We analyze two classic variants of the T RAVELING S ALESMAN P ROBLEM ( TSP ) using the toolkit of fine-grained complexity. Our first set of results is motivated by the B ITONIC TSP problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in O ( n 2 ) time. While the near-quadratic dependency of similar dynamic programs for L ONGEST C OMMON S UBSEQUENCE and D ISCRETE F réchet D istance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in O ( n log 2 n ) time and its bottleneck version in O ( n log 3 n ) time. In the more general pyramidal TSP problem, the points to be visited are labeled 1,… , n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) TSP problem also work for the pyramidal TSP problem in the plane. Our second set of results concerns the popular k - OPT heuristic for TSP in the graph setting. More precisely, we study the k - OPT decision problem, which asks whether a given tour can be improved by a k - OPT move that replaces k edges in the tour by k new edges. A simple algorithm solves k - OPT in O ( n k ) time for fixed k . For 2- OPT , this is easily seen to be optimal. For k =3, we prove that an algorithm with a runtime of the form Õ( n 3−ɛ ) exists if and only if A LL -P AIRS S HORTEST P ATHS in weighted digraphs has such an algorithm. For general k - OPT , it is known that a runtime of f ( k ) · n o ( k / log k ) would contradict the Exponential Time Hypothesis. The results for k =2,3 may suggest that the actual time complexity of k - OPT is Θ ( n k ). We show that this is not the case, by presenting an algorithm that finds the best k -move in O ( n ⌊ 2 k /3 ⌋+1 ) time for fixed k ≥ 3. This implies that 4- OPT can be solved in O ( n 3 ) time, matching the best-known algorithm for 3- OPT . Finally, we show how to beat the quadratic barrier for k =2 in two important settings, namely, for points in the plane and when we want to solve 2- OPT repeatedly.

中文翻译：

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两个经典 TSP 变体的细粒度复杂性分析

我们分析了 T 的两个经典变体游荡小号艾尔斯曼磷问题(TSP) 使用细粒度复杂性的工具包。我们的第一组结果是由 B伊托尼克 TSP问题：给定一组n平面上的点，计算由两条单调链组成的最短路径。这是一个经典的动态规划练习来解决这个问题○(n 2） 时间。而 L 的类似动态程序的近似二次依赖ONGESTC欧姆蒙小号UB序列和 D伊斯克雷特F雷谢D距离最近已被证明在强指数时间假设下基本上是最优的，我们表明可以在二次时间中找到双调旅行。更准确地说，我们提出了一种解决双音 TSP 的算法○(n日志2 n) 时间及其瓶颈版本○(n日志3 n） 时间。在更一般的金字塔 TSP 问题中，要访问的点标记为 1，...，n并且要求解中的标签序列至多有一个局部最大值。我们针对双调（瓶颈）TSP 问题的算法也适用于平面中的金字塔形 TSP 问题。我们的第二组结果涉及流行的ķ-选择图形设置中 TSP 的启发式方法。更准确地说，我们研究ķ-选择决策问题，它询问给定的游览是否可以通过ķ-选择移动取代ķ游览中的边缘ķ新的边缘。一个简单的算法解决ķ-选择在○(n ķ ) 时间固定ķ. 对于 2-选择，这很容易看出是最优的。为了ķ=3，我们证明了一个运行时间为 Õ(n 3−ɛ) 存在当且仅当 A二-P航空航天局小号最危险的磷ATHS在加权有向图中有这样的算法。对于一般ķ-选择，众所周知，运行时F(ķ) ·n ○(ķ/ 日志ķ)将与指数时间假设相矛盾。结果为ķ=2,3 可能表明实际时间复杂度为ķ-选择是 Θ (n ķ ）。我们通过提出一种找到最佳算法的算法来证明情况并非如此ķ-搬进来○(n ⌊ 2ķ/3 ⌋+1) 时间固定ķ≥ 3。这意味着 4-选择可以解决○(n 3) 时间，匹配最知名的算法 3-选择. 最后，我们展示了如何克服二次障碍ķ=2 在两个重要设置中，即对于平面中的点和当我们要求解 2-选择反复。