arXiv subjects

Yael Kirkpatrick

Publications and source records attributed to Yael Kirkpatrick.

12 recordsLinked to original sources

arXiv2026

The Cost of Changing Edges for Diameter Computation and More

The sensitivity setting is a restricted setting for dynamic algorithms, particularly practical for scenarios where extensive preprocessing is feasible but responses to real-time modifications must be near-instantaneous before the data structure is eventually rebuilt. For graph problems, a sensitivity data structure is constructed with a preprocessing time P so that the following queries can be answered quickly, preferably in $O(1)$ time: given an edge $e$, return the answer to the problem on either $G \setminus e$ (decremental) or $G \cup e$ (incremental). In this paper, we almost entirely settle the decremental setting for the diameter and eccentricities problems in a variety of approximation regimes by matching P to the static runtime while supporting $O(1)$-time queries, thereby improving upon all previous results for a single failure [Bil\`o, Cohen, Friedrich, Schirneck, MFCS 2021; Bil\`o, Choudhary, Cohen, Friedrich, Krogmann, Schirneck, ICALP 2021]. More precisely: (1) We provide a tight reduction demonstrating that any exact distance sensitivity oracle can be used to efficiently solve decremental exact diameter and all-node eccentricities; (2) For the approximate setting, we match the runtime of all known static diameter algorithms across all sparsity settings, up to an additional $1+o(1)$ factor in approximation. Conversely, for the previously unexplored incremental setting of these problems: (3) We develop new lower bounds, demonstrating that no incremental algorithm can efficiently approximate diameter, radius, or eccentricity beyond a $5/3$ factor in undirected graphs or a $2$ factor in directed graphs; (4) We introduce two new instructive techniques and demonstrate how to utilize them to construct several new algorithms. Most notably, we develop incremental single-node eccentricity approximations for both directed and undirected graphs that match our new lower bounds.

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arXiv2026

Improved Approximation of Min-Distances in Near-Linear Time

We study the problem of approximating the diameter of directed graphs under the min-distance measure, defined as $d_{\min}(u,v) = \min(d(u,v), d(v,u))$. Unlike standard shortest-path distance, min-distance is not a metric, which renders many classical techniques inapplicable. Prior work has therefore focused on approximating this parameter, culminating in an approximation-runtime tradeoff by Dalirrooyfard et al. [ICALP'19] giving a $4k-1$ approximation in $\tilde{O}(mn^{1/(k+1)})$ time for any positive integer $k$ and, more recently, the first near-linear time constant approximation by Chechik and Zhang [FOCS'22], where they obtained a 4-approximation to the min-diameter. In this work we present a randomized near-linear time algorithm that achieves a $3$-approximation to the min-diameter, outperforming all known approximation-runtime tradeoffs. Our approach introduces a novel type-classification framework that may be of independent interest. We further extend our techniques to the more general setting of multimode graphs, recently introduced as a generalization of min-distance by Kirkpatrick and Vassilevska W. [MFCS'25]. For directed $2$-mode graphs, we obtain a $3$-approximation to the diameter in near-linear time, dramatically improving over the previously best known $n$-approximation. Our results significantly narrow the gap between min-distance and multimode distance approximations, and open new directions for understanding graph parameters under non-metric distance measures.

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arXiv2026

New Diameter Approximations via Distance Oracle Techniques

Computing the diameter of a graph is a problem of great interest both in general algorithms research and specifically within fine-grained complexity, where it is a cornerstone hard problem. Recent work has achieved a full conditional lower bound tradeoff curve for both directed and undirected graphs. However, the best known upper bounds do not match the lower bounds. In particular, the best known approximation scheme for undirected graph diameter has not been improved. Moreover, this scheme is randomized and no similar deterministic scheme is known. Another fundamental field of research in shortest paths computation is the construction of approximate distance oracles. Thorup and Zwick [JACM'05] provided the first such distance oracle with constant query time and (conditionally) optimal space, and in the years since many advances have led to a vast toolbox of techniques and data structures. These two areas of research seem natural to combine since they both concern approximating shortest paths. However, the known diameter approximation algorithms only use a small subset of the techniques used in distance oracles research. In this work we show that in fact approximate diameter and distance oracles are intricately connected. We first demonstrate a strong connection between the current best known diameter approximation scheme of Cairo, Grossi and Rizzi ("CGR") and the $(2k-1)$-approximate distance oracle of Thorup and Zwick. This allows us to derandomize the CGR algorithm and obtain the first deterministic diameter approximation tradeoff. We further derandomize other central techniques in the field of distance oracles and use them to achieve new deterministic diameter approximation algorithms. Finally, we show how these new techniques can be used to derandomize many current best known results in various fields of shortest paths approximations.

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arXiv2026

Preprocessed 3SUM for Unknown Universes with Subquadratic Space

We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A' \subseteq A$ and $B' \subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

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arXiv2025

Improved Additive Approximation Algorithms for APSP

The All-Pairs Shortest Paths (APSP) is a foundational problem in theoretical computer science. Approximating APSP in undirected unweighted graphs has been studied for many years, beginning with the work of Dor, Halperin and Zwick [SICOMP'01]. Many recent works have attempted to improve these original algorithms using the algebraic tools of fast matrix multiplication. We improve on these results for the following problems. For $+2$-approximate APSP, the state-of-the-art algorithm runs in $O(n^{2.259})$ time [D\"urr, IPL 2023; Deng, Kirkpatrick, Rong, Vassilevska Williams, and Zhong, ICALP 2022]. We give an improved algorithm in $O(n^{2.2255})$ time. For $+4$ and $+6$-approximate APSP, we achieve time complexities $O(n^{2.1462})$ and $O(n^{2.1026})$ respectively, improving the previous $O(n^{2.155})$ and $O(n^{2.103})$ achieved by [Saha and Ye, SODA 2024]. In contrast to previous works, we do not use the big hammer of bounded-difference $(\min,+)$-product algorithms. Instead, our algorithms are based on a simple technique that decomposes the input graph into a small number of clusters of constant diameter and a remainder of low degree vertices, which could be of independent interest in the study of shortest paths problems. We then use only standard fast matrix multiplication to obtain our improvements.

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arXiv2025

Shortest Paths in Multimode Graphs

In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple `modes' of transportation that cannot be combined. This represents real-world scenarios where different modes are not combinable, such as flights operated by different airlines. More precisely, a $k$-multimode graph is a collection of $k$ graphs on the same vertex set and the $k$-mode distance between two vertices is defined as the minimum among the distances computed in each individual graph. We focus on approximating fundamental graph parameters on these graphs, specifically diameter and radius. In undirected multimode graphs we first show an elegant linear time 3-approximation algorithm for 2-mode diameter. We then extend this idea into a general subroutine that can be used as a part of any $\alpha$-approximation, and use it to construct a 2 and 2.5 approximation algorithm for 2-mode diameter. For undirected radius, we introduce a general scheme that can compute a 3-approximation of the $k$-mode radius for any $k$. In the directed case we develop novel techniques to construct a linear time algorithm to determine whether the diameter is finite. We also develop many conditional fine-grained lower bounds for various multimode diameter and radius approximation problems. We are able to show that many of our algorithms are tight under popular fine-grained complexity hypotheses, including our linear time 3-approximation for $3$-mode undirected diameter and radius. As part of this effort we propose the first extension to the Hitting Set Hypothesis [SODA'16], which we call the $\ell$-Hitting Set Hypothesis. We use this hypothesis to prove the first parameterized lower bound tradeoff for radius approximation algorithms.

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arXiv2025

Beyond 2-approximation for k-Center in Graphs

We consider the classical $k$-Center problem in undirected graphs. The problem is known to have a polynomial-time 2-approximation. There are even $(2+\varepsilon)$-approximations running in near-linear time. The conventional wisdom is that the problem is closed, as $(2-\varepsilon)$-approximation is NP-hard when $k$ is part of the input, and for constant $k\geq 2$ it requires $n^{k-o(1)}$ time under SETH. Our first set of results show that one can beat the multiplicative factor of $2$ in undirected unweighted graphs if one is willing to allow additional small additive error, obtaining $(2-\varepsilon,O(1))$ approximations. We provide several algorithms that achieve such approximations for all integers $k$ with running time $O(n^{k-\delta})$ for $\delta>0$. For instance, for every $k\geq 2$, we obtain an $O(mn + n^{k/2+1})$ time $(2 - \frac{1}{2k-1}, 1 - \frac{1}{2k-1})$-approximation to $k$-Center. For $2$-Center we also obtain an $\tilde{O}(mn^{\omega/3})$ time $(5/3,2/3)$-approximation algorithm. Notably, the running time of this $2$-Center algorithm is faster than the time needed to compute APSP. Our second set of results are strong fine-grained lower bounds for $k$-Center. We show that our $(3/2,O(1))$-approximation algorithm is optimal, under SETH, as any $(3/2-\varepsilon,O(1))$-approximation algorithm requires $n^{k-o(1)}$ time. We also give a time/approximation trade-off: under SETH, for any integer $t\geq 1$, $n^{k/t^2-1-o(1)}$ time is needed for any $(2-1/(2t-1),O(1))$-approximation algorithm for $k$-Center. This explains why our $(2-\varepsilon,O(1))$ approximation algorithms have $k$ appearing in the exponent of the running time. Our reductions also imply that, assuming ETH, the approximation ratio 2 of the known near-linear time algorithms cannot be improved by any algorithm whose running time is a polynomial independent of $k$, even if one allows additive error.

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arXiv2024

A Note on the Conditional Optimality of Chiba and Nishizeki's Algorithms

In a seminal work, Chiba and Nishizeki [SIAM J. Comput. `85] developed subgraph listing algorithms for triangles, 4-cycle and $k$-cliques, where $k \geq 3.$ The runtimes of their algorithms are parameterized by the number of edges $m$ and the arboricity $\alpha$ of a graph. The arboricity $\alpha$ of a graph is the minimum number of spanning forests required to cover it. Their work introduces: * A triangle listing algorithm that runs in $O(m\alpha)$ time. * An output-sensitive 4-Cycle-Listing algorithm that lists all 4-cycles in $O(m\alpha + t)$ time, where $t$ is the number of 4-cycles in the graph. * A k-Clique-Listing algorithm that runs in $O(m\alpha^{k-2})$ time, for $k \geq 4.$ Despite the widespread use of these algorithms in practice, no improvements have been made over them in the past few decades. Therefore, recent work has gone into studying lower bounds for subgraph listing problems. The works of Kopelowitz, Pettie and Porat [SODA `16] and Vassilevska W. and Xu [FOCS `20] showed that the triangle-listing algorithm of Chiba and Nishizeki is optimal under the $\mathsf{3SUM}$ and $\mathsf{APSP}$ hypotheses respectively. However, it remained open whether the remaining algorithms were optimal. In this note, we show that in fact all the above algorithms are optimal under popular hardness conjectures. First, we show that the $\mathsf{4}\text{-}\mathsf{Cycle}\text{-}\mathsf{Listing}$ algorithm is tight under the $\mathsf{3SUM}$ hypothesis following the techniques of Jin and Xu [STOC `23], and Abboud, Bringmann and Fishcher [STOC `23] . Additionally, we show that the $k\text{-}\mathsf{Clique}\text{-}\mathsf{Listing}$ algorithm is essentially tight under the exact $k$-clique hypothesis by following the techniques of Dalirooyfard, Mathialagan, Vassilevska W. and Xu [STOC `24]. These hardness results hold even when the number of 4-cycles or $k$-cliques in the graph is small.

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arXiv2024

Graph Threading with Turn Costs

How should we thread a single string through a set of tubes so that pulling the string taut self-assembles the tubes into a desired graph? While prior work [ITCS 2024] solves this problem with the goal of minimizing the length of string, we study here the objective of minimizing the total turn cost. The frictional force required to pull the string through the tubes grows exponentially with the total absolute turn angles (by the Capstan equation), so this metric often dominates the friction in real-world applications such as deployable structures. We show that minimum-turn threading is NP-hard, even for graphs of maximum degree 4, and even when restricted to some special cases of threading. On the other hand, we show that these special cases can in fact be solved efficiently for graphs of maximum degree 4, thereby fully characterizing their dependence on maximum degree. We further provide polynomial-time exact and approximation algorithms for variants of turn-cost threading: restricting to threading each edge exactly twice, and on rectangular grid graphs.

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arXiv2023

Graph Threading

Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges represent tubes and vertices represent junctions where they meet), we give a polynomial-time algorithm to find a minimum-length closed walk (representing a threading of string) that induces a connected graph of string at every junction. The algorithm is based on a surprising reduction to minimum-weight perfect matching. Along the way, we give tight worst-case bounds on the length of the optimal threading and on the maximum number of times this threading can visit a single edge. We also give more efficient solutions to two special cases: cubic graphs and the case when each edge can be visited at most twice.

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arXiv2023

Fast 2-Approximate All-Pairs Shortest Paths

In this paper, we revisit the classic approximate All-Pairs Shortest Paths (APSP) problem in undirected graphs. For unweighted graphs, we provide an algorithm for $2$-approximate APSP in $\tilde O(n^{2.5-r}+n^{\omega(r)})$ time, for any $r\in[0,1]$. This is $O(n^{2.032})$ time, using known bounds for rectangular matrix multiplication $n^{\omega(r)}$ [Le Gall, Urrutia, SODA 2018]. Our result improves on the $\tilde{O}(n^{2.25})$ bound of [Roditty, STOC 2023], and on the $\tilde{O}(m\sqrt n+n^2)$ bound of [Baswana, Kavitha, SICOMP 2010] for graphs with $m\geq n^{1.532}$ edges. For weighted graphs, we obtain $(2+\epsilon)$-approximate APSP in $\tilde O(n^{3-r}+n^{\omega(r)})$ time, for any $r\in [0,1]$. This is $O(n^{2.214})$ time using known bounds for $\omega(r)$. It improves on the state of the art bound of $O(n^{2.25})$ by [Kavitha, Algorithmica 2012]. Our techniques further lead to improved bounds in a wide range of density for weighted graphs. In particular, for the sparse regime we construct a distance oracle in $\tilde O(mn^{2/3})$ time that supports $2$-approximate queries in constant time. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs. We also obtain new bounds in the near additive regime for unweighted graphs. We give faster algorithms for $(1+\epsilon,k)$-approximate APSP, for $k=2,4,6,8$. We obtain these results by incorporating fast rectangular matrix multiplications into various combinatorial algorithms that carefully balance out distance computation on layers of sparse graphs preserving certain distance information.

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arXiv2022

The $\mathbb{F}_2$-Rank and Size of Graphs

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.

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