Shortest paths between shortest paths and independent sets

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and $P_4$-free graphs)

Shortest Paths Avoiding Forbidden Subpaths

In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception-avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences, and used the STACS styl

Shortest paths in orthogonal graphs

Orthogonal graphs were introduced as a simple but powerful tool for the description and analysis of a class of interconnection networks. Routing, and hence finding shortest paths between any two nodes of an orthogonal graph, becomes an important problem. It is shown in this paper that routing in this class of graphs reduces to a node covering problem in the bipartite coverage graph of the orthogonal graph. A minimum cover clearly leads to a shortest path. In general, the problem of finding the mínimum node cover in a bipartite graph is NP-complete. However, the bipartite coverage graphs corresponding to orthogonal graphs have a regular pattern of edges. This allows the development of a routing algorithm which results in a minimum cover. The procedure executes in polynomial time in the number of bit-nodes of the bipartite graph. It therefore results in a shortest path algorithm whose time complexity is quadratic in the logarithm of the number of nodes in the original orthogonal graph

Improved Distributed Algorithms for Exact Shortest Paths

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Becker et al. [DISC'17] which deterministically computes $(1+o(1))$-approximate shortest paths in $\tilde O(D+\sqrt n)$ time, where $D$ is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC'04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC'17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we present a new single-source shortest path algorithm with complexity $\tilde O(n^{3/4}D^{1/4})$. For polylogarithmic $D$, this improves on Elkin's $\tilde{O}(n^{5/6})$ bound and gets closer to the $\tilde{\Omega}(n^{1/2})$ lower bound of Elkin [STOC'04]. For larger values of $D$, we present an improved variant of our algorithm which achieves complexity $\tilde{O}\left( n^{3/4+o(1)}+ \min\{ n^{3/4}D^{1/6},n^{6/7}\}+D\right)$, and thus compares favorably with Elkin's bound of $\tilde{O}(n^{5/6} + n^{2/3}D^{1/3} + D )$ in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graphs (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact $\kappa$-source shortest paths...Comment: 26 page