Approximation Algorithms for Min-Distance Problems in DAGs

Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no path between them in one of the two directions. So it is natural to consider the distance between two nodes as the length of the shortest path in the direction in which this path exists, motivating the definition of the min-distance. The min-distance between two nodes u and v is the minimum of the shortest path distances from u to v and from v to u. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in O?(mn) time. So it is natural to resort to approximation algorithms in O?(mn^{1-?}) time for some positive ?. Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, and Dalirrooyfard et al [ICALP 2019] gave the first constant factor approximation algorithms on general graphs for min-diameter, min-radius and min-eccentricities. Abboud et al obtained a 3-approximation algorithm for min-radius on DAGs which works in O?(m?n) time, and showed that any (2-?)-approximation requires n^{2-o(1)} time for any ? > 0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in O?(m?n) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time 2-approximation algorithm for min-diameter along with a lower bound stating that any (3/2-?)-approximation algorithm for sparse DAGs requires n^{2-o(1)} time under SETH. We close this gap for dense DAGs by obtaining a 3/2-approximation algorithm which works in O(n^{2.350}) time and showing that the approximation factor is unlikely to be improved within O(n^{? - o(1)}) time under the high dimensional Orthogonal Vectors Conjecture, where ? is the matrix multiplication exponent

Conditionally Optimal Approximation Algorithms for the Girth of a Directed Graph

It is known that a better than $2$-approximation algorithm for the girth in dense directed unweighted graphs needs $n^{3-o(1)}$ time unless one uses fast matrix multiplication. Meanwhile, the best known approximation factor for a combinatorial algorithm running in $O(mn^{1-\epsilon})$ time (by Chechik et al.) is $3$. Is the true answer $2$ or $3$? The main result of this paper is a (conditionally) tight approximation algorithm for directed graphs. First, we show that under a popular hardness assumption, any algorithm, even one that exploits fast matrix multiplication, would need to take at least $mn^{1-o(1)}$ time for some sparsity $m$ if it achieves a $(2-\epsilon)$-approximation for any $\epsilon>0$. Second we give a $2$-approximation algorithm for the girth of unweighted graphs running in $\tilde{O}(mn^{3/4})$ time, and a $(2+\epsilon)$-approximation algorithm (for any $\epsilon>0$) that works in weighted graphs and runs in $\tilde{O}(m\sqrt n)$ time. Our algorithms are combinatorial. We also obtain a $(4+\epsilon)$-approximation of the girth running in $\tilde{O}(mn^{\sqrt{2}-1})$ time, improving upon the previous best $\tilde{O}(m\sqrt n)$ running time by Chechik et al. Finally, we consider the computation of roundtrip spanners. We obtain a $(5+\epsilon)$-approximate roundtrip spanner on $\tilde{O}(n^{1.5}/\epsilon^2)$ edges in $\tilde{O}(m\sqrt n/\epsilon^2)$ time. This improves upon the previous approximation factor $(8+\epsilon)$ of Chechik et al. for the same running time.Comment: To appear in ICALP 202

Graph Partitioning With Limited Moves

In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, one aims to find a $k$-partitioning with a smaller cut value for such networks by moving only a few nodes across partitions. The number of nodes that can be moved across partitions is often a constraint forced by budgetary limitations. Motivated by such real-world applications, we introduce and study the $r$-move $k$-partitioning~problem, a natural variant of the Multiway cut problem. Given a graph, a set of $k$ terminals and an initial partitioning of the graph, the $r$-move $k$-partitioning~problem aims to find a $k$-partitioning with the minimum-weighted cut among all the $k$-partitionings that can be obtained by moving at most $r$ non-terminal nodes to partitions different from their initial ones. Our main result is a polynomial time $3(r+1)$ approximation algorithm for this problem. We further show that this problem is $W[1]$-hard, and give an FPTAS for when $r$ is a small constant.Comment: shortened version accepted in AISTATS 2024 as ora

On Diameter Approximation in Directed Graphs

Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u,v) may not be the same as d(v,u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max_(u,v) d(u,v)) and the roundtrip diameter (max_{u,v} d(u,v)+d(v,u)). In this paper we make progress on the outstanding open question for each of them. - We design the first algorithm for diameter in sparse directed graphs to achieve n^{1.5-?} time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. - We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2-?)-approximation would imply a breakthrough algorithm for approximate ?_?-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH

Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set. In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged

Approximation Algorithms and Hardness for nn-Pairs Shortest Paths and All-Nodes Shortest Cycles

We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of $n$-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer $k\ge 1$, under the combinatorial $4k$-clique hypothesis, there is no combinatorial algorithm for unweighted undirected $n$-PSP with approximation ratio better than $1+1/k$ that runs in $O(m^{2-2/(k+1)}n^{1/(k+1)-\epsilon})$ time. This nearly matches an upper bound implied by the result of Agarwal (2014). A highlight of our algorithmic results is that one can solve both $n$-PSP and ANSC in $\tilde O(m+ n^{3/2+\epsilon})$ time with approximation factor $2+\epsilon$ (and additive error that is function of $\epsilon$), for any constant $\epsilon>0$. For $n$-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for $n$-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.Comment: Abstract truncated to meet arXiv requirement. To appear in FOCS 202

Approximation Algorithms for Min-Distance Problems

We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between u and v is the minimum of the shortest path distances from u to v and from v to u. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help. By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in O~(mn) time for directed graphs on n vertices, m edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in O(mn^{1-epsilon}) time for constant epsilon>0. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off