\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{13}{9}$\end{document}-Approximation for Graphic TSP

The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{3}{2}$\end{document}, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{4}{3}$\end{document}. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{13}{9}$\end{document} on the approximation factor, as well as a bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{19}{12}+\varepsilon$\end{document} for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

two (Papadimitriou et al. [13]), and the best known approximation factor of 3 2 was obtained by Christofides [2] more than thirty years ago. However, the so-called Held-Karp LP relaxation of TSP is conjectured to have an integrality gap of 4 3 . It is known to have a gap at least that big, however the best known upper bound [14] for the gap is equal to 3 2 , and is given by Christofides's algorithm. In a more general version of the problem, called the Travelling Salesman Path Problem (TSPP), in addition to a metric (V , d) we are also given two points s, t ∈ V and the goal is to find a path from s to t visiting each point exactly once, except if s and t are the same point in which case it can be visited twice (this is when TSPP reduces to TSP). For this problem, the best approximation algorithm known is that of Hoogeveen [7] with an approximation factor of 5 3 . However, the Held-Karp relaxation of TSPP is conjectured to have an integrality gap of 3 2 . One of the natural directions of attacking these problems is to consider special cases and several attempts of this nature has been made. Among the most interesting is the graphic TSP/TSPP, where we assume that the given metric is the shortest path metric of an undirected graph. Equivalently, in graphic TSP we are given an undirected graph G = (V , E) and we need to find a shortest tour that visits each vertex at least once. Yet another equivalent formulation asks for a minimum size Eulerian multigraph spanning V and only using edges of G. Similar equivalent formulations apply to the graphic TSPP case. The reason why these special cases are interesting is that they seem to include the difficult inputs of TSP/TSPP. Not only are they APX-hard (see [5]), but also the standard examples showing that the Held-Karp LP relaxation has a gap of at least 4 3 in the TSP case and 3 2 in the TSPP case, are in fact graphic metrics (see Figs. 1 and 2).
Very recently, significant progress has been made in approximating the graphic TSP and TSPP. First, Oveis Gharan et al. [12] gave an algorithm with an approximation factor 3 2 − ε for graphic TSP. Despite ε being of the order of 10 −12 , this is considered a major breakthrough. Following that, Mömke and Svensson [8] obtained a significantly better approximation factor of 14( √ 2−1) 12 √ 2−13 ≈ 1.461 for graphic TSP, as well as factor 3 − √ 2 + ε ≈ 1.586 + ε for graphic TSPP, for any ε > 0. Their approach uses matchings in a truly ingenious way. Whereas earlier approaches (including that of Christofides [2] as well as Oveis Gharan et al. [12]) add edges of a matching to a spanning tree to make it Eulerian, the new approach is based on adding and removing the matching edges. This process is guided by a so-called removable pairing of edges which essentially encodes the information about which edges can be simultaneously removed from the graph without disconnecting it. A large removable pairing of edges is found by computing a minimum cost circulation in a certain auxiliary flow network, and the bounds on the cost of this circulation translate into bounds on the size of the resulting TSP tour/path.

Remark 1
Since the announcement of the preliminary version of this work, several improved approximation algorithms have been found. An et al. [1] gave a factor of 1+ √ 5 2 ≈ 1.618 for the general metric TSPP, as well as a factor of ≈ 1.578 for the graphic TSPP. Sebő and Vygen [10] improved the ratio for graphic TSPP to 3 2 , which is tight w.r.t. the Held-Karp LP relaxation. They also gave a 7 5 -approximation algo-rithm for graphic TSP. Finally, Sebő [9] announced an 8 5 -approximation algorithm for the general metric TSPP.

Our Results
In this paper we present an improved analysis of the cost of the circulation used by Mömke and Svensson [8] in the construction of the TSP tour/path. Our results imply a bound of 13 9 ≈ 1.444 on the approximation factor for the graphic TSP, as well as a 19 12 + ε ≈ 1.583 + ε bound for the graphic TSPP, for any ε > 0. The circulation used in [8] consists of two parts: the "core" part based on an optimal extreme point solution to the Held-Karp LP relaxation of TSP, and the "correction" part that adds enough flow to the core part to make it feasible. We improve bounds on costs of both parts, in particular we show that the second part is, in a sense, free. In particular, we obtain the same upper-bounds for the total cost of both parts as for the first part alone. As for the first part, similarly to the original proof of Mömke and Svensson, our proof exploits its knapsack-like structure. However, we use the 2-dimensional knapsack problem in our analysis, instead of the standard knapsack problem. Not only does this lead to an improved bound, it is also in our opinion a cleaner one. In particular, we also provide an essentially matching lower bound on the cost of the core part, which means that any further progress on bounding that cost has to take into account more than just the knapsack-like structure of the circulation.

Organization of the Paper
In the next section we present previous results relevant to the contributions of this paper. In particular we recall key definitions and theorems of Mömke and Svensson [8].
In Sect. 3 we present the improved upper bound on the cost of the core part of the circulation, as well as an essentially matching lower bound. In Sect. 4 we prove that the correction part of the circulation is, in a sense, free. Finally, in Sect. 5 we apply the results of the previous sections to obtain improved approximation algorithms for graphic TSP and TSPP.

Preliminaries
In this section we review some standard results concerning TSP/TSPP approximation and recall the parts of the work of Mömke and Svensson [8] relevant to the contributions of this paper.

Held-Karp LP Relaxation and the Algorithm of Christofides
The Held-Karp LP relaxation (or subtour elimination LP) for graphic TSP on graph G = (V , E) can be formulated as follows (see [4,6,8] for details on equivalence between different formulations): min e∈E x e subject to x δ(S) ≥ 2 for ∅ = S ⊂ V , where x e ≥ 0. Fig. 1 OPT LP (G) for the graph above (so-called prism graph) is 3n-simply put x e = 1 for all horizontal edges e, and x e = 1 2 for the remaining edges. On the other hand, visiting all vertices requires going through 4n − 2 edges, and so in this case the integrality gap of LP(G) can approach 4 3 with n → ∞ Here δ(S) denotes the set of all edges between S and V \ S for any S ⊆ V , and x(F ) denotes e∈F x e for any F ⊆ E. We will refer to this LP as LP(G) and denote the value of any of its optimal solutions by OPT LP (G).
The approximation ratio of the classic 3 2 -approximation algorithm for metric TSP due to Christofides [2] is related to OPT LP (G) as follows: Theorem 1 (Wolsey [15], Shmoys and Williamson [14]) The cost of the solution produced by the algorithm of Christofides on a graph G is bounded by n + OPT LP (G)/2, and so its approximation factor is at most On the other hand, the graph in Fig. 1 shows that the integrality gap of LP(G) can be as large as 4 3 . The Held-Karp LP relaxation can be generalized to the graphic TSPP in a straightforward manner. Suppose we want to solve the problem for a graph G = (V , E) and endpoints s, t. Let Φ = {S ⊆ V : |{s, t} ∩ S| = 1}. Then the relaxation can be written as We denote this generalized program by LP(G, s, t) and its optimum value by OPT LP (G, s, t). It is clear that OPT LP (G, v, v) = OPT LP (G) for any v ∈ V . The example in Fig. 2 shows that the integrality gap of integrality gap of LP(G, s, t) can be as large as 3 2 . Let G = (V , E ∪ {e }), where e = {s, t}. From any feasible solution to LP(G, s, t) we can obtain a feasible solution to LP(G ) by adding 1 to x e . Therefore

Reduction to Minimum Cost Circulation
The authors of [8] use the optimal solution of LP(G) to construct a low cost circulation in a certain auxiliary flow network. This circulation is then used to produce a small TSP tour for G. We will now describe the construction of the flow network and the relationship between the cost of the circulation and the size of the TSP tour. Let us start with the following reduction

Lemma 1 (Lemma 2.1 and Lemma 2.1(generalized) of Mömke and Svensson [8])
If there exists a polynomial time algorithm that for any 2-vertex connected graph G returns a graphic TSP solution of cost at most r · OPT LP (G), then there exists an algorithm that does the same for any connected graph. Similarly, if there exists a polynomial time algorithm that for any 2-vertex connected graph G and its two vertices s, t returns a graphic TSPP solution of cost at most r · OPT LP (G, s, t), then there exists an algorithm that does the same for any connected graph.
We will henceforth assume that the graphs we work with are all 2-vertexconnected. Let G be such graph. We now construct a certain auxiliary flow network corresponding to G.
Let T be a depth first search spanning tree of G with an arbitrary root vertex r. Direct all edges of T (called tree-edges) away from the root, and all other edges (called back-edges) towards the root. Let G be the resulting directed graph, and let T be its subgraph corresponding to T . Where necessary to avoid confusion, we will use the name arcs (and tree-arcs and back-arcs) for the edges of this directed graph. The flow network is obtained from G by replacing some of its vertices with gadgets, as described below.
Let v be any non-root vertex of G having l children: w 1 , . . . , w l in T . We introduce l new vertices v 1 , . . . , v l and replace the tree-arc (v, w j ) by tree-arcs (v, v j ) and (v j , w j ) for j = 1, . . . , l. We also redirect to v j all the back-arcs leaving the subtree rooted at w j and entering v (see Fig. 3). We will call the new vertices and the root in-vertices and the remaining vertices out-vertices. We will also denote the set of all in-vertices by I, and the set of in-vertices in the gadget corresponding to v by I v . Notice that all the back-arcs go from out-vertices to in-vertices, and that each invertex has exactly one outgoing arc (for the root vertex this follows from 2-vertex connectivity).
is the set of incoming back-arcs of v. This basically means that the cost is 0 for tree-arcs and 1 for back-arcs, except that for every in-vertex the first unit of circulation using a back-arc is free. The circulation network described above will be denoted C(G, T ). For any circulation C, we will use |C| to denote its cost as described above.
It is worth noting that the cost function of C(G, T ) can be simulated using the usual fixed-cost arcs by introducing an extra vertex v for each in-vertex v, redirecting all in-arcs of v to v and putting two arcs from v to v: one with capacity of 1 and cost 0, and the other with capacity ∞ and cost 1. Note, that this is the only place where we use arc capacities. For simplicity of presentation we will use the simpler network with a slightly unusual cost function and infinite arc capacities.
Also note that the edges of C(G, T ) minus the incoming tree edges of the invertices are in 1-to-1 correspondence with the edges of G. Similarly, all vertices of C(G, T ) except for the new in-vertices correspond to the vertices of the original graph. We will often use the same symbol to denote both edges or both vertices.
The main technical tool of [8] is given by the following theorem: Then there exists a spanning multigraph H in G, that has an Eulerian path between s and t with at most 4 3 n + 2 3 |C * | − 2 3 + dist G (s, t) edges. In particular, this means that there exists a TSP path between s and t in the shortest path metric of G with the same cost.

Remark 2
The above theorem is not just a rewording of the generalized version of Lemma 4.1 from [8]. In our version C * is a circulation in C(G , T ) and not C(G, T ). Note however, that in the proof of Theorem 1.2 of [8] the authors are in fact using the version above, and provide arguments for why it is correct.
In order to be able to apply Theorems 3 and 4, the authors of [8] use the optimal solution of LP(G) to define a circulation f in C(G, T ) as follows. Let G = (V , E) be a graph and let x * be an optimal extreme point solution of LP(G). Let E + = {e ∈ E : x * e > 0}, i.e. E + is the support of x * , and let G + = (V , E + ). It is clear that x * is also an optimal solution for LP(G + ), so an r-approximate TSP tour with respect to OPT LP (G + ) is also r-approximate with respect to OPT LP (G). Therefore, we can always assume that E + = E. The reason why this assumption is useful is given by the following theorem.
Theorem 5 (Cornuejols, Fonlupt, Naddef [3]) For any graph G, the support of any optimal extreme point solution to LP(G) has size at most 2n − 1.
Thus, we can assume that |E| ≤ 2n − 1. Moreover, we can assume that G is 2-vertex connected because of Lemma 1.
Let T used in the construction of C(G, T ) be the tree resulting from always following the edge e with the highest value of x * e . We construct a circulation f in C(G, T ) as a sum of two circulations: f and f . The circulation f corresponds to sending, for each back-arc a, flow of size min(x * a , 1) along the unique cycle formed by a and some tree-arcs. The circulation f is defined as follows, to guarantee that f = f + f satisfies all the lower bounds. Let v be an out-vertex and w an in-vertex, such that there is an arc (v, w) in C(G, T ), and the flow on (v, w) is smaller than 1. Also let a be any back-arc going from a descendant of w to an ancestor of v (in T). Such an arc always exists since G is 2-vertex connected. We push flow along all edges of the unique cycle formed by a and tree-arcs until the flow on (v, w) reaches 1.
The total cost of f can be bounded by We will denote the terms in the above expression as |f |, |f | and |f |, respectively. Note in particular, that |f | denotes the sum v∈I f (B(v)) which is not equal to the cost of f . We thus have |f | ≤ |f | + |f |. The authors of [8] provide the following bounds for the two terms of the above expression: The main theorem of [8] follows from these two bounds

New Upper Bound for |f |
In this section we describe an improved bound on |f |.
Before presenting our analysis of the cost of f let us recall some notation and basic observations introduced in [8]. For any v ∈ I let t v be the (unique) outgoing arc of v.

Fact 7 For every in-vertex
Proof Since T was constructed by always following the arc a with the highest value of x * a , we have that x * t v ≥ x a for any a ∈ B(v) and the claim follows.
Decompose f (B(v)) into two parts: The intuition behind this decomposition is that the higher u v is, the larger OPT LP (G) is. In particular, if we let u * = v∈I u v , then Proof Consider a vertex v of G which (in the construction of C(G, T )) is replaced by a gadget with a set I v of in-vertices, and let x * (v) be the fractional degree of v in x * . Since for any w ∈ I v , the tree-arc t w and all the back-arcs entering w correspond to edges of G incident to v, each such w contributes at least 2 + u w to x * (v), provided that u w > 0 (if u w = 0 we cannot bound w's contribution in any way). Since we also know that x * (v) ≥ 2 (this is one of the inequalities of the Held-Karp LP relaxation), we get the following bound Summing this over all vertices we get 2 OPT LP (G) ≥ 2n + u * , and the claim follows.
Because of Theorem 5 we have v∈I |B(v)| + n − 1 ≤ 2n − 1, and so by Fact 7 Note that in terms of l v and u v the total cost of f is given by the following formula v∈I max(0, l v + u v − 1).
Our goal is to upper-bound this cost as a function of n and u * . Instead of working directly with G and the solution x * to the corresponding LP(G), we abstract out the key properties of x * t v , l v and u v and work in this restricted setting. (x, l, u), where x, l, u : {1, . . . , n} → R ≥0 such that for all i = 1, . . . , n

Definition 1 A configuration of size n is a triple
Let C = (x, l, u) be a configuration. We call the triple (x i , l i , u i ) the i-th element of C. We say that the i-th element of C uses l i +u i x i edges and denote this number by e i (C), or e i if it is clear what C is. We also say that C uses n i=1 e i edges. Note that by the definition of a configuration, the number of edges used by C is at most n.
The value of the i-th element of C is defined as val i = val i (C) = max(0, l i +u i −1) and the value of C as val(C) = n i=1 val i (C).

Remark 3
The values x i , l i and u i correspond to x * t v , l v and u v , respectively. The properties enforced on the former are clearly satisfied by the latter with the exception of the inequalities x i ≤ 1. The reason for introducing these inequalities is the following. Without them, the natural definition of the number of edges used by the i-th element of C would be l i +u i min(x i , 1) . However, in that case, for any configuration C there would exists a configuration C with val(C ) = val(C) and x i ≤ 1 for all i = 1, . . . , n. In order to construct C simply replace all x i > 1 with ones. If as a result we get l i < 2 − x i and u i > 0 for some i, simultaneously decrease u i and increase l i at the same rate until one of these inequalities becomes an equality.
For that reason, we prefer to simply assume x i ≤ 1 and be able to use a (slightly) simpler definition of e i . As we will see, the inequalities x i ≤ 1 turn out to be quite useful as well.
We denote by CONF(n, u * ) the set of all configurations (x, l, u) of size n such that n i=1 u i = u * . We also use OPT(n, u * ) to denote any maximum value element of CONF(n, u * ), and VAL(n, u * ) to denote its value. We clearly have Fact 9 |f | ≤ VAL(n, u * ).
Notice that determining VAL(n, u * ) for given n and u * is a 2-dimensional knapsack problem. Here, items are the possible elements (x i , l i , u i ) satisfying the configuration definition. The value of element (x i , l i , u i ) is equal to max(0, l i + u i − 1), i.e. its contribution to the configuration value, if used in one. Also, the "mass" of (x i , l i , u i ) is u i and its "volume" is e i . We want to maximize the total item value, while keeping the total mass ≤ u * and total volume ≤ n.
Lemma 5 For any n ∈ N, u * ∈ R ≥0 , there exists an optimal configuration in CONF(n, u * ) such that: Proof We prove each property by showing a way to transform any C ∈ CONF(n, u * ) into C ∈ CONF(n, u * ) such that val(C ) ≥ val(C) and C satisfies the property.
Let us start with the first property, which basically says that all edges are fully saturated. Assume we have e i > l i +u i To prove the second property, let us assume that for some i ∈ {1, . . . , n} we have 0 < l i < 2 − x i . We also assume that our configuration already satisfies the first property, in particular we have e i = l i x i (u i = 0 since l i < 2 − x i ). We increase x i and keep l i = e i x i until l i + x i = 2. This increases the value of the configuration and keeps u i and e i unchanged. To see that x i ≤ 1, note that x i = l i /e i ≤ l i and x i + l i = 2.
Proof It is enough to prove the bound for optimal configurations satisfying the properties in Lemma 5. Let C be such a configuration. We will prove that for all i = 1, . . . , n we have: Summing this bound over all i gives the desired claim. If u i = l i = e i = 0, then the bound clearly holds. It follows from Lemma 5 that the only other case to consider is when l i = 2 − x i and e i = l i +u i x i (notice that since we only consider x i ≤ 1, we have l i + u i − 1 ≥ 0 in this case, and so val i = l i + u i − 1). It follows from these two equalities that e i x i = l i + u i = 2 − x i + u i and so Using this expression to bound val i we get We need to prove that or equivalently Since u i ≤ e i (this follows from property 1 in Lemma 5 and the fact that x i ≤ 1), we have two cases to consider.
Case 1: 1 6 − 1 1+e i ≥ 0. In this case the whole expression is clearly nonnegative. Case 2: 1 6 − 1 1+e i < 0, meaning that e i ∈ {1, 2, 3, 4}. In this case we proceed as follows: The first term is clearly nonnegative and the second one can be checked to be nonnegative for e i ∈ {1, 2, 3, 4}. Note that integrality of e i plays a key role here, as the second term is negative for e i ∈ (2, 3).
We can show that the above bound is essentially tight To handle the case of u * > 0 we need another (almost) tight case in the proof of Theorem 10 which occurs when u i is close to e i and e i is relatively large. In this case the value of the expression (e i − u i )( 1 6 − 1 1+e i ) + 1 1+e i is clearly close to 0. This corresponds to using items of the form x i = 1, l i = 1 and arbitrary u i . For such elements we have e i = u i + 1 and so Fig. 4 For each of the three tight cases, a corresponding part of G and the solution to LP(G) is shown.
The bold edges are the tree edges, the remaining edges are back-edges. ε is a very small number so the difference between the two is at most 1 3 . By combining the three types of items described, we can clearly construct C as required for any n and u * . Figure 4 illustrates the three tight cases directly in terms of the corresponding solutions of LP(G).
We are now ready to prove the Lemma 4.
Proof of Lemma 4 It follows from Theorem 10 and Fact 9 that Using Fact 8 we get:

New Upper Bound for |f |
In this section we give a new bound for |f |. We do not bound it directly, as in Lemma 2. Instead, we show the following.

Lemma 6
|f | ≤ 5 What this says is basically that f can be fully paid for by ( 5 6 of) the slack we get in Fact 8. To better understand this bound, and in particular the constant 5 6 , before we proceed to prove it, let us first show how it can be used.
Proof We have |f | ≤ |f | + |f | ≤ ( 5 6 u * + 1 6 n) + 5 6 (2 OPT LP −2n − u * ) =  There are several interesting things to note here. First of all, we got the exact same bound as in Lemma 4, which means that |f | can be fully paid for by the slack in Fact 8, as suggested earlier. In particular, this means that improving the constant 5 6 in Lemma 6 is pointless, since we would still be getting the same bound on |f | when |f | = 0. Therefore, we do not try to optimize this constant, but instead make the proof of the Lemma as straightforward as possible.
Let us now proceed to prove Lemma 6. For any non-root in-vertex w let z w = x * t w + x * (B(w)). Basically, if v is the parent of w in T, then z w is the total value of x * over all edges connecting v with vertices in the subtree T w of T determined by w. By equality (1) we have Also, let γ w be the total of x * over all edges connecting vertices in T w with vertices above v. Note that max(0, 1 − γ v ) is essentially by how much f falls short of reaching the lower-bound of 1 on arc (v, w). The definitions of z w and γ w are illustrated in Fig. 5. We can formulate the following local version of Lemma 6.

Lemma 7 For every non-root vertex
Notice that Lemma 6 follows from Lemma 7 by summing over all non-root vertices.

Proof of Lemma 7
Let v be a non-root vertex of G. We define 3 types of vertices in I v : w ∈ I v is heavy if γ w < 1 and z w > 2, w ∈ I v is light if γ w < 1 and z w ≤ 2, w ∈ I v is trivial otherwise (i.e. γ w ≥ 1).
We denote by H v and L v the sets of heavy and light vertices in I v , respectively. Intu- The last inequality holds because we have z w − u w = 2 for heavy w and z w − u w = z w ≥ 2 − γ w for light w, where the second step follows from the first observation of Lemma 8. This proves Lemma 7 for the case where there is at least one trivial vertex w with z w > 2. Hence it remains to prove the lemma for the case where all trivial vertices have z w ≤ 2 (and hence u w = 0 using equality (2)).
Note that using the second observation of Lemma 8, and the fact that for trivial vertices we have γ w ≥ 1 and hence max(0, 1 − γ w ) = 0, it suffices to prove the following inequality: and since we now assume that all trivial vertices have z w ≤ 2, it is enough to prove: Clearly, if all w ∈ I v are trivial, both sides of the bound are 0 and so it trivially holds. Otherwise, we consider the following two cases: Case 1: w∈H v ∪L v γ w > 2. Notice that this implies |H v | + |L v | ≥ 3. In this case the RHS of (3) becomes The ratio of the above expression and the LHS of (3) is lower-bounded by the ratio of these same expressions with all γ w = 0, i.e. 5 6 · 2(|L v |+|H v |−1) |L v |+|H v | , which is definitely at least 1, since |L v | + |H v | ≥ 3. Case 2: w∈H v ∪L v γ w ≤ 2. In this case the RHS of (3) becomes By rearranging the terms and using the inequality z w ≥ 2 − γ w we lower-bound this expression by The claim now follows by observing that (2 − 2γ w ) = 2(1 − γ w ) and 2 − γ w ≥ 2(1 − γ w ).

Applications to Graphic TSP and TSPP
As a consequence of Corollary 1, we get improved approximation factors for graphic TSP and graphic TSPP. and 2n − 2 − d OPT LP −1 .
For a fixed value of OPT LP the first of these expressions is increasing and the second is decreasing in d. Therefore the worst case bound we get for an algorithm that picks the best of the two solutions occurs when 10 9 OPT which leads to OPT LP −1 + 5 6 .
Since OPT LP ≥ n this is at most which proves the claim.

Remark 5
One might ask why the improvement for the graphic TSP is much bigger than the one for graphic TSPP. The reason for this is that while for large values of OPT /n our bound on |f | is significantly better than the one in [8], it is only slightly better when OPT = n. As it turns out, this is exactly the worst case for TSPP, both in our analysis and in the one in [8]. For TSP however, the worst case value of OPT for the analysis in [8] is larger than n.