We give a new, strongly polynomial-time algorithm and improved analysis for the metric $s-t$ path TSP. It finds a tour of cost less than 1.53 times the optimum of the subtour elimination LP, while known examples show that 1.5 is a lower bound for the integrality gap.
A key new idea is the deletion of some edges of Christofides' trees, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, which are then partly reconnected by `parity correction'. We show that the arising `connectivity correction' can be achieved for a minor extra cost.
On the one hand this algorithm and analysis extend previous tools such as the best-of-many Christofides algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees can replace the convex combination of spanning trees in the best-of-may Christofides algorithm.
These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as it is already sketched in the preliminary shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, and details of proofs and explanations have been added.
We consider the integrality gap of the subtour LP relaxation of the Traveling Salesman Problem restricted to circulant instances. De Klerk and Dobre conjectured that the value of the optimal solution to the subtour LP on these instances is equal to an entirely combinatorial lower bound from Van der Veen, Van Dal, and Sierksma. We prove this conjecture by giving an explicit optimal solution to the subtour LP. We then use it to show that the integrality gap of the subtour LP is 2 on circulant instances, making such instances one of the few non-trivial classes of TSP instances for which the integ…
We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If $ρ_{\text{ATSP}}$ and $ρ_{\text{ATSPP}}$ denote the integrality ratios for the asymmetric TSP and its path version, then $ρ_{\text{ATSPP}}\le 4ρ_{\text{ATSP}}-3$. We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is $ρ_{\text{ATSP}}^{\text{NW}}$, then the integrality ratio for ATSPP on node-weighted instances is at most $2ρ_{\text{ATSP}}^{\text NW}-1$. Moreover, we show that for ATSP node-weig…
In this paper we investigate instances with high integrality ratio of the subtour LP. We develop a procedure to generate families of Euclidean TSP instances whose integrality ratios converge to $\frac{4}{3}$ and may have a different structure than the instances currently known from the literature. Moreover, we compute the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices. Based on these instances we give families of instances whose integrality ratio converge to $\frac{4}{3}$ for Rectilinear, Multidimensional Rectilinear and Euclidean TSP that have similar st…
The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, e.g., algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [9] present an SDP based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program, but generally dominates it on small instances. We provide a family of \emp…
The well known $4/3$ conjecture states that the integrality ratio of the subtour LP is at most $4/3$ for metric Traveling Salesman instances. We present a family of Euclidean Traveling Salesman instances for which we prove that the integrality ratio of the subtour LP converges to $4/3$. These instances (using the rounded Euclidean norm) turn out to be hard to solve exactly with Concorde, the fastest existing exact TSP solver. For a 200 vertex instance from our family of Euclidean Traveling Salesman instances Concorde needs several days of CPU time. This is more than 1,000,000 times the runtime…
The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components
Scuola Universitaria Professionale della Svizzera Italiana (SUPSI)（瑞士意大利语区应用科学大学）；Dalle Molle Institute for Artificial Intelligence USI-SUPSI（USI-SUPSI 达莱莫勒人工智能研究所）；Università della Svizzera Italiana (USI)（瑞士意大利语区大学）
Finding the exact integrality gap $α$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq α\leq 3/2$, and a famous conjecture states $α= 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $ρ$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simp…
Among various variants of the traveling salesman problem, the s-t-path graph TSP has the special feature that we know the exact integrality ratio, 3/2, and an approximation algorithm matching this ratio. In this paper, we go below this threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP with approximation ratio 1.497. Our algorithm can be viewed as a refinement of the 3/2-approximation algorithm by Sebő and Vygen [2014], but we introduce several completely new techniques. These include a new type of ear-decomposition, an enhanced ear induction that reveals a novel conne…
In this paper we investigate the integrality ratio of the standard LP relaxation for the metric $s-t$ Path TSP. We make a near-optimal choice for an auxiliary function used in the analysis of Traub and Vygen which leads to an improved upper bound on the integrality ratio of 1.5273.
We give an improved analysis of the best-of-many Christofides algorithm with lonely edge deletion, which was proposed by Sebő and van Zuylen [2016]. This implies an improved upper bound on the integrality ratio of the standard LP relaxation for the $s$-$t$-path TSP.
Finding the exact integrality gap $α$ for the LP relaxation of the 2-edge-connected spanning multigraph problem (2EC) is closely related to the same problem for the Held-Karp relaxation of the metric traveling salesman problem (TSP). While the former problem seems easier than the latter, since it is less constrained, currently the upper bounds on the respective integrality gaps for the two problems are the same. An approach to proving integrality gaps for both of these problems is to consider fundamental classes of extreme points. For 2EC, better bounds on the integrality gap are known for cer…
We revisit the constant-factor approximation algorithm for the asymmetric traveling salesman problem by Svensson, Tarnawski, and Végh. We improve on each part of this algorithm. We avoid the reduction to irreducible instances and thus obtain a simpler and much better reduction to vertebrate pairs. We also show that a slight variant of their algorithm for vertebrate pairs has a much smaller approximation ratio. Overall we improve the approximation ratio from $506$ to $22+ε$ for any $ε> 0$. This also improves the upper bound on the integrality ratio from $319$ to $22$.
We design a $1.49993$-approximation algorithm for the metric traveling salesperson problem (TSP) for instances in which an optimal solution to the subtour linear programming relaxation is half-integral. These instances received significant attention over the last decade due to a conjecture of Schalekamp, Williamson and van Zuylen stating that half-integral LP solutions have the largest integrality gap over all fractional solutions. So, if the conjecture of Schalekamp et al. holds true, our result shows that the integrality gap of the subtour polytope is bounded away from $3/2$.
Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to opti…
The paper proposes a quantum algorithm for the traveling salesman problem (TSP) based on the Grover Adaptive Search (GAS), which can be successfully executed on IBM's Qiskit library. Under the GAS framework, there are at least two fundamental difficulties that limit the application of quantum algorithms for combinatorial optimization problems. One difficulty is that the solutions given by the quantum algorithms may not be feasible. The other difficulty is that the number of qubits of current quantum computers is still very limited, and it cannot meet the minimum requirements for the number of …
We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge $e$ is considered covered by the first time when $k_e$ many of its vertices appear in the ordering. We give a $4.642$ approximation algorithm for GMSSC, coming close to the best possible bound of $4$, already for the classical special case (with all $k_e=1$) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali, and improving u…
