LP-Based Approximation Algorithms for Traveling Salesman Path Problems

The traveling salesman problem (TSP) has been widely studied in a variety of settings [10, 5, 13, 7, 17, 14, 12, 4, 2, 1], but we shall focus on the approximation results for the traveling salesman path problem with two prespecified endpoints. In this variant, two vertices in the graph are specified as a part of the input, and we need to find a Hamiltonian path (in contrast to Hamiltonian circuit found in other variants) between these two vertices. We will assume a symmetric metric cost: costs are defined over the complete graph; the costs between two vertices in opposite directions do not differ; and the costs satisfy the triangle inequality. The triangle inequality can be assumed without loss of generality by allowing vertices to be visited more than once; furthermore, it is directly satisfied by many cost functions of interest. This assumption is necessary, in that without the triangle inequality the problem cannot be approximated within any polynomial approximation factor unless $\mathrm{P}=\mathrm{NP}$.

For the traveling salesman circuit problem, a $3/2$-approximation algorithm due to Christofides [5] attains the best performance guarantee known; the matching upper bound on the integrality gap of the Held-Karp relaxation [10], a standard LP relaxation of the problem, is obtained by an analysis of this algorithm [19, 18]. In contrast, for the traveling salesman path problem with two prespecified endpoints, the best approximation ratio known is only $5/3$, as achieved by Hoogeveen’s algorithm [11]; no upper bound on the integrality gap of the (accordingly modified) Held-Karp relaxation is known. Recently, Oveis Gharan, Saberi and Singh [16] studied a special case of the traveling salesman circuit problem where the cost function is a shortest-path metric defined by an unweighted undirected graph, and gave a $(\frac{3}{2}-\epsilon_{0})$-approximation algorithm; its analysis constructively proves that the Held-Karp relaxation under this special case has an integrality gap of at most $\frac{3}{2}-\epsilon_{0}$. In this paper, we first present an LP-based new analysis of Hoogeveen’s algorithm that establishes an upper bound of $5/3$ on the integrality gap of the path-variant Held-Karp relaxation under the general metric. We then modify Oveis Gharan, Saberi and Singh’s algorithm in order to apply it to the path problem; our new analysis of Hoogeveen’s algorithm reveals that the combination of these two algorithms yields a $(\frac{5}{3}-\epsilon)$-approximation algorithm for some constant $\epsilon>0$. This analysis would also constructively prove the matching integrality gap upper bound. We note that recent independent work due to Mömke and Svensson [15] yields a better performance guarantee for the same special case of both the circuit- and path-variants, but shows the matching integrality gap upper bound only for the circuit-variant Held-Karp relaxation; it instead shows for the path-variant case that, for any $\epsilon^{\prime}>0$, there exists $\tau(\epsilon^{\prime})$ that depends on $\epsilon^{\prime}$ such that every graph with more than $\tau(\epsilon^{\prime})$ vertices has an integral optimal solution of value no more than $3-\sqrt{2}+\epsilon^{\prime}$ times the Held-Karp optimum.

Proposed by Held and Karp [10] originally for the symmetric traveling salesman circuit problem, the Held-Karp relaxation is a linear program relaxation of the traveling salesman polytope. It consists of the degree-2 constraints, and the edge connectivity constraints stating that the graph needs to be 2-edge-connected; the asymmetric Held-Karp relaxation is similarly formulated. These relaxations have proven to be useful in designing approximation algorithms in many settings (see, e.g., [2, 1, 16, 15]). In the LP-based design of an approximation algorithm, one important measure of the strength of a particular LP relaxation is its integrality gap, i.e., the worst-case ratio between the integral and fractional optimal values. There is a significant gap between currently known lower and upper bounds on the integrality gap of the Held-Karp relaxation. For the asymmetric circuit case, the best lower bound known, due to Charikar, Goemans and Karloff [3], is $2$, whereas the upper bound, constructively proven by the best known approximation algorithm of Asadpour, Goemans, Mądry, Oveis Gharan and Saberi [2], is $O(\log n/\log\log n)$. For the symmetric circuit case, the best lower bound known is $4/3$, achieved by the family of graphs depicted in Figure 1(a) under the shortest-path metric [8]; yet, the best upper bound known, also constructively proven [19, 18] based on Christofides’ algorithm [5], is $3/2$. Christofides’ algorithm does not directly compute the LP optimum, but the analyses due to Wolsey [19] and Shmoys & Williamson [18] establish that the output solution value can be bounded using the LP optimum.

For the path problem, Hoogeveen [11] presents Christofides-like approximation algorithms for three cases, depending on the number of prespecified endpoints. When both endpoints are prespecified, Hoogeveen [11] gives a $5/3$-approximation algorithm. However, the analysis compares the output solution value to the optimal (integral) solution, and therefore it is unclear whether the algorithm yields an integrality gap bound for the Held-Karp relaxation analogously formulated for the path problem. We observe that the family of graphs in Figure 1(b) demonstrates the integrality gap lower bound of $3/2$. Note that this lower bound is strictly greater than the known upper bound on the integrality gap of the circuit-variant Held-Karp relaxation under the shortest-path metric defined by an unweighted undirected graph; this suggests that the lack of a performance guarantee known matching the $3/2$ for all other TSP variants has a true structural cause.

We present in this paper a new analysis of Hoogeveen’s algorithm, bounding the output solution value in terms of the LP optimum. In particular, we produce two different bounds that both depend on the cost of the direct edge between the two endpoints but with opposite signs; taking the minimum of these two bounds will establish the integrality gap upper bound of $5/3$.

These two bounds coincide when the edge cost is exactly $1/3$ times the LP optimum; this in fact is the critical case that determines the integrality gap bound proven, and the analysis provides a better ratio on all other cases. Thus, the question whether this critical case can be improved, possibly by using a different algorithm only in the critical case or formulating a third bound that does not coincide with the other two at the critical point, could naturally follow. We will illustrate the former approach in a special case where the cost function is a shortest-path metric defined by an unweighted undirected graph: we will show how Oveis Gharan, Saberi and Singh’s algorithm can be modified for the path case and prove that its performance guarantee near the critical point is slightly better than Hoogeveen’s algorithm. This analysis would also serve as a constructive integrality gap proof for the special case.

Section 2 of this paper introduces some definitions and notation; Section 3 presents the new analysis yielding a constructive integrality gap proof for the general metric. Section 4 gives the improved approximation algorithm for the special case of the shortest-path metric defined by an unweighted undirected graph, proving the matching integrality gap upper bound.

In this section, we introduce some definitions and notation to be used throughout this paper.

Let $G=(V,E)$ be the input complete graph with cost function $c:E\to\mathbb{R}_{+}$. Let $s$ and $t$ be the two prespecificed endpoints; we call the other vertices internal points. For $U\subset V$, $\delta(U)$ denotes the set of cut edges: i.e., $\delta(U)=\left\{\{u,v\}\in E:|\{u,v\}\cap U|=1\right\}$. Let $G(U)$ denote the subgraph induced by $U\subset V$.

For $F\subset E$ and $x\in\mathbb{R}^{E}$, $x(F)$ is a shorthand for $\sum_{f\in F}x_{f}$; $c(x)$ is for $\sum_{e\in E}x_{e}c(e)$.

Both linear programs can be solved in polynomial time via the ellipsoid method using a min-cut algorithm to solve the separation problem [9]. The following observation gives an equivalent formulation of (2).

The following theorem gives a nice polyhedral representation of $T$-joins.

Following are from Oveis Gharan, Saberi and Singh [16]. In these, the parameters $\epsilon_{1},\epsilon_{2},\gamma,\delta$ and $\rho$ can be chosen as follows: $\epsilon_{1}=1.875\cdot 10^{-12}$, $\epsilon_{2}=5\cdot 10^{-2}$, $\gamma=10^{-7}$, $\delta=6.25\cdot 10^{-16}$, $\rho=1.5\cdot 10^{-24}$, and $n$ denotes $|V|$.

In this section, we present a new analysis of Hoogeveen’s algorithm for the traveling salesman path problem with two prespecified endpoints. This analysis compares the output solution value to the LP optimum of the path-variant Held-Karp relaxation, although the LP optimum is never computed by the algorithm.

Hoogeveen’s algorithm, when both endpoints are prespecified, is shown in Algorithm 1.

Let $x^{*}\in\mathbb{R}^{E}$ be the LP optimum of the path-variant Held-Karp relaxation, and $\mathsf{ALG}$ be the output of the Algorithm.

Lemmas 3 and 4 give two different bounds on the cost of $M$.

We illustrate in this section how the critical case discussed in Section 3 can be improved by introducing another algorithm that bears a better performance guarantee near the critical case. In particular, we show how the present results combine with the recent work due to Oveis Gharan, Saberi and Singh [16] on a special case of the traveling salesman circuit problem, to yield an improved constructive integrality gap proof of the path-variant Held-Karp relaxation under a similar special case.

Oveis Gharan et al. consider the case where the metric is a shortest path metric defined by an unweighted undirected graph, and give a $\left(\frac{3}{2}-\epsilon_{0}\right)$-approximation algorithm for some constant $\epsilon_{0}>0$. We will consider the traveling salesman path problem with two specified endpoints under the same class of metric, and show how to modify Oveis Gharan, Saberi and Singh’s algorithm for the path case and that, when $c(s,t)$ is close to $\frac{1}{3}c(x^{*})$, this modified algorithm carries a performance guarantee that is slightly better than $5/3$.

Algorithm 2 gives the entire algorithm for the traveling salesman path problem under the special case. It first computes the LP optimum $x^{*}$. If $c(s,t)$ is close to $\frac{1}{3}c(x^{*})$, we run a modified version of Oveis Gharan, Saberi and Singh’s algorithm (Cases A1 and A2); otherwise, we invoke Hoogeveen’s algorithm (Case B). Let $\mathsf{ALG}$ be the output of the algorithm. Parameters $\sigma_{l},\sigma_{u}$ and $\epsilon^{\prime}_{2}$ are to be chosen later.

First we show that we can have a Structure Theorem analogous to Theorem 2 by adjusting $\epsilon_{2}$ and replacing $n$ with $(n-1)$ in Case 2. The following corollary states that either there are good edges of significant weight with respect to $x^{*}_{\mathrm{circuit}}$ or there are many nearly integral edges with respect to $x^{*}$.

Lemmas 5, 6 and 7 yield the following theorem.

Note that this proof also establishes the matching integrality gap upper bound.