An Improved Integrality Gap for Asymmetric TSP Paths††thanks: An extended abstract of this paper appears in the Proceedings of the 16th Conference on Integer Programming and Combinatorial Optimization, 2013.
The Asymmetric Traveling Salesperson Path Problem (ATSPP) is one where, given an asymmetric metric space with specified vertices and , the goal is to find an - path of minimum length that passes through all the vertices in .
This problem is closely related to the Asymmetric TSP (ATSP), which seeks to find a tour (instead of an - path) visiting all the nodes: for ATSP, a -approximation guarantee implies an -approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming relaxations for these problems: the current-best approximation algorithm for ATSPP is , whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elimination LP) for ATSPP is .
In this paper, we close this gap, and improve the current best bound on the integrality gap from to . The resulting algorithm uses the structure of narrow - cuts in the LP solution to construct a (random) tree spanning tree that can be cheaply augmented to contain an Eulerian - walk.
We also build on a result of Oveis Gharan and Saberi and show a strong form of Goddyn’s conjecture about thin spanning trees implies the integrality gap of the subtour elimination LP relaxation for ATSPP is bounded by a constant. Finally, we give a simpler family of instances showing the integrality gap of this LP is at least .
In the Asymmetric Traveling Salesperson Path Problem (ATSPP), we are given an asymmetric metric space (i.e., one where the distances satisfy the triangle inequality, but potentially not the symmetry condition), and also specified source and sink vertices and , and the goal is to find an - Hamilton path of minimum length.
ATSPP is a close relative of Asymmetric TSP (ATSP), where the goal is to find a Hamilton tour instead of an - path. For ATSP, the -approximation of Frieze, Galbiati, and Maffioli  from 1982 was the best result known for more than two decades, until it was finally improved by constant factors in [4, 13, 9]. A breakthrough on this problem was an -approximation due to Asadpour, Goemans, M ‘ a dry, Oveis Gharan, and Saberi ; they also bounded the integrality gap of the subtour elimination linear programming relaxation for ATSP by the same factor.
Somewhat surprisingly, the study of ATSPP has been of a more recent vintage: the first approximation algorithms appeared only around 2005 [15, 6, 9]. It is easily seen that the ATSP reduces to ATSPP in an approximation-preserving fashion (by guessing two consecutive nodes on the tour). In the other direction, Feige and Singh  show that a -approximation for ATSP implies an -approximation for ATSPP. Using the above-mentioned -approximation for ATSP , this implies an -approximation for ATSPP as well.
The subtour elimination linear program generalizes simply to ATSPP and is given in Section 2. However, prior to our work, the best integrality gap known for this LP for ATSPP was still . In this paper we show the following result.
The integrality gap of the subtour elimination linear program for ATSPP is .
We also explore the connection between integrality gaps for ATSPP and the so-called “thin trees conjecture”. In particular, if Goddyn’s conjecture regarding thin trees holds with strong-enough quantitative bounds then the integrality gap of the subtour elimination LP for ATSPP is bounded by a constant. The precise statement of the conjecture and of our result can be found in Section 5. This is analogous to a similar statement made by Oveis Gharan and Saberi regarding the integrality gap of the subtour elimination LP for ATSP .
Finally, we give a simple construction showing that the integrality gap of this LP is at least ; this example is simpler than previous known integrality gap instance showing the same lower bound, due to Charikar, Goemans, and Karloff .
Given the central nature of linear programs in approximation algorithms, it is useful to understand the integrality gaps for linear programming relaxations of optimization problems. Not only does this study give us a deeper understanding into the underlying problems, but upper bounds on the integrality gap of LPs are often useful in approximating related problems. For example, the polylogarithmic approximation guarantees in the work of Nagarajan and Ravi  for Directed Orienteering and Minimum Ratio Rooted Cycle, and those in the work of Bateni and Chuzhoy  for Directed -Stroll and Directed -Tour were all improved by a factor of following the improved bound of on the integrality gap of the subtour LP relaxation for ATSP. We emphasize that these improvements required the integrality gap bound improvement for ATSP, not merely improved approximation guarantees.
1.1 Our Approach
Our approach to bound the integrality gap for ATSPP is similar to that for ATSP [2, 18], but with some crucial differences. To prove Theorem 1.1, we sample a random spanning tree in the underlying undirected multigraph and then augment the directed version of this tree to an integral circulation using Hoffman’s circulation theorem while ensuring the - edge is only used once. The support of this circulation is weakly connected, so it can be used to obtain an Eulerian circuit with no greater cost. Deleting the - edge from this walk results in a spanning - walk.
However, the non-Eulerian nature of ATSPP makes it difficult to satisfy the cut requirements in Hoffman’s circulation theorem if we sample the spanning tree directly from the distribution given by the LP solution. It turns out that the problems come from the - cuts that are nearly-tight: i.e., which satisfy for some small constant — these give rise to problems when the sampled spanning tree includes more than one edge across this cut. Such problems also arise in the symmetric TSP paths case (studied in the recent papers of An, Kleinberg, and Shmoys  and Sebő ): their approach is again to take a random tree directly from the distribution given by the optimal LP solution, but in some cases they need to boost the narrow cuts, and they show that the loss due to this boosting is small.
In our case, the asymmetry in the problem means that boosting the narrow cuts might be prohibitively expensive. Hence, our idea is to preprocess the distribution given by the LP solution to tighten the narrow cuts, so that we never pick two edges from a narrow cut. Since the original LP solution lies in the spanning tree polytope, lowering the fractional value on some edges means we need to raise the fractional value on other edges. This would cause the costs to increase, and the technical heart of the paper is to ensure this can be done with a small increase in the cost.
Our approach for proving an integrality gap bound under the thin trees conjecture is similarly inspired by related work for ATSP , but, again, we must be careful to ensure that the thin tree crosses each narrow cut exactly once. We do this by finding a cheap thin tree “between” narrow cuts (which we will prove are nested) and then chaining these thin together trees by selecting a single edge across each narrow cut. The resulting tree will have the desired thinness properties.
1.2 Other Related Work
The first non-trivial approximation for ATSPP was an -approximation by Lam and Newman . This was improved to by Chekuri and Pál , and the constant was further improved in . The paper  also showed that a -approximation algorithm for ATSP can be used to obtain an -approximation algorithm for ATSPP. All these results are combinatorial and do not bound integrality gap of ATSPP. A bound of on the integrality gap of ATSPP was given by Nagarajan and Ravi , and was improved to by Friggstad, Salavatipour and Svitkina . Note that there is still no result known that relates the integrality gaps of subtour elimination relaxations for ATSP and ATSPP in a black-box fashion.
In the symmetric case (where the problems become TSPP and TSP respectively), constant factor approximations and integrality gaps have long been known. We do not survey the rich body of literature on TSP here, instead pointing the reader to, e.g., the recent paper on graphical TSP by Sebő and Vygen . An exception is a result of An, Kleinberg, and Shmoys , who give an upper bound of on integrality gap of the LP relaxation for the TSPP problem; their algorithm also proceeds via studying the narrow - cuts, and the connections to our work are discussed in Section 1.1. This bound on the integrality gap was subsequently improved to via a more refined analysis by Sebő .
1.3 Notation and Preliminaries
Given a directed graph , and two disjoint sets , let . We use the standard shorthand that , and . When the set is a singleton (say ), we use or instead of or . For undirected graph , we use to denote edges crossing between and , and to denote the edges with exactly one endpoint in (which is the same as . For any subset we let denote , the set of arcs with both endpoints in . If we are discussing subsets of arcs of , we add subscripts to the notation to indicate we only consider those arcs crossing the cut that in are . For example, denotes and so on. A collection of subsets of , say is a partition if each element of occurs in exactly one part of . Given a graph and a partition of , we let to be the set of edges in which have endpoints in different sets of .
For a digraph , a set of arcs is weakly connected if the undirected version of forms a connected graph that spans all vertices in .
For values for all , and a set of arcs , we let denote the sum .
Given an undirected graph and a subset of edges , we let denote the characteristic vector . The spanning tree polytope is the convex hull of . See, e.g., [20, Chapter 50] for several equivalent linear programming formulations of this polytope. We sometimes abuse notation and call a set of directed arcs a tree if the undirected version of is a tree in the usual sense.
A directed metric graph on vertices has arcs where the non-negative arc costs satisfy the triangle inequality for all . However, arcs and need not have the same cost. An instance of ATSPP is a directed metric graph along with distinguished vertices .
2 The Rounding Algorithm
In this section, we give the linear programming relaxation for ATSPP, and show how to round a feasible solution to this LP to get a path of cost times the cost of . We then give the proof, with some of the details being deferred to the following sections.
Given a directed metric graph with arc costs , we use the following standard linear programming relaxation for ATSPP which is also known as the subtour elimination linear program.
Constraints (4) can be separated over in polynomial time using standard min-cut algorithms, so this LP can be solved in polynomial time using the ellipsoid method. We begin by solving the above LP to obtain an optimal solution . Consider the undirected (multi)graph obtained by removing the orientation of the arcs of . That is, create precisely two edges between every two nodes in , one having cost and the other having cost . (Hence, .) For a point , let denote the corresponding point in , and view as the “undirected” version of .
We will use the following definition: An - cut is a subset such that . The following fact will be used throughout the paper.
Let be a feasible solution to LP (ATSPP). For any cut , . Also, for every nonempty .
For any nonempty subset of vertices we have
If is an cut, then the first sum in the last expression is and the second sum is by Constraints (1), (2), and (3). If , then both sums are equal to by Constraints (3). ∎
Definition 2.2 (Narrow cuts).
Let . An - cut is -narrow if (or equivalently, ).
The main technical lemma is the following:
For any , one can find, in polynomial-time, a vector (over the directed arcs) such that:
its undirected version lies in the spanning tree polytope for ,
(where the inequality denotes component-wise dominance), and
and for every -narrow - cut .
Before we prove the lemma (in Section 2.1), let us sketch how it will be useful to get a cheap ATSPP solution. Since (or more correctly, its undirected version ) lies in the spanning tree polytope, it can be represented as a convex combination of spanning trees. Using some recently-developed algorithms (e.g., those due to [2, 7]) one can choose a (random) spanning tree that crosses each cut only times more than the LP solution. Finally, we can use times the LP solution to patch this tree to get an - path. Since the LP solution is “weak” on the narrow cuts and may contribute very little to this patching (at most ), it is crucial that by property (c) above, this tree will cross the narrow cuts only once, and that too, it crosses in the “right” direction, so we never need to use the LP when verifying the cut conditions of Hoffman’s circulation theorem on narrow cuts. The details of these operations appear in Section 3.
We will assume to ensure all of our arguments work. For , we use the known integrality gap bound of from  to ensure the gap is bounded for all .
2.1 The Structure of Narrow Cuts
We now prove Lemma 2.3: it says that we can take the LP solution and find another vector such that if an - cut is narrow in (i.e. ), then . Moreover, the undirected version of can be written as a convex combination of spanning trees, and is not much larger than for any arc .
The undirected version of itself can be written as a convex combination of spanning trees, so if we force to cross the narrow cuts to an extent less than (loosely, this reduces the connectivity), we had better increase the value on other arcs. To show we can perform this operation without changing any of the coordinates by very much, we need to study the structure of narrow cuts more closely. (Such a study is done in the symmetric TSP path paper of An et al. , but our goals and theorems are somewhat different.)
First, say two - cuts and cross if and are non-empty.
For , no two -narrow - cuts cross.
Suppose and are crossing -narrow - cuts. Then
where the last inequality follows from the first three terms being cuts excluding and hence having at least unit -value crossing them (by the LP constraints), the fourth term being non-negative, and the last term being the -value of a subset of the arcs in and remembering that and are -narrow. However, this contradicts . ∎
Lemma 2.4 says that the -narrow cuts form a chain with . For . let . We also define and . Let and . For the rest of this paper, we will use to denote a value in the range . Ultimately, we will set for the final bound but we state the lemmas in their full generality for .
Next, we show that out of the (at most) mass of across each -narrow cut , most of it comes from the “local” arcs in .
For each ; .
If then so in fact . In this case, and and the LP constraints clearly imply .
Now consider the case . For , since we have from the LP constraints. We also have because is -narrow, and therefore . A similar argument for shows . So it remains to consider . Define the following quantities, some of which can be zero.
because and is -narrow. Similarly
Summing these two inequalities yields where we have used . Rearranging shows . ∎
Now, recall that denotes the assignment of arc weights to the graph from the previous section obtained by “removing” the directions from arcs in . We prove that the restriction of to any almost satisfies the partition inequalities that characterize the convex hull of connected spanning subgraphs of . This characterization was given by Edmonds ; see also Chapter 50, Corollary 50.8(a) in Schrijver . We state it here for completeness.
 Let be a graph. Then the convex hull of all connected spanning subgraphs of is given by . Moreover, the convex hull of spanning trees of is given by .
For a partition of a subset of , we let denote the set of edges whose endpoints lie in two different sets in the partition. To be clear, does not contain any edge that has at least one endpoint in .
For any and any partition of , we have .
Since and , there is nothing to prove for or . So, we suppose .
Consider the quantity . On one hand because neither nor lie in for any , so . On the other hand, counts each arc between two parts in exactly twice and each arc with one end in and the other not in precisely once. So, .
Notice that and are disjoint subsets of . So, since both and are -narrow, . This shows which, after rearranging, is what we wanted to show. ∎
For any partition of , we have .
From Lemma 2.7, we have for any . ∎
Finally, to efficiently implement the arguments in the proof of Lemma 2.3, we need to be able to efficiently find all -narrow cuts . This is done by a standard recursive algorithm that exploits the fact that the cuts are nested.
There is a polynomial-time algorithm to find all -narrow cuts.
Consider following recursive algorithm. As input, the routine is given a directed graph with arc weights and distinct nodes where both and are -narrow. Say a -narrow cut in is non-trivial if and . The claim is that the procedure will find all non-trivial -narrow cuts of , provided that they are nested.
The procedure works as follows. If there are non-trivial -narrow cuts in , then there are nodes such that some -narrow cut has . So, the procedure tries all pairs of distinct nodes , contracts both and to a single node and determines if the minimum cut separating these contracted nodes has -capacity less than . If such a cut was found for some , the algorithm makes two recursive calls, one with the contracted graph with start node being the contraction of and end node being , and the other with the contracted graph with start node and end node being the contraction of . After both recursive calls complete, the algorithm returns all -narrow cuts found by these two recursive calls (of course, after expanding the contracted nodes) and the -narrow cut itself. If such a cut was not found over all choices of , then the algorithm returns nothing because there are no non-trivial -narrow cuts in .
It is easy to see that a non-trivial -narrow cut in either contracted graph corresponds to a -narrow cut in . On the other hand, if the -narrow cuts are nested in , then every non-trivial -narrow cut apart from itself corresponds to a non-trivial -narrow cut in exactly one of or . Also, the -narrow cuts in both contracted graphs remain nested. So, the recursive procedure finds all non-trivial -narrow cuts of . The number of recursive calls is at most the number of non-trivial -narrow cuts, and this is at most because the cuts are nested so it is an efficient algorithm. We call this algorithm initially with graph , start node and end node . Lemma 2.4 implies the -narrow cuts of are nested so the recursive procedure finds all non-trivial -narrow cuts of . Adding these to and gives all -narrow cuts of . ∎
Proof of Lemma 2.3.
The claimed vector can be described by linear constraints: indeed, consider the following polytope on the variables .
Consider the vector given as follows.
Next we show Constraints (5) holds. It suffices to show that can be decomposed as a convex combination of characteristic vectors of connected graphs. For , let denote the restriction of to edges whose endpoints are both contained in . Then Corollary 2.8, Constraints (10), and [20, Corollary 50.8a] imply that can be decomposed as a convex combination of integral vectors, each of which corresponds to an edge set that is connected on . Next, let denote the restriction of to edges whose endpoints are both contained in some common . Since the sets are disjoint, we have that (where the addition is component-wise). Furthermore, , being the sum of the vectors, can be decomposed as a convex combination of integral vectors corresponding to edge sets such that the connected components of the graph are precisely the sets .
Next, let denote the restriction of to edges contained in one such . We also note that the sets are disjoint. By construction, we have for each so we may decompose as a convex-combination of integral vectors, each of which includes precisely one edge across each .
Adding any integral point in the decomposition of to any integral point in the decomposition of results in an integral vector that corresponds to a connected graph: each is connected by and consecutive are connected by . By construction of , we have so we may write as a convex combination of characteristic vectors of connected graphs, each of which satisfies Constraints (5).
Finally, we modify slightly to ensure constraint (6) holds while maintaining the other constraints. From [20, Corollary 50.8a] and Constraints (5) and (10), lies in the convex hull of incidence vectors corresponding to connected (multi)graphs. Decompose into a convex combination of such vectors, drop edges from the corresponding connected graphs to get spanning tree, and recombine these spanning trees to get a point in the spanning tree polytope. Note that we only decreased values so Constraints (7) and (9) continue to hold. Finally, since now lies in the spanning tree polytope then each tree must still cross each narrow cut, so Constraints (8) still hold.
3 Obtaining an - Path
Having transformed the optimal LP solution into the new vector (as in Lemma 2.3) without increasing it too much in any coordinate, we now sample a random tree such that it has a small total cost, and that the tree does not cross any cut much more than prescribed by . Finally we add some arcs to this tree (without increasing its cost much) so that every has equal indegree and outdegree while ensuring that has outdegree 1 and indegree 0. This gives us an Eulerian - walk.
By the triangle inequality, shortcutting this walk past repeated nodes yields a Hamiltonian path of no greater cost. While this general approach is similar to that used in , some new ideas are required because we are working with the LP for ATSPP—in particular, only one unit of flow is guaranteed to cross - cuts, which is why we needed to deal with narrow cuts in the first place. The details appear in the rest of this section.
3.1 Sampling a Tree
For a digraph and a collection of arcs , we say is -thin with respect to if for every . The set is also -approximate with respect to if the total cost of all arcs in is at most times the cost of , i.e., . The reason we are deviating from the undirected setting used in  to the directed setting is that the orientation of the arcs across each -narrow cut will be important when we sample a random “tree”.
Let . Let and . For , there is a randomized, polynomial time algorithm that, with probability at least 1/2, finds an -thin and -approximate (with respect to ) collection of arcs that is weakly connected and satisfies and for each -narrow - cut .
Let be a vector as promised by Lemma 2.3. From , randomly sample a set of arcs whose undirected version is a spanning tree on . This should be done from any distribution with the following two properties:
(Negative Correlation) For any subset of edges ,
This can be obtained using, for example, the swap rounding approach for the spanning tree polytope given by Chekuri et al. . As in , the negative correlation property implies the following theorem. The proof is found in Section 4.
For , the tree is -thin with probability at least .
By Lemma 2.3(b), property (i) of the random sampling, and Markov’s inequality, we get that (from Lemma 3.1) is -approximate with respect to with probability at least 2/3. By a trivial union bound, for we have with probability at least 1/2 that is both -thin and -approximate with respect to . It is also weakly connected—i.e., the undirected version of (namely, ) connects all vertices in .
The statement for -narrow - cuts follows from the fact that satisfies Lemma 2.3(c). That is, contains no arcs of , since (for being a -narrow - cut). But since is a spanning tree, must contain at least one arc from . Finally, since is exactly 1, then any set of arcs supported by this distribution we use must have precisely one arc from . ∎
3.2 Augmenting to an Eulerian - Walk
We wrap up by augmenting the set of arcs to an Eulerian - walk. Specifically, we prove the following for general .
Suppose we are given a collection of arcs that is weakly connected, -thin, and satisfies and for any -narrow cut . We can find a Hamiltonian path with cost at most in polynomial time.
Given a directed flow network , with each arc having a lower bound and an upper bound (and ), there exists a circulation satisfying for all arcs if and only if for all . Moreover, if the and are integral, then the circulation can be taken integral.
Proof of Theorem 3.3.
Set lower bounds on the arcs by:
For now, we set an upper bound of 1 on arc and leave all other arc upper bounds at . We compute the minimum cost circulation satisfying these bounds (we will soon see why one must exist). Since the bounds are integral and since is weakly connected, this circulation gives us a directed Eulerian graph. Furthermore, since , the arc must appear exactly once in this Eulerian graph. Our final Hamiltonian - path is obtained by following an Eulerian circuit, removing the single arc from this circuit to get an Eulerian - walk, and finally shortcutting this walk past repeated nodes. The cost of this Hamiltonian path will be, by the triangle inequality, at most the cost of the circulation minus the cost of the arc.
Finally, we need to bound the cost of the circulation (and also to prove one exists). To that end, we will impose stronger upper bounds as follows:
We use Hoffman’s circulation theorem to show that a circulation exists satisfying these bounds and (The calculations appear in the next paragraph.) Since is no longer integral, the circulation might not be integral, but it does demonstrate that a circulation exists where each arc is assigned at most more flow in the circulation than the number of times it appears in . Consequently, it shows that the minimum cost circulation in the setting where we only had a non-trivial upper bound of on the arc can be no more expensive (since there are fewer constraints), and that circulation can be chosen to be integral. The cost of circulation is at most the cost of , which is at most
Subtracting the cost of the arc (since we drop it to get the Hamilton path), we get that the final Hamiltonian path has cost at most
One detail remains: we need to verify the conditions of Theorem 3.4 for the bounds and . Firstly, it is clear by definition that for each arc . Now we need to check for each cut . This is broken into four cases.
is a -narrow - cut. Then , since contains only one arc in . But .
is an - cut, but not -narrow. Then by the -thinness of and Claim 2.1,
On the other hand,
where the last inequality used the fact that .
is a - cut. Then by the -thinness of and Claim 2.1,
the last inequality using that . Moreover
does not separate from . Then
4 Guaranteeing -Thinness
We prove Theorem 3.2 in this section. Recall that -thin means the number of arcs chosen from should not exceed (so a directed version). Let where the logarithm is the natural logarithm. Recall that is the set of arcs found with corresponding undirected spanning tree . By the first property of the distribution (preservation of marginals on singletons) we have for each that .
We have negative correlation on subsets of items, so we can apply standard concentration bounds. Specifically, we use the following inequality.
[19, Theorem 3.4] Let be given 0-1 random variables with and such that for all , . Then for any we have
For notational simplicity, let . Theorem 4.1 immediately shows
However, for any graph, there are at most cuts whose capacity is at most times the capacity of the minimum cut . Note that the minimum cut with capacities is 1, so there are at most cuts of the undirected graph with capacity (under ) at most . Another way to view this is that there are at most cuts whose capacity is between and . For each such cut , the previous analysis shows that probability that is at most . Thus, by the union bound, the probability that for some is bounded by
Since , then we have just seen that with probability at least that there is no with . This is close to what we want, except we should bound against the -capacity of . That is, we ultimately want to show for every