Wesley Pegden, Anish Sevekari
We prove that even in average case, the Euclidean Traveling Salesman Problem exhibits an integrality gap of $(1+ε)$ for $ε>0$ when the Held-Karp Linear Programming relaxation is augmented by all comb inequalities of bounded size. This implies that large classes of branch-and-cut algorithms take exponential time for the Euclidean TSP, even on random inputs.
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Edge Elimination in TSP Instances
旅行商问题是组合优化中研究最多的NP-难问题之一。在过去的60年里，已经开发出了强大的方法来寻找大型TSP实例的最佳解决方案。迄今为止，最大的TSP实例已被最佳解决，有85，900个顶点。它的解决方案需要超过136年的总CPU时间使用分支和切割基于协和TSP代码[1]。在本文中，我们提出了图论的结果，可以证明，一些边缘的TSP实例不能出现在任何最佳TSP巡回赛。基于这些结果，我们提出了一个组合算法来识别这样的边缘。我们的算法的主要部分的运行时间是$O(n^2 \log n)$的n顶点TSP实例。通过将我们的方法与Concorde TSP求解器相结合，我们能够比单独使用Concorde解决大型TSPLIB实例快11倍以上。
Combining Approximation Algorithms for the Prize-Collecting TSP
本文提出了一个求解有奖旅行商问题的1.91457-近似算法。这是通过将Bienstock等人的舍入算法的随机变体与Goemans和Williamson的原始对偶算法相结合来获得的。
We present a black-box reduction from the path version of the Traveling Salesman Problem (Path TSP) to the classical tour version (TSP). More precisely, we show that given an $α$-approximation algorithm for TSP, then, for any $ε>0$, there is an $(α+ε)$-approximation algorithm for the more general Path TSP. This reduction implies that the approximability of Path TSP is the same as for TSP, up to an arbitrarily small error. This avoids future discrepancies between the best known approximation factors achievable for these two problems, as they have existed until very recently. A well-studied spec…
Given a metric $(V,d)$ and a $\textsf{root} \in V$, the classic $\textsf{$k$-TSP}$ problem is to find a tour originating at the $\textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of $\textsf{$k$-TSP}$. In Stoch-Reward $k$-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex $v$ in the given metric $(V,d)$ contains a stochastic reward $R_v$. The goal is to adaptively find a tour of minimum expected length that collects at least re…
TSP is a classical NP-hard combinatorial optimization problem with many practical variants. LKH is one of the state-of-the-art local search algorithms for the TSP. LKH-3 is a powerful extension of LKH that can solve many TSP variants. Both LKH and LKH-3 associate a candidate set to each city to improve the efficiency, and have two different methods, $α$-measure and POPMUSIC, to decide the candidate sets. In this work, we first propose a Variable Strategy Reinforced LKH (VSR-LKH) algorithm, which incorporates three reinforcement learning methods (Q-learning, Sarsa, Monte Carlo) with LKH, for th…
Linear Programming based Reductions for Multiple Visit TSP and Vehicle Routing Problems
本研究提出基于线性规划的归约方法，可将多旅行商问题（mTSP）的LP类算法转换为多次访问多旅行商问题（MV-mTSP）的同近似因子算法，改进或匹配了该问题多类变体的当前最优近似因子，证明访问需求不会提升近似难度。
Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, s-t-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the sp…
Improved space-time tradeoff for TSP via extremal set systems
本文改进了TSP的空间时间折衷，通过构造稀疏集合系统，实现了更优的折衷，同时推翻了相关猜想，拓展到其他排列问题。
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…
A 5/4 bound for graphic $s$-$t$ path TSP on subcubic graphs
该研究针对次立方图上的图论s-t路径旅行商问题，消除了s、t需为边端点的限制，得到5/4的最优渐近界，还提出了O(n²)算法，为立方图路径旅行商问题提供了首个直接证明的5/4界。
An Improved ACS Algorithm for the Solutions of Larger TSP Problems
如何有效地求解大型旅行商问题是计算机科学研究人员面临的一个挑战。针对组合优化问题中最突出的一类TSP问题，提出了一种改进的蚁群算法--红黑蚁群算法（RB-ACS）。RB-ACS算法利用蚁群系统的思想和遗传算法的并行搜索机制，快速获得最优解。在本文中，它表明，建议RB-ACS算法产生显着更好的性能比现有的最知名的算法。
A Speed-up for Helsgaun's TSP Heuristic by Relaxing the Positive Gain Criterion
针对旅行商问题的LKH启发式算法，本研究通过放宽正增益准则发现新改进步骤，经多基准库仿真验证，大规模算例下平均比最新版LKH快13%。
In the Bounded Degree Matroid Basis Problem, we are given a matroid and a hypergraph on the same ground set, together with costs for the elements of that set as well as lower and upper bounds $f(\varepsilon)$ and $g(\varepsilon)$ for each hyperedge $\varepsilon$. The objective is to find a minimum-cost basis $B$ such that $f(\varepsilon) \leq |B \cap \varepsilon| \leq g(\varepsilon)$ for each hyperedge $\varepsilon$. Király et al. (Combinatorica, 2012) provided an algorithm that finds a basis of cost at most the optimum value which violates the lower and upper bounds by at most $2 Δ-1$, where …
The cost due to delay in services may be intrinsically different for various applications of vehicle routing such as medical emergencies, logistical operations, and ride-sharing. We study a fundamental generalization of the Traveling Salesman Problem, namely $L_p$ TSP, where the objective is to minimize an aggregated measure of the delay in services, quantified by the Minkowski $p$-norm of the delay vector. We present efficient combinatorial and Linear Programming algorithms for approximating $L_p$ TSP on general metrics. We provide several approximation algorithms for the $L_p$ TSP problem, i…
We introduce the $L_p$ Traveling Salesman Problem ($L_p$-TSP), given by an origin, a set of destinations, and underlying distances. The objective is to schedule a destination visit sequence for a traveler of unit speed to minimize the Minkowski $p$-norm of the resulting vector of visit/service times. For $p = \infty$ the problem becomes a path variant of the TSP, and for $p = 1$ it defines the Traveling Repairman Problem (TRP), both at the center of classical combinatorial optimization. We provide an approximation preserving polynomial-time reduction of $L_p$-TSP to the segmented-TSP Problem […
We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a $θ$-cyclic point: $x_e \in \{0,θ, 1-θ, 1\}$, where the support graph is subcubic and each vertex is incident to at least one…
Approximating Prize-Collecting Variants of TSP
本文针对同时推广Prize-collecting TSP和Ordered TSP的PCOTSP问题，通过结合树采样、概率顶点选取和剪枝奇偶性修正技术，提出了2.097近似算法，并扩展至Prize-collecting Multi-Path TSP获得2.41近似比。
For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today's computation, we employ one of the most successful models of such precomputation -- the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations. We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also cons…
Maximum Entropy is a 10/7-Approximation Algorithm for the TSP on Half-Integral Cycle Cut Instances
针对TSP的半整数环割实例，证明最大熵算法达到10/7近似比，该类实例包含Subtour LP积分间隙为4/3的案例，且算法性能不低于11/8。
Efficient Local Search Algorithms for Known and New Neighborhoods for the Generalized Traveling Salesman Problem
广义旅行商问题（GTSP）是一个著名的组合优化问题，有着广泛的应用。它是旅行推销员问题（TSP）的扩展，其中城市集被划分为所谓的集群，推销员必须访问每个集群一次。虽然GTSP是一个非常重要的组合优化问题，并在许多方面得到了很好的研究，但文献中使用的局部搜索算法大多是简单TSP算法的基本适应。因此，需要对GTSP的邻域和局部搜索算法进行全面深入的研究。我们形式化的TSP邻域的GTSP和分类所有其他现有的和一些新的GTSP邻域的适应过程。对于每个邻域，我们提供了有效的探索算法，通常比文献中已知的算法要快得多。最后，我们比较不同的本地搜索实现经验。
We describe a hybrid procedure for solving the traveling salesman problem (TSP) to provable optimality. We first sparsify the instance, and then use a hybrid algorithm that combines a branch-and-cut TSP solver with a Hamiltonian cycle problem solver. We demonstrate that this procedure enables us to solve difficult instances to optimality, including one which had remained unsolved since its construction in 2002.
Quantum Space-Time Tradeoffs for TSP and Hypercube Path via Extremal Set Systems
研究通过极值集系统改善TSP等排列问题的时空权衡，提出量子类似物，利用量子最小值查找，给出TSP量子算法的时空复杂度，结合改进构造得到权衡曲线，在\(1 < S \leq 1.657\)时优于已知量子权衡。
A Novel Feature-Based Approach to Characterize Algorithm Performance for the Traveling Salesman Problem
元算法常用于解决NP难的组合优化问题。在本文中，我们有助于了解成功的2-选择为基础的局部搜索算法解决旅行商问题（TSP）。虽然2-opt在实践中被广泛使用，但从理论角度很难理解它的成功。我们采取统计方法，并检查TSP的情况下，使问题很难或很容易解决的功能。作为2-opt的问题难度的度量，我们使用它在给定实例上实现的近似比。我们的调查指出了重要的功能，使TSP的实例很难或很容易被近似的2-选择。
In the maximum traveling salesman problem (Max TSP) we are given a complete undirected graph with nonnegative weights on the edges and we wish to compute a traveling salesman tour of maximum weight. We present a fast combinatorial $\frac 45$ - approximation algorithm for Max TSP. The previous best approximation for this problem was $\frac 79$. The new algorithm is based on a novel technique of eliminating difficult subgraphs via half-edges, a new method of edge coloring and a technique of exchanging edges. A half-edge of edge $e=(u,v)$ is informally speaking "a half of $e$ containing either $u…
The Kernighan-Lin Search Algorithm
本文将Lin-Kernighan和Kernighan-Lin算法中共享的可变深度搜索机制抽象为通用的Kernighan-Lin搜索算法，实验证明其在解质量和运行时间上优于遗传算法等通用搜索算法。
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying more tractable special cases of the problem. In this paper, we focus on the fundamental special case of regions that are hyperplanes in the $d$-dimensional Euclidean space. This case contrasts the much-better understood case of so-called fat regions. While for $d=2$ an exact algorithm with r…
We consider the {\em MST-interdiction} problem: given a multigraph $G = (V, E)$, edge weights $\{w_e\geq 0\}_{e \in E}$, interdiction costs $\{c_e\geq 0\}_{e \in E}$, and an interdiction budget $B\geq 0$, the goal is to remove a set $R\subseteq E$ of edges of total interdiction cost at most $B$ so as to maximize the $w$-weight of an MST of $G-R:=(V,E\setminus R)$. Our main result is a $4$-approximation algorithm for this problem. This improves upon the previous-best $14$-approximation~\cite{Zenklusen15}. Notably, our analysis is also significantly simpler and cleaner than the one in~\cite{Zenk…
Vector TSP: A Traveling Salesperson Problem with Racetrack-like Acceleration Constraints
本文研究带赛道式离散加速度约束的新旅行商问题VectorTSP，证明其NP难，提出轨迹预言机的A*适配实现，实验表明朴素方法次优于专用启发式算法。
Multi-armed Bandit and Backbone boost Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problems
针对 LKH 中 α 值未利用历史信息且引导单一的问题，本文提出动态骨干信息，并通过多臂老虎机自适应组合骨干、α 值与距离，显著提升 LKH/LKH-3 在 TSP、CTSP 和 CVRPTW 上的性能与泛化性。
On modeling NP-Complete problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers"
该研究针对NP完全问题能否建模为多项式规模线性规划的疑问，以TSP为例，证明TSP多面体无多项式扩展表述不代表无法用多项式规模线性规划求解TSP，可通过不用传统城市间变量绕开相关壁垒。
TSP (Traveling Salesman Problem), a classic NP-complete problem in combinatorial optimization, is of great significance in multiple fields. Exact algorithms for TSP are not practical due to their exponential time cost. Thus, approximate algorithms become the research focus and can be further divided into two types, tour construction algorithms and tour improvement algorithms. Researches show that the latter type of algorithms can obtain better solutions than the former one. However, traditional tour improvement algorithms have shortcomings. They converge very slowly and tend to be trapped in l…
The traveling salesman problem (TSP) is one of the most challenging NP-hard problems. It has widely applications in various disciplines such as physics, biology, computer science and so forth. The best known approximation algorithm for Symmetric TSP (STSP) whose cost matrix satisfies the triangle inequality (called $\triangle$STSP) is Christofides algorithm which was proposed in 1976 and is a $\frac{3}{2}$-approximation. Since then no proved improvement is made and improving upon this bound is a fundamental open question in combinatorial optimization. In this paper, for the first time, we prop…
In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t. In this paper we identify a new polynomially solvable case of the Path-TSP where the distance matrix of the cities is a so-called Demidenko matrix. We identify a number of crucial combinatorial properties of the optimal solution, and we design a dynamic program with time complexity $O(n^6)$.
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