly, for an objective function c ∈ R S ,the greedy algorithm considers the elements of S in decreasing order of the absolutevalues and attempts to put a considered s ∈ S into A if c ( s ) ≥ A , and to B if c ( s ) ≤ B , where “it is possible”means a YES answer of the extendibility oracle with the attempted update of A and B (see precisely below).The following theorem is a generalization of Lemma 2 and therefore a direct proofof it. Given a square graph ( G , M ) , let R = R ( G ) ⊆ E ( G ) \ M be a reference set contain-ing exactly one edge from each square of E ( G ) \ M ; | R | = | V ( G ) | /
4. Let H = H ( G ) be the set of Hamiltonian cycles of G containing M . Theorem 11.
Let ( G , M ) be a square graph. Then H (cid:54) = /0 , and D : = { H ∩ R : H is a Hamiltonian cycle of G containing M } is a delta-matroid. Proof.
To prove the first part, replace one by one, one after the other in arbitrary order,the squares by one of their two perfect matchings.
Claim : With at least one of the two choices the graph remains connected.It follows that at the end of the procedure we get a connected graph with all degreesequal to 2 containing M , that is, a Hamiltonian cycle containing M .To prove the claim note that the graph G C we get after deleting the square C hasat most 2 components. So one of the two perfect matchings of C must join the twocomponents, since otherwise the graph obtained by adding back C is also not connected.The claim is proved.So we proved that D (cid:54) = /0. In order to prove that it is a delta-matroid, we have tocheck that for any two Hamiltonian cycles H (cid:54) = H , and square C of G − M where H and H do not use the same perfect matching of C, either H (cid:52) C is also a Hamiltoniancycle or there exists a square D of G − M so that H (cid:52) C (cid:52) D is a Hamiltonian cycle.
To prove this, suppose H (cid:52) C is not a Hamiltonian cycle. It is still a 2 -factor – subsetof edges with all degrees equal to 2 – with two components, with the cut Q ⊆ E ( G ) between the two components. Since H is connected, it contains a square D so that forone of the two perfect matchings of D : H ∩ D ∩ Q (cid:54) = /0. But then clearly, H and H donot use the same perfect matching of D , and H (cid:52) C (cid:52) D is again connected, and thus aHamiltonian cycle containing M . (cid:117)(cid:116) Let us call the delta-matroid D of the theorem square . It is the same delta-matroidas Bouchet’s delta-matroid of transitions in Eulerian trails [2]. Lemma 4.
For square delta-matroids the extendability oracle can be computed in poly-nomial time.Proof.
Assume A , B ⊆ R . If A ∩ B (cid:54) = /0 the answer of the extendability oracle is NO. Solet A ∩ B = /0.For edges a ∈ A , choose in the square of a the perfect matching containing a , andfor b ∈ B in the square of b the one that does not contain b .If the obtained graph is not connected, then clearly, the extendability oracle gives aNO answer. In case it is connected, replace each of the remaining squares (those disjointfrom A ∪ B ) one after the other by one of its two perfect matchings, so that it remainsconnected. One of the two choices does indeed keep the graph connected, since if not,adding both perfect matchings, (like in the proof of Theorem 11) it would also not beconnected. (cid:117)(cid:116) General Greedy Algorithm (GREEDY) for Delta-Matroids [3]
Input : Delta-matroid ( S , D ) given with the extendability oracle, and cost vector c ∈ R S . Task : Determine D ∈ D , equivalently the vector χ D , of minimum cost.1. Order the S decreasingly in the absolute values of c , that is, | c | ≥ | c | ≥ . . . | c | S | | , where we can suppose S = { , . . . , | S |} . Define i : = A : = B : = /0.2. i:=i+1 ; while i ≤ n do: - If c i ≤ A i : = A i − ∪ { i } and B i : = B i − , then keep this definition of A i , B i .In case the extendability oracle gives a NO answer, A i : = A i − , B i : = B i − ∪ { i } .(Since ( A i , B i ) is extendable, the answer for the latter choice is YES in this case.)- If c i > A i : = A i − and B i : = B i − ∪ { i } , then keep this definition of A i , B i .In case the extendability oracle gives a NO answer, A i : = A i − ∪ { i } , B i : = B i − .3. Output D : = A n . end It can be readily checked that { A n , B n } is a partition of S, A n ∈ D , and it is also notdifficult to check that c ( A n ) = min { c ( D ) : D ∈ D } . Moreover, (HAM) is a special case,and we have:
Theorem 12.
The output of (HAM) for a square graph ( G , M ) is a Hamiltonian cycleof G containing M of minimum cost. Acknowledgements
We are indebted to Michel Goemans for an email from his sailboat with a pointer toTheorem 6; to Alantha Newman, Frans Schalekamp, Kenjiro Takazawa and Anke vanZuylen for helpful discussions.
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