Search for: All records

Given a weighted graph G = (V,E,w), a (β, ε)-hopset H is an edge set such that for any s,t ∈ V, where s can reach t in G, there is a path from s to t in G ∪ H which uses at most β hops whose length is in the range [dist_G(s,t), (1+ε)dist_G(s,t)]. We break away from the traditional question that asks for a hopset H that achieves small |H| and small diameter β and instead study the sensitivity of H, a new quality measure. The sensitivity of a vertex (or edge) given a hopset H is, informally, the number of times a single hop in G ∪ H bypasses it; a bit more formally, assuming shortest paths in G are unique, it is the number of hopset edges (s,t) ∈ H such that the vertex (or edge) is contained in the unique st-path in G having length exactly dist_G(s,t). The sensitivity associated with H is then the maximum sensitivity over all vertices (or edges). The highlights of our results are: - A construction for (Õ(√n), 0)-hopsets on undirected graphs with O(log n) sensitivity, complemented with a lower bound showing that Õ(√n) is tight up to polylogarithmic factors for any construction with polylogarithmic sensitivity. - A construction for (n^o(1), ε)-hopsets on undirected graphs with n^o(1) sensitivity for any ε > 0 that is at least inverse polylogarithmic, complemented with a lower bound on the tradeoff between β, ε, and the sensitivity. - We define a notion of sensitivity for β-shortcut sets (which are the reachability analogues of hopsets) and give a construction for Õ(√n)-shortcut sets on directed graphs with O(log n) sensitivity, complemented with a lower bound showing that β = Ω̃(n^{1/3}) for any construction with polylogarithmic sensitivity. We believe hopset sensitivity is a natural measure in and of itself, and could potentially find use in a diverse range of contexts. More concretely, the notion of hopset sensitivity is also directly motivated by the Differentially Private All Sets Range Queries problem [Deng et al. WADS 23]. Our result for O(log n) sensitivity (Õ(√n), 0)-hopsets on undirected graphs immediately improves the current best-known upper bound on utility from Õ(n^{1/3}) to Õ(n^{1/4}) in the pure-DP setting, which is tight up to polylogarithmic factors. 

In the k-Disjoint Shortest Paths (k-DSP) problem, we are given a graph G (with positive edge weights) on n nodes and m edges with specified source vertices s_1, … , s_k, and target vertices t_1, … , t_k, and are tasked with determining if G contains vertex-disjoint (s_i,t_i)-shortest paths. For any constant k, it is known that k-DSP can be solved in polynomial time over undirected graphs and directed acyclic graphs (DAGs). However, the exact time complexity of k-DSP remains mysterious, with large gaps between the fastest known algorithms and best conditional lower bounds. In this paper, we obtain faster algorithms for important cases of k-DSP, and present better conditional lower bounds for k-DSP and its variants. Previous work solved 2-DSP over weighted undirected graphs in O(n⁷) time, and weighted DAGs in O(mn) time. For the main result of this paper, we present optimal linear time algorithms for solving 2-DSP on weighted undirected graphs and DAGs. Our linear time algorithms are algebraic however, and so only solve the detection rather than search version of 2-DSP (we show how to solve the search version in O(mn) time, which is faster than the previous best runtime in weighted undirected graphs, but only matches the previous best runtime for DAGs). We also obtain a faster algorithm for k-Edge Disjoint Shortest Paths (k-EDSP) in DAGs, the variant of k-DSP where one seeks edge-disjoint instead of vertex-disjoint paths between sources and their corresponding targets. Algorithms for k-EDSP on DAGs from previous work take Ω(m^k) time. We show that k-EDSP can be solved over DAGs in O(mn^{k-1}) time, matching the fastest known runtime for solving k-DSP over DAGs. Previous work established conditional lower bounds for solving k-DSP and its variants via reductions from detecting cliques in graphs. Prior work implied that k-Clique can be reduced to 2k-DSP in DAGs and undirected graphs with O((kn)²) nodes. We improve this reduction, by showing how to reduce from k-Clique to k-DSP in DAGs and undirected graphs with O((kn)²) nodes (halving the number of paths needed in the reduced instance). A variant of k-DSP is the k-Disjoint Paths (k-DP) problem, where the solution paths no longer need to be shortest paths. Previous work reduced from k-Clique to p-DP in DAGs with O(kn) nodes, for p = k + k(k-1)/2. We improve this by showing a reduction from k-Clique to p-DP, for p = k + ⌊k²/4⌋. Under the k-Clique Hypothesis from fine-grained complexity, our results establish better conditional lower bounds for k-DSP for all k ≥ 4, and better conditional lower bounds for p-DP for all p ≤ 4031. Notably, our work gives the first nontrivial conditional lower bounds 4-DP in DAGs and 4-DSP in undirected graphs and DAGs. Before our work, nontrivial conditional lower bounds were only known for k-DP and k-DSP on such graphs when k ≥ 6. 

The aspect ratio of a (positively) weighted graph G is the ratio of its maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as a complexity measure in graph algorithms, especially related to the computation of shortest paths. Popular paradigms are to interpolate between the settings of weighted and unweighted input graphs by incurring a dependence on aspect ratio, or by simply restricting attention to input graphs of low aspect ratio. This paper studies the effects of these paradigms, investigating whether graphs of low aspect ratio have more structured shortest paths than graphs in general. In particular, we raise the question of whether one can generally take a graph of large aspect ratio and reweight its edges, to obtain a graph with bounded aspect ratio while preserving the structure of its shortest paths. Our findings are: - Every weighted DAG on n nodes has a shortest-paths preserving graph of aspect ratio O(n). A simple lower bound shows that this is tight. - The previous result does not extend to general directed or undirected graphs; in fact, the answer turns out to be exponential in these settings. In particular, we construct directed and undirected n-node graphs for which any shortest-paths preserving graph has aspect ratio 2^{Ω(n)}. We also consider the approximate version of this problem, where the goal is for shortest paths in H to correspond to approximate shortest paths in G. We show that our exponential lower bounds extend even to this setting. We also show that in a closely related model, where approximate shortest paths in H must also correspond to approximate shortest paths in G, even DAGs require exponential aspect ratio. 

We construct n-node graphs on which any O(n)-size spanner has additive error at least +Ω(n^{3/17}), improving on the previous best lower bound of Ω(n^{1/7}) [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for O(n)-size additive emulators to +Ω(n^{1/14}).