1 Introduction
In the traveling salesman problem (TSP) we are given a complete graph
together with a vector
of edge costs satisfying the triangle inequality: for . The goal is to find a minimum cost Hamiltonian cycle of . The following formulation is a classic linear programming relaxation for TSP [DFJ54].Let denote the feasible region of this relaxation. We will refer to as the objective function. A tour of is a connected, spanning, Eulerian multisubgraph of . It is well known that due to the triangle inequality on the edge costs, a tour of can be turned into a Hamiltonian cycle of of no greater cost. For any , the vector can be decomposed into a convex combination of tours of . This follows from a polyhedral analysis of Christofides’ famous approximation algorithm [Chr76, Wol80, SW90]. For a point , define to be the support graph of . Let be the convex hull of characteristic vectors of tours of . The following conjecture is wellknown and widely studied and implies a approximation algorithm for TSP.
Conjecture 1 (The FourThirds Conjecture).
If , then .
However, more than four decades after the publication of Christofides’ algorithm, there is still no approximation algorithm known for TSP. For special cases, there has been some progress in the past few years. For example, in the unweighted case where the edge costs correspond to the shortest path metric of an unweighted graph, a series of papers improved the factor to [OSS11, MS16, SV14].
One interesting special case of weighted TSP is when the solution that minimizes the objective function is halfinteger. In the unweighted case, if a halfinteger point minimizes the objective function, then there is a approximation algorithm for TSP [MS16].
Problem 1 (Halfinteger TSP).
For , henceforth a halfinteger point, show for constant .
Consider a halfinteger point and let and . Carr and Vempala showed that in Problem 1, we can assume without loss of generality a stronger condition for : a halfinteger CarrVempala point is a halfinteger point such that the support graph is a cubic graph and for every vertex , there is exactly one edge incident on with and two edges incident on with . Moreover, forms a Hamilton cycle of , and forms a perfect matching of . If for any halfinteger CarrVempala point we have , then for any halfinteger point we have [CV04, BS17].
We consider a generalization of a halfinteger CarrVempala point called a halfcycle point, which is a halfinteger point such that the graph is a cubic graph and for every vertex , there is exactly one edge incident on with and two edges incident on with . This implies that , the halfedges in , forms a 2factor of (in which the minimum cycle length is three). Formally, we define a halfcycle point as follows.
Definition 1.
A vector is called a halfcycle point if the support graph of is cubic and 2edgeconnected and for all .
Halfcycle points have been studied in restricted cases when all cycles in the 2factor are triangles [BC11, BL17] or squares [BS17, HN18]. Schalekamp, Williamson and van Zuylen conjectured that the largest gap between and
occurs for halfcycle points in which the 2factor consists of oddcycles.
^{1}^{1}1Their precise conjecture is that instances of TSP that have an optimal solution that is also an optimal fractional 2matching exhibit the largest integrality gap for . The extreme points of the fractional 2matching polytope are halfcycle points in which all cycles in the 2factor are odd [Bal65]. We can restate Problem 1 as follows.Problem 2 (Halfinteger TSP).
Let be a halfcycle point. Show for constant .
We can also state Problem 2 in different way.
Problem 3 (Halfinteger TSP).
Let be a halfcycle point. Define vector as follows: for and for . Show there exists constants such that .
The aforementioned polyhedral analysis of Christofides algorithm implies the following theorem.
Theorem 1.1 ([Chr76, Wol80, Sw90]).
Let be a halfcycle point. Define vector as follows: for and for . Then .
Our main result is the following.
Theorem 1.2.
Let be a halfcycle point. Define vector as follows: for and for . Then .
While Theorem 1.2 is not strong enough to resolve Problem 3 (and therefore Problem 2), it does have several applications. For example, given an edge cost function for which a halfcycle point minimizes the objective function, if the total edge costs of the 1edges is a constant fraction of the total cost of the halfedges, then by Theorem 1.2, we obtain an approximation factor better than .
Another application is related to the problem of uniform covers posed by Sebő [SBS14]. Let be a cubic point if . Observe that is cubic and 3edgeconnected.
Problem 4 (Uniform cover problem).
Let be a cubic point. Show that for constant .
Recently, Haddadan, Newman and Ravi gave a positive answer to Problem 4 and showed [HNR17]. Previously, Boyd and Sebő had shown that if is additionally Hamiltonian [BS17]. In fact, Theorem 1.2 gives an alternative way to answer Problem 4.
Lemma 1.3.
Let be a halfcycle point. Define vector as follows: for and for for constants . Suppose . Then for any cubic point , we have for .
In other words, suppose that we can save either on the 1edges or on the halfedges. Then we can solve the uniform cover problem. Moreover, Theorem 1.2 can be used to slightly improve the currently bestknown factors for Problem 4. The proofs of Lemma 1.3 and Theorem 1.4 can be found in Section 5.
Theorem 1.4.
Let be a cubic point. Then for . If is Hamiltonian, then .
On a high level, our proof of Theorem 1.2 is based on Christofides’ algorithm: We show that a halfcycle point can be written as a convex combination of spanning subgraphs with certain properties and then we show that vector , where for and for , can be used for parity correction. Our main new tool is a procedure to glue tours over critical cuts. For , let denote the subset of edges crossing the cut .
Definition 2.
Let be a halfcycle point. A proper cut^{2}^{2}2A cut is proper if and . in is called critical if and contains exactly one edge with . Moreover, for each pair of edges in , their endpoints in (and in ) are distinct.
Observe that a critical cut in is a proper 3edge cut that is tight: the values of the three edges crossing the cut sum to 2. Thus, critical cuts are difficult to handle using an approach based on Christofides’ algorithm. In particular, using would be insufficient for parity correction of a critical cut if it is crossed by an odd number of edges in the spanning subgraph.
Applying our gluing procedure, we can reduce TSP on halfcycle points to a problem (i.e., base case) where there are only two types of tight 3edge cuts. The first type of cut is a vertex cut, which we show are easier to handle. In particular, the parity of vertex cuts can be addressed with a key tool used by Boyd and Sebő [BS17] called rainbow trees (see Theorem 1.6). We refer to the second type of cut as a degenerate tight cut, which is a cut such that , and and the two halfedges in share an endpoint in either or . (Observe that for every degenerate tight cut in , there is a 2edge cut in .) These cuts are also easier to handle. Using this in combination with a decomposition of the 1edges into few induced matchings (see Definition 6), which have some additional required properties, we can prove Theorem 1.2 for the base case. We discuss gluing procedures in more detail in Section 1.1.
Let us look back at Problem 1. Let be a quartic point if . Observe that is 4regular and 4edgeconnected. Yet another equivalent version of Problem 1 is as follows.
Problem 5 (Halfinteger TSP).
Let be a quartic point. Show for constant .
If we assume that the only 4edge cuts of are its vertex cuts and the number of vertices is even, we can answer this problem.
Theorem 1.5.
Let be a quartic point. If has an even number of vertices, and does not have any proper 4edge cuts, then .
In the case of a quartic point, Theorem 1.5 could serve as the base case for if we were able to glue over proper 4edge cuts of . However, the main difference here is that the gluing arguments we presented for halfcycle points can not easily be extended to this case due to the increased complexity of the distribution of patterns. The proof of Theorem 1.5 can be found in Section 4.
1.1 Gluing tours over cuts
The approach of gluing solutions over (often) 3edge cuts and thereby reducing to an instance without such cuts has been used previously for TSP (e.g., [CNP85]) and extensively in the case of two related problems: the 2edgeconnected multigraph problem (2EC) and the 2edgeconnected subgraph problem (2ECSS). In 2EC, we want to find a minimum cost 2edgeconnected spanning multisubgraph (henceforth, multigraph for brevity), and in 2ECSS, we want to find a minimum cost 2edgeconnected spanning subgraph (i.e., we are not allowed to double edges). Let and denote that convex hulls of characteristic vectors of 2edgeconnected multigraphs and subgraphs, respectively, of . Observe that and .
For example, consider the problem of showing for a cubic point [BL17]. Here, we can assume that is essentially 4edgeconnected due to the following commonly used observation. Let be a subset of vertices such that in . We construct graphs, and by contracting the sets and , respectively, in to a pseudovertex. Suppose that the graphs and contain no proper 3edge cuts and suppose we can write restricted to the edge set of each graph as a convex combination of 2edgeconnected subgraphs of the respective graph. Let us consider the patterns around the pseudovertices; each vertex can be adjacent to two or three edges and therefore, there are only four possible patterns around a vertex. Moreover, since each pattern appears the same percentage of time (in the respective convex combinations) for each pseudovertex, tours with corresponding patterns can be glued over the 3edge cut. (For a more formal presentation of this argument, see Lemma 3.3 in [HN18] or Case 2 in Section 3.1.2 in [Leg17].) Thus, for 2ECSS, this gluing procedure is quite straightforward. Gluing has also been used for 2EC, but here it is necessary to make certain extra assumptions to control the number of patterns around a vertex, due to the fact that the distribution of possible patterns is more complex. Carr and Ravi proved that the vector for a halfinteger point [CR98]. To control the number of patterns so that they can use gluing, they require some strong assumptions on the multigraphs in their convex combinations: for example, no edge with is doubled and some arbitrarily chosen edge is never used.
In contrast, it appears that no such gluing procedure has been used in approximation algorithms for TSP. Indeed, gluing proofs for 2ECSS and 2EC [CR98, BL17, Leg17] can not be easily extended to TSP for several reasons: (1) As just discussed, they are used for gluing subgraphs (no doubled edges), while for multigraphs, there are often too many different patterns around a vertex. (For TSP, we must allow doubled edges.) (2) They do not necessarily preserve parity of the vertex degrees. Finally, (3) many of the results for 2ECSS and 2EC based on gluing do not result in polynomialtime algorithms.
The main technical contribution of this paper is to show that for a carefully chosen set of tours, we can design a gluing procedure over critical cuts. In particular, we can fix a critical cut in and find a convex combination of tours for . Then we can find a set of tours for such that the distribution of patterns around the pseudovertex corresponding to matches that of the pseudovertex corresponding to in . This is done by separately matching the pattern for the spanning subgraphs and for the parity correction. In fact, while each vertex may have a different set of patterns around it, we show that the patterns around each vertex can be encapsulated by a single parameter: the fraction of times in the convex combination of spanning subgraphs that a vertex is a leaf. There can be some flexibility in this degree distribution for some arbitrarily chosen vertex, and this is what we exploit to sufficiently control the patterns around a pseudovertex to enable gluing.
1.2 Definitions, tools and notation
Definition 3.
Let be a graph. For a vertex , a tree of is a subset of such that and induces a spanning tree of .
Denote by the convex hull of incidence vectors of trees of . The is characterized by the following linear inequalities.
(1.1) 
Definition 4.
Let and be a vertex of . Let a collection of disjoint subsets of . A rainbow tree of , is a tree of such that for .
Definition 5.
Let and let be a vector . Let denote a set of subgraphs of (i.e., each for each
). If there is a probability distribution
such that , then we say is a convex combination for . If such a probability distribution exists, then we say that can be decomposed into (or written as) a convex combination of subgraphs in .Theorem 1.6 (Boyd, Sebő [Bs17]).
Let and be a collection of disjoint subsets of such that for . Then, can be decomposed into a convex combination of rainbow trees of for any .
Definition 6.
Given a graph , a set of edges forms an induced matching in if the subgraph of induced on the endpoints of forms a matching (i.e., if edges and belong to an induced matching , then there is no 3edge path in containing both and ).
Consider a halfcycle point . For a vertex in we denote by the unique 1edge incident on and by the two vertices that are the other endpoints of the halfedges incident on . In other words, suppose and suppose that and are the other endpoints of and , respectively. Then .
2 Saving on 1edges for halfcycle points
Let be a halfcycle point. In this section, we present an algorithm to write as a convex combination of tours of . Following Christofides’ algorithm, we first construct a convex combination of spanning subgraphs in Section 2.1. Next, we address parity correction in Section 2.2. We combine these two steps in Section 2.3 for the base case, in which contains no critical cuts. In Section 2.4, we show how to iteratively glue tours for base cases together to construct tours for general .
2.1 Convex combinations of spanning subgraphs
Definition 7.
Let be a halfcycle point and let be a vertex of . Suppose is a subset of 1edges of . Let . Let be a set of spanning connected subgraphs of and let be a probability distribution such that is a convex combination for . Then we say holds for the convex combination if it has the following properties.

and .

For each edge , for .

induces a spanning subgraph on .
Lemma 2.1.
Suppose forms an induced matching in and edge . Then there is a set of spanning connected subgraphs of and a probability distribution such that is a convex combination for for which holds.
Proof.
For each , pair the halfedges incident on and pair those incident on to obtain disjoint subsets of edges and decompose into a convex combination of rainbow trees (i.e., ) via Theorem 1.6. This is the desired convex combination since for all , we have and for all endpoints of edges in . Thus, the first and second conditions are satisfied. The third condition holds by definition of trees. ∎
Lemma 2.2.
Let and let be any constant such that . If forms an induced matching in , and . Then there is a set of spanning connected subgraphs of and a probability distribution such that is a convex combination for for which holds.
Proof.
As in the proof of Lemma 2.1, for each , pair the halfedges incident on and pair those incident on to obtain a collection of disjoint subsets of edges . Apply Theorem 1.6 to obtain which is a convex combination for , where is a set of rainbow trees (i.e., ). Notice that this convex combination clearly satisfies the second requirement in Definition 7.
Now let , where and are the other endpoints of and , respectively. Without loss of generality, assume . Since , we have for since . In addition, we have for all by the definition of trees. Hence, . Without loss of generality, assume and for , and and for , where and .
We can also assume that there are subsets and such that and , since . For , replace with . Similarly, for , replace with . For all , keep as is. Observe that still induces a spanning subgraph on since we did not remove any edge in from the tree . We want to show that the new convex combination is the desired convex combination for . Notice that
So . Also, is a connected subgraph of since each is obtained by removing an edge incident on , which does not disconnect it. Finally, for each vertex with , we have for all . To observe this, notice that the initial convex combination satisfies this property for vertex (since the convex combination is obtained via Theorem 1.6). In the transformation of the convex combination we only change edges incident on and , so if the property clearly still holds after the transformation. If or , we only remove or add an edge incident on if . ∎
2.2 Tools for parity correction
Let be an arbitrary graph and where is even. An join of is a subgraph of in which the set of odddegree vertices of are exactly . The convex hull of characteristic vectors of joins of , denoted by can be described as follows.
for  (2.1) 
Lemma 2.3.
Let be a halfcycle point and assume that has no critical cuts. Let be a subset of 1edges of such that each 3edge cut in contains at most one edge from . Let be a subset of vertices such that is even and for all , neither nor is in . Also for any set such that , both and are even. Define vector as follows: if and , and otherwise. Then vector .
Proof.
By definition, . Now we will show that satisfies the constraint (2.1). If , then for some . Let . We consider two cases. If , then . So . If , then we need to consider even. If , then . If , we have . Hence . If , then we consider odd. If , then . Finally, if , then , and .
On the other hand, if , we have and . So we need to consider odd. If , then we have . If , then .
Now assume and . If , then . Hence, . We now consider the case where . If , then is odd as is even by assumption. Hence, . Also by assumption is even. Observe that in this case, we have . This implies that .
Finally, we consider the case when . In this case, since does not contain any critical cuts, there are two possibilities: (i) , or (ii) is a degenerate tight cut. In case (i), since and , we have . Hence, . For case (ii), suppose and are halfedges that share endpoint and without loss of generality, suppose . Observe that edge and form a 2edge cut in . Therefore, by assumption, either or . In the former case, we have and . When , (2.1) is satisfied, as . When , we have . The latter case is when . Here, and . Observe that can be either even or odd. When it is even and , the lefthand side of (2.1) is always nonnegative and righthand size is zero. When , we have . When is odd, then since , we satisfy (2.1) when . When , we have . Thus, in all instances we conclude that (2.1) is satisfied. ∎
Observation 2.1.
Let be a cubic graph, and let be a subset of vertices such that is even. Let , and for all . Then there exists a set of joins of , namely , and a probability distribution such that is a convex combination for . Moreover, for each vertex , the following properties hold.

If , then we have for each . (Notice that in this case we must have .)

If and , then we have the following (four) cases. (Notice that sum of the right hand sides is exactly 1.)
The proof of this observation follows from the fact that if , then it can be efficiently decomposed into a convex combination of joins of [EJ73].
2.3 Convex combinations of tours: Base case
Let be a halfcycle point such that has no critical cuts. Let be a fixed vertex in and let . Let be a partition of into induced matchings such that for all , , each 3edge cut of contains at most one edge from each , and each 2edge cut of contains an even number of edges from each . Let and be some constant where .
For , let be a set of spanning subgraphs of and let be a convex combination for for which holds (by Lemma 2.1). For , let be a set of spanning subgraphs of and let be a convex combination for for which holds (by Lemma 2.2). Notice that since . Let .
We can write as a convex combination of the spanning subgraphs in , by weighting each set by . In particular, we have . For each , let . Then is a convex combination for . From Definition 7 and Lemmas 2.1 and 2.2, we observe the following.
Claim 1.
For each , induces a connected, spanning subgraph on .
For each , define if and otherwise. For each , let be the set of odddegree vertices of . By construction, we have . By Lemma 2.3, we have , so there exists a set of joins and a probability distribution such that is a convex combination for . This implies that can be written as a convex combination of tours of . We denote this set of tours by and we let . We claim that can be written as a convex combination of tours of in using the probability distribution , constructed as follows: For a tour that is the union of and , set .
Claim 2.
Let be a halfcycle point such that contains no critical cuts. Define vector as for and for . Then is a convex combination for .
Proof..
We need to show that . First, let be a 1edge of and be the induced matching that contains . Then, , for and . Hence,
For a halfedge of , we have and for , so .
Now we prove some additional useful properties of the convex combination . For a vertex such that (i.e., where is a 1edge and and are halfedges), let denote the following set of patterns of edges such that has even degree and the 1edge is included at least once.
Let . For , define the function as follows.
(2.2) 
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