Abstract
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.
Highlights
1.1 MotivationWe analyze two classic variants of the Traveling Salesman Problem by applying the modern toolkit of fine-grained complexity analysis
Before we turn our attention to the minimization version of the bottleneck pyramidal TSP, we first show an Ω(n log n) time lower bound for the decision version of the problem in the Euclidean plane, in the algebraic computation-tree model [9]. (The algebraic computation-tree model is similar to a real RAM, the main difference being that it excludes the floor function.) The reduction even applies to the bitonic setting where the points are ordered from left to right
Revisiting the worst-case complexity of k-opt and pyramidal TSP led to a number of new results on these classic problems
Summary
We analyze two classic variants of the Traveling Salesman Problem (tsp) by applying the modern toolkit of fine-grained complexity analysis. One might wonder whether O(n2) runtime is best possible for this problem. The second TSP variant concerns k-opt, a popular local search heuristic that attempts to improve a suboptimal solution by a k-opt move (or: k-move for short), which is an operation that removes k edges from the current tour and reconnects the resulting pieces into a new tour by inserting k new edges. The decision problem associated with k-opt asks, given a tour in an edge-weighted graph, whether it is possible to obtain a tour of smaller weight by replacing k edges. As the weight change for each reconnection pattern can be evaluated in O(k ) time, this simple algorithm finds the best k-opt improvement in time O(nk ) for each fixed k. One might wonder whether there are other, faster algorithmic approaches that proceed without enumerating all moves.
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