Fine-grained complexity for sparse graphs

Abstract

We consider the fine-grained complexity of sparse graph problems that currently haveÕ(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (minimum weight cycle), and Eccentricities, which is the problem of computing, for each vertex in the graph, the length of a longest shortest path starting at that vertex.We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in thẽ O(mn) class. Surprisingly, very few of the known nontrivial reductions between problems in thẽ O(mn) class are sparse reductions. In the directed case, our results give a partial order on a large collection of problems in theÕ(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to a natural variant of APSP. The latter reduction also gives an improved algorithm for ANSC (for dense graphs).We propose the MWC Conjecture, a new conditional hardness conjecture that the weight of a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in theÕ(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, and Eccentricities, are MWCC hard. We also identify Eccentricities as a key problem in theÕ(mn) class which is simultaneously MWCC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than n k .Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which theÕ(mn) time algorithms are the ones typically used in practice, and not theÕ(n 3 ) time algorithms.1Õ andΘ can hide sub-polynomial factors; in our new results they only hide polylog factors.

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

“…Let G be the graph in which we want to find the Shortest Cycle. Compute APSP in G, and then for every edge add d(u, v) + w(v, u) and take the minimum -this is the weight of the Shortest Cycle in G. Reducing directed Shortest Cycle to APSP in undirected graphs seems more problematic, as noted by Agarwal and Ramachandran [AR16]. In our paper, however, we were able to reduce Min Weight k-Cycle in a directed graph to Radius in undirected graphs.…”

“…Agarwal and Ramachandran [AR16] build on these reductions to show many sparsity-preserving reductions from Shortest Cycle to various fundamental graph problems.…”

“…THEOREM 1.1. ( [AR16]) Suppose that there is a constant ε > 0 such that one of the following problems on n-node, m-edge weighted graphs can be solved in O(mn 1−ε + n 2 ) time:…”

“…We also create novel reductions from the k-Cycle problem, in Section B, and these give us novel hardness results for many new problems. The main new contributions are reductions to Radius in undirected graphs (the result in [AR16] is only for directed) and to the Wiener Index problem which asks for the sum of all distances in the graph. Together all these pieces give us the following theorem.…”

“…The bold edges represent a subset of the reductions in this paper. The dashed edges are reductions from Agarwal and Ramachandran [AR16].…”