Improving on Best-of-Many-Christofides for T -tours
aa r X i v : . [ c s . D M ] S e p Improving on Best-of-Many-Christofides for T -tours Vera Traub ∗ Abstract
The T -tour problem is a natural generalization of TSP and Path TSP. Given a graph G = ( V, E ), edge cost c : E → R ≥ , and an even cardinality set T ⊆ V , we want to computea minimum-cost T -join connecting all vertices of G (and possibly containing parallel edges).In this paper we give an -approximation for the T -tour problem and show that theintegrality ratio of the standard LP relaxation is at most . Despite much progress for thespecial case Path TSP, for general T -tours this is the first improvement on Sebő’s analysisof the Best-of-Many-Christofides algorithm (Sebő [2013]). The traveling salesman problem (TSP) is one of the most classical problems in combinatorialoptimization. Given a set V of vertices and a metric c on V , we want to find an order v , . . . , v n of the vertices in V minimizing c ( v n , v ) + P ni =2 c ( v i − , v i ). Another definition of theTSP, which can be easily shown to be equivalent, is the following. Given a connected graph G = ( V, E ) and nonnegative edge costs c : E → R ≥ , find a minimum cost multi-subset F of E such that ( V, F ) is connected and Eulerian, i.e. every vertex has even degree. For many yearsbest known approximation algorithm for the TSP was the classical -approximation algorithmdue to Christofides [4] and Serdjukov [17]. Only very recently this approximation ratio wasimproved to − ε for some small ε > V of vertices and ametric c on V , we are given a start-vertex s ∈ V and an end-vertex t ∈ V . The task is to findan order s = v , . . . , v n = t of the vertices in V minimizing P ni =2 c ( v i − , v i ). As for the TSPwe can also formulate the Path TSP as a graph problem: given a connected graph G = ( V, E ),vertices s, t ∈ V ( s = t ), and nonnegative edge costs c : E → R ≥ , find a minimum costmulti-subset F of E such that ( V, F ) is connected and the set odd( F ) of odd-degree vertices in( V, F ) contains precisely the vertices s and t . In other words, s and t have odd degree, whileall other vertices have even degree.Christofides’ algorithm for the TSP can be generalized to the path version, but then it hasan approximation ratio of only as shown by Hoogeveen [7]. However in contrast to TSP, forthe path version we do know better approximation algorithms than Christofides’ algorithm.The first such approximation algorithm was given by An, Kleinberg, and Shmoys [1], whoproposed and analyzed the Best-of-Many-Christofides algorithm, which we will discuss in moredetail later in this paper. Subsequently, there has been a line of work [14, 22, 6, 16, 18, 24]improving the approximation ratio further. Moreover, there is a black-box reduction from the ∗ Department of Mathematics, ETH Zurich, Zurich, Switzerland. Email: [email protected]. Sup-ported by Swiss National Science Foundation grant 200021_184622. α -approximation algorithm for TSP, there also is an( α + ε )-approximation algorithm for the path version, for any fixed ε >
0. Combining thisblack-box reduction with their new approximation algorithm for TSP, Karlin, Klein, and OveisGharan [10] obtain a ( − ε )-approximation algorithm for Path TSP for some small ε > T -tour problem which is a natural generalization of TSP andits path version. An instance consists of a connected graph G = ( V, E ), a set T ⊆ V with | T | even, and nonnegative edge costs c : E → R ≥ . The task is to compute a a minimum costmulti-subset F of E such that ( V, F ) is connected and odd( F ) = T , i.e. vertices in T have odddegree and vertices in V \ T have even degree. In other words, F is a T -join (with possiblyparallel edges) and connects all vertices of G . The TSP is the special case T = ∅ and the PathTSP is the special case | T | = 2.However, many of the results for the Path TSP do not generalize to the T -tour problem.Cheriyan, Friggstad, and Gao [3] extended the Best-of-Many-Christofides algorithm by An,Kleinberg, and Shmoys [1] from | T | = 2 to general T and proved an approximation ratio of , which is slightly worse than the ratio √ obtained in [1] for | T | = 2. Then Sebő [14]improved the analysis of the same algorithm and showed that the Best-of-Many-Christofidesalgorithm yields an -approximation for the T -tour problem, which was also an improvementfor the Path TSP. Despite much further progress regarding the approximability of Path TSP,this result is the best previously known approximation ratio for the T -tour problem.The results from [6, 16, 19, 25] apply only to Path TSP, i.e. the case | T | = 2. The reasonfor this is that they all rely on a structural theorem by Gottschalk and Vygen [6], which cannotbe extended for the case | T | ≥ T -tour problem with | T | constant, but not to thegeneral case. This yields a ( − ε )-approximation algorithm for some small ε > | T | constant. For unit-weight graphs, i.e. the special case where c ( e ) = 1 for every edge e ∈ E , a -approximation algorithm is known for general | T | [15].In this paper we give the first improvement of the approximation guarantee for the gen-eral T -tour problem over Sebő’s -approximation algorithm [14]. Our main result is an -approximation algorithm. We analyze the algorithm with respect to the standard LP relaxation(see (1) in Section 2) and we therefore also prove an upper bound of on the integrality ratioof this relaxation. We will analyze our algorithm with respect to the following LP relaxation.min c ( x ) s.t. x ( δ ( U )) ≥ ∅ 6 = U ( V with | T ∩ U | even x ( δ ( W )) ≥ |W| − W of Vx e ≥ e ∈ E, (1)where δ ( U ) is the set of edges with exactly one endpoint in U and δ ( W ) denotes the set ofedges with endpoints in different elements of the partition W . The LP (1) can be solved inpolynomial time using the ellipsoid method. Barahona and Conforti [2] show that one can2eparate the even cut constraints x ( δ ( U )) ≥ ∅ 6 = U ( V with | T ∩ U | even in polynomialtime. The other constraints define the connector polyhedron n x ∈ R E ≥ : x ( δ ( W )) ≥ |W| − W of V o , (2)which is the convex hull of all multi-subsets F of E for which ( V, F ) is connected. (See e.g.Section 50.5 in [13].) Since one can optimize over (2) in polynomial time, one can also separateits constraints in polynomial time.Let x ∗ be an optimum solution to (1). In the following we denote by S the set of alledge sets of spanning trees of our given graph G . Since the constraints of (1) imply that x ∗ is contained in the connector polyhedron (2), the vector x ∗ dominates a convex combinationof incidence vectors of spanning trees, i.e. there are coefficients p S ≥ S ∈ S such that P S ∈S p S = 1 and x ∗ ≥ X S ∈S p S · χ S . (3)To prove our main result, we will use two different algorithms and bound the cost ofthe better of the two resulting T -tours. One of these two algorithms is the Best-of-Many-Christofides algorithm, proposed by An, Kleinberg, and Shmoys [1] for the Path TSP andextended to T -tours by Cheriyan, Friggstad, and Gao [3]. The algorithm proceeds as follows. Algorithm 1:
Best-of-Many-Christofides1. Compute an optimum solution x ∗ to the LP (1).2. Find a convex combination (3) of spanning trees dominated by x ∗ .3. For every S ∈ S with p S >
0, compute a cheapest (odd( S ) △ T )-join J ∗ S .4. Return the cheapest of the resulting T -tours S . ∪ J ∗ S .The cost of the tour computed by Algorithm 1 ismin S ∈S : p S > ( c ( S ) + c ( J ∗ S )) ≤ X S ∈S p S · ( c ( S ) + c ( J ∗ S )) ≤ c ( x ∗ ) + X S ∈S p S · c ( J ∗ S ) . To bound the cost of the (odd( S ) △ T )-join J ∗ S , both [1] and [14] follow Wolsey’s analysis [23]of Christofides’ algorithm for TSP. Since every vector contained in the (odd( S ) △ T )-joinpolyhedron n y ∈ R E ≥ : y ( δ ( U )) ≥ U with | U ∩ (odd( S ) △ T ) | odd o (4)dominates a convex combination of incidence vectors of (odd( S ) △ T )-joins [5], we have c ( J ∗ S ) ≤ c ( y S ) for every vector y S in (4). We call a vector in (4) a parity correction vector . The maindifficulty in the analysis of the Best-of-Many-Christofides algorithm is to construct a cheapparity correction vector.If T = ∅ (which is the special case TSP), the vector x ∗ is a feasible parity correctionvector. To see this, note that in this case | T ∩ U | is even for every ∅ 6 = U ( V and hence3 ∗ ( C ) ≥ C . However, for T = ∅ this is not necessarily the case. We call a cut C with x ∗ ( C ) < narrow and denote by N := { δ ( U ) : x ∗ ( δ ( U )) < } the set of narrow cuts . By the constraints of (1), every narrow cut C is a T -cut, i.e. C = δ ( U )for some U ⊂ V with | U ∩ T | odd.Recall that for a parity correction vector y we require y ( C ) ≥ C is an (odd( S ) △ T )-cut. Let U ⊆ V such that δ ( U ) ∈ N is a narrow cut. Then | T ∩ U | is odd. Hence, | U ∩ (odd( S ) △ T ) | is odd if and only if | U ∩ odd( S ) | is even. This is the case if and only if | δ ( U ) ∩ S | is even. In particular, a narrow cut C with | S ∩ C | = 2 is an (odd( S ) △ T )-cut,while a narrow cut C with | S ∩ C | = 1 is not.If | S ∩ C | = 1 for a narrow cut C , we say that C is lonely for S . Then we also say thatthe unique edge e ∈ C ∩ S is lonely at C . Lonely cuts and edges play a special role in Sebő’s[14] analysis of the Best-of-Many-Christofides algorithm in two ways. First, the lonely cutsof a tree ( V, S ) are important since they don’t need parity correction, meaning that they areno (odd( S ) △ T )-cuts. Second, the incidence vectors χ e of lonely edges are used to constructcheap parity correction vectors. Here, the vector χ e for an edge e that is lonely in a tree ( V, S )is used to construct the parity correction vectors y S ′ for other trees ( V, S ′ ). (See Section 4 formore details.)Besides the Best-of-Many-Christofides algorithm we will analyze another algorithm for the T -tour problem to prove our main result. This algorithm builds on an algorithm by Sebőand van Zuylen [16] for the case | T | = 2. They start with a particular convex combinationof incidence vectors of spanning trees, which was shown to exist by Gottschalk and Vygen [6](see also [12]). For every spanning tree S contributing to the convex combination, they nowdelete some edges to obtain a forest F S . Then they compute an (odd( F S ) △ T )-join J ∗ S . Nowodd( F S . ∪ J ∗ S ) = T , but ( V, F
S . ∪ J ∗ S ) might be disconnected. Therefore, they finally reconnectby adding two copies of some edges to obtain a T -tour.Sebő and van Zuylen [16] show that on average they save more by deleting edges than theyneed to pay in the final reconnection step. The reason why their result does not carry over togeneral T -tours is that their analysis crucially relies on the structure of the convex combinationof incidence vectors of spanning trees which one cannot achieve for | T | ≥ Let us now explain our new approximation algorithm for the T -tour problem and outline itsanalysis. We proceed as in the algorithm by Sebő and van Zuylen [16] for the case | T | = 2, butwe start with an arbitrary convex combination (3) instead of one with additional structure. Inthe following we denote by L S the set of lonely cuts of a spanning tree ( V, S ), i.e. the set ofnarrow cuts C ∈ N with | S ∩ C | = 1. Moreover, L S denotes the set of lonely edges.4igure 1: The left picture shows a spanning tree ( V, S ), where the edges in F S are solid andthe lonely edges are dashed. The squares with white interior represent the set T and the filledcircles represent V \ T . The right picture shows a T -tour resulting from this tree. It consists ofthe forest F S , an (odd( F S ) △ T )-join J S (dotted), and 2 R S (the doubled edges). For every edge { v, w } ∈ J S , the set R S contains two copies of all but one of the lonely edges of the v - w -pathin ( V, S ). Algorithm 2:
Best-of-Many-Christofides with lonely edge deletion1. Compute an optimum solution x ∗ to the LP (1).2. Find a convex combination (3) of spanning trees dominated by x ∗ andcompute the set N of narrow cuts.3. For every S ∈ S with p S >
0, let L S = S C ∈L S ( S ∩ C ) and F S := S \ L S .Compute an (odd( F S ) △ T )-join J ∗ S with minimum c S ( J ∗ S ), where for an edge ec S ( e ) := c ( e ) + 2 · X C ∈L S : e ∈ C c ( S ∩ C ) − max C ∈L S : e ∈ C c ( S ∩ C ) , where max ∅ := 0.4. Compute a cheapest set R S of edges such that ( V, F S ∪ J ∗ S ∪ R S ) is connected.5. Return the cheapest of the resulting T -tours S . ∪ J ∗ S . ∪ R S . ∪ R S .In order to implement step 2 in polynomial time one can use any polynomial-time algorithmfor enumerating all near-minimum cuts [8, 9, 11, 21].The cost function c S is chosen to anticipate the cost for reconnection. More precisely, wecan observe the following, which is shown in [16] for | T | = 2. See Figure 1. Lemma 1.
In Algorithm 2 we have for every S ∈ S with p S > c ( J ∗ S ) + 2 · c ( R S ) ≤ c S ( J ∗ S ) . Proof.
Since J ∗ S is an (odd( F S ) △ T )-join and every lonely cut of S is an (odd( F S ) △ T )-cut,we have | C ∩ J ∗ S | ≥ C ∈ L S . For an edge e ∈ J ∗ S let R e := S C ∈L S : e ∈L S ( C ∩ S ). Then S ⊆ F S ∪ S e ∈ J ∗ S R e . 5or e ∈ J ∗ S let R ′ e result from R e by removing its most expensive element. Consider a lonelyedge l in the unique cycle in S ∪ { e } . Then there is a unique (lonely) cut C with { l } = C ∩ S .For this cut C we have e ∈ C , implying l ∈ R e . Therefore, F S ∪ R e ∪ { e } contains a cycle.This implies that F S ∪ J ∗ S ∪ S e ∈ J ∗ S R ′ e is connected. Hence, c ( J ∗ S ) + 2 · c ( R S ) ≤ c ( J ∗ S ) + 2 · X e ∈ J ∗ S c ( R ′ e ) = c S ( J ∗ S ) . As in [16] we construct a vector ¯ y S in the (odd( F S ) △ T )-join polyhedron to bound the cost c S ( J ∗ S ) of parity correction and reconnection. However, we now have the following difficulty.Like Sebő [14] and Gottschalk and Vygen [6], Sebő and van Zuylen [16] also use the incidencevector χ e of a lonely edge of tree S to construct the parity correction vectors ¯ y S ′ for other trees S ′ . To bound the resulting expected reconnection cost c S ′ ( e ) − c ( e ), Sebő and van Zuylen exploitthe particular structure of the convex combination they work with. Without this structure wecannot give a sufficiently good bound anymore and hence we will not use the incidence vectorsof lonely edges to construct parity correction vectors for other trees.Instead, our parity correction vector ¯ y S will consist only of a fraction of x ∗ and incidencevectors of edges of the tree S itself. This will allow us to control the reconnection cost: tobound the cost c S ( x ∗ ) − c ( x ∗ ) we can generalize an argument from [16] (see Lemma 6) and foran edge e ∈ S we have c S ( e ) = c ( e ). Since using a parity correction vector consisting not onlyof a fraction of x ∗ and incident vectors of edges of the tree S itself was crucial in all of theimprovements [1, 14, 22, 6, 16] over Christofides’ algorithm for | T | = 2, we need new insightsto obtain a good bound anyways.The key idea is that the deletion of lonely edges can help parity correction in the followingsense. If a narrow cut C contains exactly two edges of a tree ( V, S ), then this cut is an(odd( S ) △ T )-cut. Now suppose one of the two edges in C ∩ S is a lonely edge of S . Then | F S ∩ C | = 1 and hence C is not an (odd( F S ) △ T )-cut. (Note that one can show that it isimpossible that both edges in C ∩ S are lonely edges of S .)Of course, it might happen that a narrow cut C with | C ∩ S | = 2 does not contain a lonelyedge. However, in this case Sebő’s analysis of the Best-of-Many-Christofides algorithm is nottight as we will show in Section 4. The overall approximation algorithm that we analyze is thefollowing. Apply Algorithm 1 and Algorithm 2 and return the cheaper of the two resulting T -tours. In this section we present the details of the analysis of the Best-of-Many-Christofides algorithmfrom [14], which builds on [1]. We include this analysis here for completeness. Moreover, whileanalyzing the Best-of-Many-Christofides algorithm, we will introduce some notation and usefulfacts that we also need later on.For S ∈ S we denote by I S the unique T -join contained in S . Moreover, we define J S := S \ I S . Then J S is the unique ( T △ odd( S ))-join in S . In the following we write I p := P S ∈S p S · χ I S and J p := P S ∈S p S · χ J S . Then we have x ∗ = I p + J p . The following is well-known. 6 emma 2.
The Best-of-Many-Christofides algorithm (Algorithm 1) returns a solution to the T -tour problem of cost at most c ( x ∗ ) + c ( J p ) . Proof.
Recall that J S is an ( T △ odd( S ))-join in S for every S ∈ S . Hence, the cheapest of the T -tours S . ∪ J ∗ S with p S > S ∈S : p S > ( c ( S ) + c ( J ∗ S )) ≤ X S ∈S p S ( c ( S ) + c ( J ∗ S )) ≤ X S ∈S p S ( c ( S ) + c ( J S )) = c ( x ∗ ) + c ( J p ) . If this does not yields an approximation ratio of better than , we have c ( J p ) ≥ c ( x ∗ ) andhence c ( I p ) ≤ c ( x ∗ ).One important observation by An, Kleinberg, and Shmoys [1] is that for a narrow cut C with x ∗ ( C ) much smaller than 2, a large fraction of the spanning trees will be lonely at C .More precisely, by (3) we have x ∗ ( C ) ≥ − X S : C ∈L S p S (5)for every narrow cut C and hence X S : C ∈N \L S p S ≤ x ∗ ( C ) − . (6)Let now ( V, S ) be a spanning tree. Sebő [14] uses the following parity correction vector tobound the cost of a cheapest (odd( S ) △ T )-join: y S := 12 x ∗ + α · χ I S + X C ∈N \L S max { − x ∗ ( C ) − α, } · v C , where α ≥ v C := 12 − x ∗ ( C ) · X S ∈S : C ∈L S p S · χ S ∩ C . Note that the T -join I S intersects all narrow cuts because all narrow cuts are T -cuts. Moreover, v C ( C ) ≥ C because of (5).We now show that y S is indeed a parity correction vector for S , i.e. it is contained in the(odd( S ) △ T )-join polyhedron. Lemma 3.
For every (odd( S ) △ T ) -cut C we have y S ( C ) ≥ .Proof. Let C be a cut. If C / ∈ N , we have y S ( C ) ≥ x ∗ ( C ) ≥
1. Otherwise, the constraintsof (1) imply that C is a T -cut.If C ∈ L S ⊆ N , we have | S ∩ C | = 1 and hence C is also an odd( S )-cut. Therefore C isnot an (odd( S ) △ T )-cut.Now consider the remaining case C ∈ N \ L S . Since C is a T -cut, we have | I S ∩ C | ≥ v C ( C ) ≥
1, this implies y S ( C ) ≥ x ∗ ( C ) + α + max { − x ∗ ( C ) − α, } ≥ . emma 4. Let α ≥ . Then the Best-of-Many-Christofides algorithm (Algorithm 1) returnsa solution to the T -tour problem of cost at most c ( x ∗ ) + α · c ( I p ) + X C ∈N ( x ∗ ( C ) − · max { − x ∗ ( C ) − α, } · c ( v C ) . Proof.
By Lemma 3, Algorithm 1 returns a T -tour of cost at mostmin S ∈S : p S > (cid:16) c ( S ) + c ( y S ) (cid:17) ≤ X S ∈S p S · (cid:16) c ( S ) + c ( y S ) (cid:17) ≤ c ( x ∗ ) + α · c ( I p ) + X C ∈N X S ∈S : C / ∈L S p S · max { − x ∗ ( C ) − α, } · c ( v C ) ≤ c ( x ∗ ) + α · c ( I p ) + X C ∈N ( x ∗ ( C ) − · max { − x ∗ ( C ) − α, } · c ( v C ) , where we used (6) in the last inequality.Sebő [14] now completes the analysis as follows. If an edge e of S is lonely at a narrow cut C , we have 1 = | C ∩ S | ≥ | C ∩ I S | ≥ C is a T -cut and I S is a T -join) and hence e ∈ I S . This shows L S ⊆ I S . Therefore, with L p := P S ∈S p S · χ L S , we have X C ∈N (2 − x ∗ ( C )) · v C = L p ≤ I p . Using this and ( x − · (1 − x − )2 − x ≤ for 1 ≤ x <
2, one can show that Lemma 4 for α = impliesan approximation ratio of if c ( I p ) ≤ · c ( x ∗ ). Otherwise, Lemma 2 implies the approximationratio . This analysis is only tight if c ( L p ) = c ( I p ) = c ( x ∗ ). In this section we show the following.
Lemma 5.
Algorithm 2 returns a solution to the T -tour problem of cost at most c ( x ∗ ) + 15 c ( I p ) − c ( L p ) − X C ∈N (2 − x ∗ ( C )) · c ( v C ) . Note that the bound in Lemma 5 is smaller than c ( x ∗ ) in the case where Sebő’s [14] analysisof the Best-of-Many-Christofides algorithm is tight: then we have c ( L p ) = c ( I p ) = c ( x ∗ ).In the rest of this section we prove Lemma 5. Let ( V, S ) be a spanning tree and F S := S \ L S .The following lemma bounds the average cost for reconnection. It is essentially due to Sebőand van Zuylen [16] who proved it for the Path TSP. Their proof can be generalized to the T -tour problem as we show below. Lemma 6.
We have c S ( x ∗ ) − c ( x ∗ ) ≤ · P C ∈L S ( x ∗ ( C ) − · c ( S ∩ C ) . roof. We consider the directed bipartite auxiliary graph with vertex set E . ∪ L S and arc set A := { ( e, C ) : e ∈ E, C ∈ L S , e ∈ C } . We claim that there exists a function f : A → R ≥ inthis auxiliary graph such that f ( δ + ( e )) ≤ x ∗ e for all e ∈ E and f ( δ − ( C )) ≥ C ∈ L S .By the Hall condition such a function f exists if and only if for every subset L ′ ⊆ L S x ∗ [ C ∈L ′ C ≥ |L ′ | . (7)Let L ′ ⊆ L S and let L ′ ⊆ L S be the set of lonely edges of S that are contained in of thecuts L S . Note that every lonely edge l ∈ L ′ is contained in only one lonely cut, namely thefundamental cut of l in the tree S . Since every cut in L S contains exactly one lonely edge of S , this implies | L ′ | = |L ′ | . Let W be the partition of V that consists of the vertex sets of theconnected components of ( V, S \ L ′ ). Then |W| = |L ′ | + 1. Moreover, for every cut C ∈ L ′ and every edge { v, w } ∈ C , the unique edge l ∈ C ∩ S is contained in the unique v - w -path in S . Since l ∈ L ′ , this implies that v and w are contained in different connected components of( V, S \ L ′ ) and hence { v, w } ∈ δ ( W ). Using the constraints of the LP (1), we therefore get x ∗ [ C ∈L ′ C ≥ x ∗ ( δ ( W )) ≥ |W| − |L ′ | . This shows that a function f as claimed does indeed exist. Thus we have c S ( x ∗ ) − c ( x ∗ ) = X e ∈ E x ∗ e · · X C ∈L S : e ∈ C c ( S ∩ C ) − max C ∈L S : e ∈ C c ( S ∩ C ) ≤ X e ∈ E X C ∈L S : e ∈ C x ∗ e · c ( S ∩ C ) − X e ∈ E · X C ∈L S : e ∈ C f ( e, C ) · max C ∈L S : e ∈ C c ( S ∩ C ) ≤ X e ∈ E X C ∈L S : e ∈ C x ∗ e · c ( S ∩ C ) − X e ∈ E · X C ∈L S : e ∈ C f ( e, C ) · c ( S ∩ C )= X C ∈L S · c ( S ∩ C ) · X e ∈ C ( x ∗ e − f ( e, C )) ≤ X C ∈L S · c ( S ∩ C ) · ( x ∗ ( C ) − . We now construct a parity correction vector ¯ y S for F S . Let¯ y S := 25 x ∗ + 15 χ S + 15 χ I S \ L S + X C ∈L S
25 (2 − x ∗ ( C )) · χ S ∩ C . Before formally proving that ¯ y S is a parity correction vector for F S , let us briefly describe thepurpose of the different parts. The term x ∗ + χ S is used to ensure ¯ y S ( C ) ≥ S or is not narrow. A narrow cut C that containsprecisely two edges of S will either have the “correct parity” after deleting the lonely edges,i.e. it contains exactly one edge of F S and is therefore not an (odd( F S ) △ T )-cut, or it contains9o lonely edge. In the latter case, the term χ I S \ L S will contribute to ¯ y S ( C ). Finally, the lastterm is used for the lonely cuts of S . These are all (odd( F S ) △ T )-cuts.We now formally prove that ¯ y S is indeed contained in the (odd( F S ) △ T )-join polyhedron. Lemma 7.
For every (odd( F S ) △ T ) -cut C we have ¯ y S ( C ) ≥ .Proof. Let C be an (odd( F S ) △ T )-cut. If C is not narrow, we have¯ y S ( C ) ≥ x ∗ ( C ) + | S ∩ C | ≥ · · . Let now C ∈ N . Then C is a T -cut. Since C is an (odd( F S ) △ T )-cut, this implies | F S ∩ C | even. Hence it suffices to consider the following three cases. Case 1: | F S ∩ C | = 0 and | S ∩ C | ≤ C is narrow, it is a T -cut and hence | I S ∩ C | is odd. This implies | I S ∩ C | = 1.Recall that S \ F S = L S ⊆ I S . Therefore, | ( S \ F S ) ∩ C | ≤ | I S ∩ C | = 1. Using | F S ∩ C | = 0,we conclude | S ∩ C | = 1 and hence C ∈ L S . Thus,¯ y S ( C ) ≥ x ∗ ( C ) + | S ∩ C | + (2 − x ∗ ( C )) | S ∩ C | = x ∗ ( C ) + + (2 − x ∗ ( C )) = 1 . Case 2: | S ∩ C | = | F S ∩ C | = 2.Because F S = S \ L S , we have L S ∩ C = ∅ . Recall that C ∈ N and hence C is a T -cut,implying | C ∩ I S | ≥
1. Therefore, | ( I S \ L S ) ∩ C | ≥
1. We conclude¯ y S ( C ) ≥ x ∗ ( C ) + | S ∩ C | + | ( I S \ L S ) ∩ C | ≥ · · · . Case 3: | S ∩ C | ≥ y S ( C ) ≥ x ∗ ( C ) + | S ∩ C | ≥ · · e ∈ S we have c S ( e ) = c ( e ). Therefore, byLemma 6 we have c S (¯ y S ) − c (¯ y S ) ≤ c S ( x ∗ ) − c ( x ∗ ) ≤ X C ∈L S · ( x ∗ ( C ) − · c ( S ∩ C ) . (8)Hence, c S (¯ y S ) ≤ c ( x ∗ ) + c ( S ) + c ( I S \ L S ) + X C ∈L S (cid:16) (2 − x ∗ ( C )) + ( x ∗ ( C ) − (cid:17) c ( S ∩ C )= c ( x ∗ ) + c ( S ) + c ( I S \ L S ) + X C ∈L S x ∗ ( C ) · c ( S ∩ C )= c ( x ∗ ) + c ( S ) + c ( I S \ L S ) + c ( L S ) − X C ∈L S (2 − x ∗ ( C )) · c ( S ∩ C ) . Moreover, c ( F S ) = c ( S ) − c ( L S ), implying c ( F S ) + c S (¯ y S ) ≤ c ( x ∗ ) + c ( S ) + c ( I S \ L S ) − c ( L S ) − X C ∈L S (2 − x ∗ ( C )) · c ( S ∩ C )= c ( x ∗ ) + c ( S ) + c ( I S ) − c ( L S ) − X C ∈L S (2 − x ∗ ( C )) · c ( S ∩ C ) . T -tours has cost at mostmin S ∈S : p S > (cid:16) c ( F S ) + c S (¯ y S ) (cid:17) ≤ X S ∈S p S · (cid:16) c ( F S ) + c S (¯ y S ) (cid:17) ≤ X S ∈S p S · c ( x ∗ ) + c ( S ) + c ( I S ) − c ( L S ) − X C ∈L S (2 − x ∗ ( C )) · c ( S ∩ C ) = c ( x ∗ ) + c ( I p ) − c ( L p ) − X C ∈N (2 − x ∗ ( C )) · c ( v C ) . This completes the proof of Lemma 5.
We now combine the results from the previous sections to prove our main result.
Theorem 8.
There is a polynomial-time algorithm that computes for every instance of the T -tour problem a solution of cost at most times the value of the LP (1) .Proof. The algorithm we analyze is the following. Run Algorithm 1 and Algorithm 2 and returnthe better of the two resulting T -tours. Then each of Lemma 2, Lemma 4, and Lemma 5 yieldsan upper bound on the cost of the resulting T -tour. We will take a convex combination of thesethree upper bounds. The bounds from Lemma 2, Lemma 4, and Lemma 5 will be weighted with λ , λ , λ , respectively. We set λ := , λ := , λ := , and α := . Then λ , λ , λ ≥ λ + λ + λ = 1. Hence, our algorithm computes a T -tour of cost at most λ · (cid:16) c ( x ∗ ) + c ( J p ) (cid:17) + λ · c ( x ∗ ) + α · c ( I p ) + X C ∈N ( x ∗ ( C ) − · max { − x ∗ ( C ) − α, } · c ( v C ) ! + λ · c ( x ∗ ) + c ( I p ) − c ( L p ) − X C ∈N (2 − x ∗ ( C )) · c ( v C ) ! = (cid:16) λ + λ (cid:17) · c ( x ∗ ) + λ · c ( J p ) + (cid:16) α · λ + λ (cid:17) · c ( I p ) − λ · c ( L p )+ X C ∈N (cid:16) ( x ∗ ( C ) − · max { − x ∗ ( C ) − α, } · λ − (2 − x ∗ ( C )) · λ (cid:17) · c ( v C )= · c ( x ∗ ) + · c ( J p ) + · c ( I p ) − · c ( L p )+ X C ∈N (cid:16) ( x ∗ ( C ) − · max { − x ∗ ( C ) , } · − · (2 − x ∗ ( C )) (cid:17) · c ( v C )= · c ( x ∗ ) − · c ( L p )+ X C ∈N (cid:16) ( x ∗ ( C ) − · max { − x ∗ ( C ) , } · − · (2 − x ∗ ( C )) (cid:17) · c ( v C ) , x ∗ = J p + I p . 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