Towards improving Christofides algorithm for half-integer TSP

07/03/2019 ∙ by Arash Haddadan, et al. ∙ Grenoble Institute of Technology Carnegie Mellon University 0

We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: x_e ∈{ 0 ,1/2 , 1 } where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides famous algorithm. Proving an integrality gap of α for the subtour relaxation is equivalent to showing that α x can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides bound, our goal is to show that (2-ϵ)x can be written as a convex combination of tours for some positive constant ϵ. Let y_e = 2-ϵ when x_e=1 and y_e= 3/4 when x_e = 1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x , thus reducing the problem to a base case in which such cuts do not occur.

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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 multi-subgraph 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 well-known and widely studied and implies a -approximation algorithm for TSP.

Conjecture 1 (The Four-Thirds 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 half-integer. In the unweighted case, if a half-integer point minimizes the objective function, then there is a -approximation algorithm for TSP [MS16].

Problem 1 (Half-integer TSP).

For , henceforth a half-integer point, show for constant .

Consider a half-integer point and let and . Carr and Vempala showed that in Problem 1, we can assume without loss of generality a stronger condition for : a half-integer Carr-Vempala point is a half-integer 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 half-integer Carr-Vempala point we have , then for any half-integer point we have  [CV04, BS17].

We consider a generalization of a half-integer Carr-Vempala point called a half-cycle point, which is a half-integer 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 half-edges in , forms a 2-factor of (in which the minimum cycle length is three). Formally, we define a half-cycle point as follows.

Definition 1.

A vector is called a half-cycle point if the support graph of is cubic and 2-edge-connected and for all .

Half-cycle points have been studied in restricted cases when all cycles in the 2-factor are triangles [BC11, BL17] or squares [BS17, HN18]. Schalekamp, Williamson and van Zuylen conjectured that the largest gap between and

occurs for half-cycle points in which the 2-factor consists of odd-cycles.

111Their precise conjecture is that instances of TSP that have an optimal solution that is also an optimal fractional 2-matching exhibit the largest integrality gap for . The extreme points of the fractional 2-matching polytope are half-cycle points in which all cycles in the 2-factor are odd [Bal65]. We can restate Problem 1 as follows.

Problem 2 (Half-integer TSP).

Let be a half-cycle point. Show for constant .

We can also state Problem 2 in different way.

Problem 3 (Half-integer TSP).

Let be a half-cycle 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 half-cycle point. Define vector as follows: for and for . Then .

Our main result is the following.

Theorem 1.2.

Let be a half-cycle 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 half-cycle point minimizes the objective function, if the total edge costs of the 1-edges is a constant fraction of the total cost of the half-edges, 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 3-edge-connected.

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 half-cycle 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 1-edges or on the half-edges. Then we can solve the uniform cover problem. Moreover, Theorem 1.2 can be used to slightly improve the currently best-known 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 half-cycle 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 half-cycle point. A proper cut222A 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 3-edge 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 half-cycle points to a problem (i.e., base case) where there are only two types of tight 3-edge 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 half-edges in share an endpoint in either or . (Observe that for every degenerate tight cut in , there is a 2-edge cut in .) These cuts are also easier to handle. Using this in combination with a decomposition of the 1-edges 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 4-regular and 4-edge-connected. Yet another equivalent version of Problem 1 is as follows.

Problem 5 (Half-integer TSP).

Let be a quartic point. Show for constant .

If we assume that the only 4-edge 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 4-edge 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 4-edge cuts of . However, the main difference here is that the gluing arguments we presented for half-cycle 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) 3-edge 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 2-edge-connected multigraph problem (2EC) and the 2-edge-connected subgraph problem (2ECSS). In 2EC, we want to find a minimum cost 2-edge-connected spanning multi-subgraph (henceforth, multigraph for brevity), and in 2ECSS, we want to find a minimum cost 2-edge-connected spanning subgraph (i.e., we are not allowed to double edges). Let and denote that convex hulls of characteristic vectors of 2-edge-connected 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 4-edge-connected 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 3-edge cuts and suppose we can write restricted to the edge set of each graph as a convex combination of 2-edge-connected 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 3-edge 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 half-integer 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 polynomial-time 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 3-edge path in containing both and ).

Consider a half-cycle point . For a vertex in we denote by the unique 1-edge incident on and by the two vertices that are the other endpoints of the half-edges incident on . In other words, suppose and suppose that and are the other endpoints of and , respectively. Then .

2 Saving on 1-edges for half-cycle points

Let be a half-cycle 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 half-cycle point and let be a vertex of . Suppose is a subset of 1-edges 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.

  1. and .

  2. For each edge , for .

  3. 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 half-edges 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 half-edges 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 odd-degree 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 half-cycle point and assume that has no critical cuts. Let be a subset of 1-edges of such that each 3-edge 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 half-edges that share endpoint and without loss of generality, suppose . Observe that edge and form a 2-edge 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 left-hand side of (2.1) is always nonnegative and right-hand 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.

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

  2. 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 half-cycle 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 3-edge cut of contains at most one edge from each , and each 2-edge 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 odd-degree 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 half-cycle 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 1-edge of and be the induced matching that contains . Then, , for and . Hence,

For a half-edge 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 1-edge and and are half-edges), let denote the following set of patterns of edges such that has even degree and the 1-edge is included at least once.

Let . For , define the function as follows.

(2.2)