Near-Shortest and K-Shortest Simple Paths

2005, Networks, 46, pp. 98-109

Fine-Grained Complexity Analysis of Two Classic TSP Variants

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic 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 Longest Common Subsequence and Discrete Frechet Distance 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. 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 $\tilde{O}(n^{3-\epsilon})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $\Theta(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. 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.Comment: Extended abstract appears in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016

Replacement Paths via Row Minima of Concise Matrices

Matrix $M$ is {\em $k$-concise} if the finite entries of each column of $M$ consist of $k$ or less intervals of identical numbers. We give an $O(n+m)$-time algorithm to compute the row minima of any $O(1)$-concise $n\times m$ matrix. Our algorithm yields the first $O(n+m)$-time reductions from the replacement-paths problem on an $n$-node $m$-edge undirected graph (respectively, directed acyclic graph) to the single-source shortest-paths problem on an $O(n)$-node $O(m)$-edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacement-paths problem is no harder than the single-source shortest-paths problem on undirected graphs and directed acyclic graphs. Moreover, our linear-time reductions lead to the first $O(n+m)$-time algorithms for the replacement-paths problem on the following classes of $n$-node $m$-edge graphs (1) undirected graphs in the word-RAM model of computation, (2) undirected planar graphs, (3) undirected minor-closed graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete Mathematic

Approximating the Diameter of Planar Graphs in Near Linear Time

We present a $(1+\epsilon)$-approximation algorithm running in $O(f(\epsilon)\cdot n \log^4 n)$ time for finding the diameter of an undirected planar graph with non-negative edge lengths