arXiv subjects

Yinzhan Xu

Publications and source records attributed to Yinzhan Xu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Shaving Logs via Large Sieve Inequality: Faster Algorithms for Sparse Convolution and More

In sparse convolution-type problems, a common technique is to hash the input integers modulo a random prime $p\in [Q/2,Q]$ for some parameter $Q$, which reduces the range of the input integers while preserving their additive structure. However, this hash family suffers from two drawbacks, which led to bottlenecks in many state-of-the-art algorithms: (1) The collision probability of two elements from $[N]$ is $O(\frac{\log N}{Q})$ rather than $O(\frac{1}{Q})$; (2) It is difficult to derandomize the choice of $p$; known derandomization techniques lead to super-logarithmic overhead [Chan, Lewenstein STOC'15]. In this paper, we partially overcome these drawbacks in certain scenarios, via novel applications of the large sieve inequality from analytic number theory. Consequently, we obtain the following improved algorithms for various problems (in the standard word RAM model): Sparse Nonnegative Convolution: We obtain an $O(t\log t)$-time Las Vegas algorithm that computes the convolution $A\star B$ of two nonnegative integer vectors $A,B$, where $t$ is the output sparsity $\|A\star B\|_0$. Moreover, our algorithm terminates in $O(t\log t)$ time with $1-1/\mathrm{poly}(t)$ probability. Text-to-Pattern Hamming Distances: Given a length-$m$ pattern $P$ and a length-$n$ text $T$, we obtain a deterministic $O(n\sqrt{m\log \log m})$-time algorithm that exactly computes the Hamming distance between $P$ and every length-$m$ substring of $T$. Sparse General Convolution: We also give a Monte Carlo $O(t\log t)$ time algorithm for sparse convolution with possibly negative input in the restricted case where the length $N$ of the input vectors satisfies $N\le t^{1.99}$.

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arXiv2024

Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques

We study finding and listing $k$-cliques in a graph, for constant $k\geq 3$, a fundamental problem of both theoretical and practical importance. Our main contribution is a new output-sensitive algorithm for listing $k$-cliques in graphs, for arbitrary $k\geq 3$, coupled with lower bounds based on standard fine-grained assumptions, showing that our algorithm's running time is tight. Previously, the only known conditionally optimal output-sensitive algorithms were for the case of $3$-cliques by Björklund, Pagh, Vassilevska W. and Zwick [ICALP'14]. Typical inputs to subgraph isomorphism or listing problems are measured by the number of nodes $n$ or the number of edges $m$. Our framework is very general in that it gives $k$-clique listing algorithms whose running times are measured in terms of the number of $\ell$-cliques $Δ_\ell$ in the graph for any $1\leq \ell<k$. This generalizes the typical parameterization in terms of $n$ (the number of $1$-cliques) and $m$ (the number of $2$-cliques). If the matrix multiplication exponent $ω$ is $2$, and if the size of the output, $Δ_k$, is sufficiently large, then for every $\ell<k$, the running time of our algorithm for listing $k$-cliques is $$\tilde{O}\left(Δ_\ell^{\frac{2}{\ell (k - \ell)}}Δ_k^{1-\frac{2}{k(k-\ell)}}\right).$$ For sufficiently large $Δ_k$, we prove that this runtime is in fact {\em optimal} for all $1 \leq \ell < k$ under the Exact $k$-Clique hypothesis. In the special cases of $k = 4$ and $5$, our algorithm in terms of $n$ is conditionally optimal for all values of $Δ_k$ if $ω= 2$. Moreover, our framework is powerful enough to provide an improvement upon the 19-year old runtimes for $4$ and $5$-clique detection in $m$-edge graphs, as a function of $m$ [Eisenbrand and Grandoni, TCS'04].

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arXiv2023

New Bounds for Matrix Multiplication: from Alpha to Omega

The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou, FOCS 2023], the new method introduces several new ingredients that not only yield an improved bound on the matrix multiplication exponent $ω$, but also improve the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia, SODA 2018]. In particular, the new bound on $ω$ is $ω\le 2.371552$ (improved from $ω\le 2.371866$). For the dual matrix multiplication exponent $α$ defined as the largest $α$ for which $ω(1,α,1)=2$, we obtain the improvement $α\ge 0.321334$ (improved from $α\ge 0.31389$). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.

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arXiv2023

Simpler and Higher Lower Bounds for Shortcut Sets

We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of the graph. Our results are based on a vast simplification of the recent construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an $\widetildeΩ(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set problem. We highlight that our simplification completely removes the use of the convex sets by Bárány and Larman [Math. Ann. 1998] used in all previous lower bound constructions. Our simplification also removes the need for randomness and further removes some log factors. This allows us to generalize the construction to higher dimensions, which in turn can be used to show the following results. For $O(m)$-sized shortcut sets, we show an $Ω(n^{1/5})$ lower bound, improving on the previous best $Ω(n^{1/8})$ lower bound. For all $\varepsilon > 0$, we show that there exists a $δ> 0$ such that there are $n$-vertex $O(n)$-edge graphs $G$ where adding any shortcut set of size $O(n^{2-\varepsilon})$ keeps the diameter of $G$ at $Ω(n^δ)$. This improves the sparsity of the constructed graph compared to a known similar result by Hesse [SODA 2003]. We also consider the sourcewise setting for shortcut sets: given a graph $G=(V,E)$, a set $S\subseteq V$, how much can we decrease the sourcewise diameter of $G$, $\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty} \text{dist}(s,v)$ by adding a set of edges $H$ from the transitive closure of $G$? We show that for any integer $d \ge 2$, there exists a graph $G=(V, E)$ on $n$ vertices and $S \subseteq V$ with $|S| = \widetildeΘ(n^{3/(d+3)})$, such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is $\widetildeΩ(|S|^{1/3})$.

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arXiv2023

Simpler Reductions from Exact Triangle

In this paper, we provide simpler reductions from Exact Triangle to two important problems in fine-grained complexity: Exact Triangle with Few Zero-Weight $4$-Cycles and All-Edges Sparse Triangle. Exact Triangle instances with few zero-weight $4$-cycles was considered by Jin and Xu [STOC 2023], who used it as an intermediate problem to show $3$SUM hardness of All-Edges Sparse Triangle with few $4$-cycles (independently obtained by Abboud, Bringmann and Fischer [STOC 2023]), which is further used to show $3$SUM hardness of a variety of problems, including $4$-Cycle Enumeration, Offline Approximate Distance Oracle, Dynamic Approximate Shortest Paths and All-Nodes Shortest Cycles. We provide a simple reduction from Exact Triangle to Exact Triangle with few zero-weight $4$-cycles. Our new reduction not only simplifies Jin and Xu's previous reduction, but also strengthens the conditional lower bounds from being under the $3$SUM hypothesis to the even more believable Exact Triangle hypothesis. As a result, all conditional lower bounds shown by Jin and Xu [STOC 2023] and by Abboud, Bringmann and Fischer [STOC 2023] using All-Edges Sparse Triangle with few $4$-cycles as an intermediate problem now also hold under the Exact Triangle hypothesis. We also provide two alternative proofs of the conditional lower bound of the All-Edges Sparse Triangle problem under the Exact Triangle hypothesis, which was originally proved by Vassilevska Williams and Xu [FOCS 2020]. Both of our new reductions are simpler, and one of them is also deterministic -- all previous reductions from Exact Triangle or 3SUM to All-Edges Sparse Triangle (including Pătraşcu's seminal work [STOC 2010]) were randomized.

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arXiv2023

Fredman's Trick Meets Dominance Product: Fine-Grained Complexity of Unweighted APSP, 3SUM Counting, and More

In this paper we carefully combine Fredman's trick [SICOMP'76] and Matoušek's approach for dominance product [IPL'91] to obtain powerful results in fine-grained complexity: - Under the hypothesis that APSP for undirected graphs with edge weights in $\{1, 2, \ldots, n\}$ requires $n^{3-o(1)}$ time (when $ω=2$), we show a variety of conditional lower bounds, including an $n^{7/3-o(1)}$ lower bound for unweighted directed APSP and an $n^{2.2-o(1)}$ lower bound for computing the Minimum Witness Product between two $n \times n$ Boolean matrices, even if $ω=2$, improving upon their trivial $n^2$ lower bounds. Our techniques can also be used to reduce the unweighted directed APSP problem to other problems. In particular, we show that (when $ω= 2$), if unweighted directed APSP requires $n^{2.5-o(1)}$ time, then Minimum Witness Product requires $n^{7/3-o(1)}$ time. - We show that, surprisingly, many central problems in fine-grained complexity are equivalent to their natural counting versions. In particular, we show that Min-Plus Product and Exact Triangle are subcubically equivalent to their counting versions, and 3SUM is subquadratically equivalent to its counting version. - We obtain new algorithms using new variants of the Balog-Szemerédi-Gowers theorem from additive combinatorics. For example, we get an $O(n^{3.83})$ time deterministic algorithm for exactly counting the number of shortest paths in an arbitrary weighted graph, improving the textbook $\widetilde{O}(n^{4})$ time algorithm. We also get faster algorithms for 3SUM in preprocessed universes, and deterministic algorithms for 3SUM on monotone sets in $\{1, 2, \ldots, n\}^d$.

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arXiv2023

Removing Additive Structure in 3SUM-Based Reductions

Our work explores the hardness of $3$SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving $3$SUM on a size-$n$ integer set that avoids solutions to $a+b=c+d$ for $\{a, b\} \ne \{c, d\}$ still requires $n^{2-o(1)}$ time, under the $3$SUM hypothesis. Such sets are called Sidon sets and are well-studied in the field of additive combinatorics. - Combined with previous reductions, this implies that the All-Edges Sparse Triangle problem on $n$-vertex graphs with maximum degree $\sqrt{n}$ and at most $n^{k/2}$ $k$-cycles for every $k \ge 3$ requires $n^{2-o(1)}$ time, under the $3$SUM hypothesis. This can be used to strengthen the previous conditional lower bounds by Abboud, Bringmann, Khoury, and Zamir [STOC'22] of $4$-Cycle Enumeration, Offline Approximate Distance Oracle and Approximate Dynamic Shortest Path. In particular, we show that no algorithm for the $4$-Cycle Enumeration problem on $n$-vertex $m$-edge graphs with $n^{o(1)}$ delays has $O(n^{2-\varepsilon})$ or $O(m^{4/3-\varepsilon})$ pre-processing time for $\varepsilon >0$. We also present a matching upper bound via simple modifications of the known algorithms for $4$-Cycle Detection. - A slight generalization of the main result also extends the result of Dudek, Gawrychowski, and Starikovskaya [STOC'20] on the $3$SUM hardness of nontrivial 3-Variate Linear Degeneracy Testing (3-LDTs): we show $3$SUM hardness for all nontrivial 4-LDTs. The proof of our main technical result combines a wide range of tools: Balog-Szemer{é}di-Gowers theorem, sparse convolution algorithm, and a new almost-linear hash function with almost $3$-universal guarantee for integers that do not have small-coefficient linear relations.

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arXiv2023

Optimal Bounds for Noisy Sorting

Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy Sorting problem, where each comparison result is flipped independently with probability $p$ for some fixed $p\in (0, \frac 12)$. As our main result, we show that $$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$ noisy comparisons are both necessary and sufficient to sort $n$ elements with error probability $o(1)$ using noisy comparisons, where $I(p)=1 + p\log_2 p+(1-p)\log_2 (1-p)$ is capacity of BSC channel with crossover probability $p$. This simultaneously improves the previous best lower and upper bounds (Wang, Ghaddar and Wang, ISIT 2022) for this problem. For the related Noisy Binary Search problem, we show that $$ (1\pm o(1)) \left((1-δ)\frac{\log_2(n)}{I(p)} + \frac{2 \log_2 \left(\frac 1δ\right)}{(1-2p)\log_2\left(\frac {1-p}p\right)}\right) $$ noisy comparisons are both necessary and sufficient to find the predecessor of an element among $n$ sorted elements with error probability $δ$. This extends the previous bounds of (Burnashev and Zigangirov, 1974), which are only tight for $δ= 1/n^{o(1)}$.

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arXiv2022

Near-Tight Algorithms for the Chamberlin-Courant and Thiele Voting Rules

We present an almost optimal algorithm for the classic Chamberlin-Courant multiwinner voting rule (CC) on single-peaked preference profiles. Given $n$ voters and $m$ candidates, it runs in almost linear time in the input size, improving the previous best $O(nm^2)$ time algorithm of Betzler et al. (2013). We also study multiwinner voting rules on nearly single-peaked preference profiles in terms of the candidate-deletion operation. We show a polynomial-time algorithm for CC where a given candidate-deletion set $D$ has logarithmic size. Actually, our algorithm runs in $2^{|D|} \cdot poly(n,m)$ time and the base of the power cannot be improved under the Strong Exponential Time Hypothesis. We also adapt these results to all non-constant Thiele rules which generalize CC with approval ballots.

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arXiv2022

Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failures

This paper considers a natural fault-tolerant shortest paths problem: for some constant integer $f$, given a directed weighted graph with no negative cycles and two fixed vertices $s$ and $t$, compute (either explicitly or implicitly) for every tuple of $f$ edges, the distance from $s$ to $t$ if these edges fail. We call this problem $f$-Fault Replacement Paths ($f$FRP). We first present an $\tilde{O}(n^3)$ time algorithm for $2$FRP in $n$-vertex directed graphs with arbitrary edge weights and no negative cycles. As $2$FRP is a generalization of the well-studied Replacement Paths problem (RP) that asks for the distances between $s$ and $t$ for any single edge failure, $2$FRP is at least as hard as RP. Since RP in graphs with arbitrary weights is equivalent in a fine-grained sense to All-Pairs Shortest Paths (APSP) [Vassilevska Williams and Williams FOCS'10, J.~ACM'18], $2$FRP is at least as hard as APSP, and thus a substantially subcubic time algorithm in the number of vertices for $2$FRP would be a breakthrough. Therefore, our algorithm in $\tilde{O}(n^3)$ time is conditionally nearly optimal. Our algorithm implies an $\tilde{O}(n^{f+1})$ time algorithm for the $f$FRP problem, giving the first improvement over the straightforward $O(n^{f+2})$ time algorithm. Then we focus on the restriction of $2$FRP to graphs with small integer weights bounded by $M$ in absolute values. Using fast rectangular matrix multiplication, we obtain a randomized algorithm that runs in $\tilde{O}(M^{2/3}n^{2.9153})$ time. This implies an improvement over our $\tilde{O}(n^{f+1})$ time arbitrary weight algorithm for all $f>1$. We also present a data structure variant of the algorithm that can trade off pre-processing and query time. In addition to the algebraic algorithms, we also give an $n^{8/3-o(1)}$ conditional lower bound for combinatorial $2$FRP algorithms in directed unweighted graphs.

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arXiv2022

Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and Fine-Grained Lower Bounds

The AP-LCA problem asks, given an $n$-node directed acyclic graph (DAG), to compute for every pair of vertices $u$ and $v$ in the DAG a lowest common ancestor (LCA) of $u$ and $v$ if one exists. In this paper we study several interesting variants of AP-LCA, providing both algorithms and fine-grained lower bounds for them. The lower bounds we obtain are the first conditional lower bounds for LCA problems higher than $n^{ω-o(1)}$, where $ω$ is the matrix multiplication exponent. Some of our results include: - In any DAG, we can detect all vertex pairs that have at most two LCAs and list all of their LCAs in $O(n^ω)$ time. This algorithm extends a result of [Kowaluk and Lingas ESA'07] which showed an $\tilde{O}(n^ω)$ time algorithm that detects all pairs with a unique LCA in a DAG and outputs their corresponding LCAs. - Listing $7$ LCAs per vertex pair in DAGs requires $n^{3-o(1)}$ time under the popular assumption that 3-uniform 5-hyperclique detection requires $n^{5-o(1)}$ time. This is surprising since essentially cubic time is sufficient to list all LCAs (if $ω=2$). - Counting the number of LCAs for every vertex pair in a DAG requires $n^{3-o(1)}$ time under the Strong Exponential Time Hypothesis, and $n^{ω(1,2,1)-o(1)}$ time under the $4$-Clique hypothesis. This shows that the algorithm of [Echkardt, Mühling and Nowak ESA'07] for listing all LCAs for every pair of vertices is likely optimal. - Given a DAG and a vertex $w_{u,v}$ for every vertex pair $u,v$, verifying whether all $w_{u,v}$ are valid LCAs requires $n^{2.5-o(1)}$ time assuming 3-uniform 4-hyperclique requires $n^{4 - o(1)}$ time. This defies the common intuition that verification is easier than computation since returning some LCA per vertex pair can be solved in $O(n^{2.447})$ time [Grandoni et al. SODA'21].

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arXiv2022

Hardness for Triangle Problems under Even More Believable Hypotheses: Reductions from Real APSP, Real 3SUM, and OV

The $3$SUM hypothesis, the APSP hypothesis and SETH are the three main hypotheses in fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real $3$SUM" hypotheses, which assert that the APSP and $3$SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts. Under the very believable hypothesis that at least one of the Integer $3$SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires $n^{3-o(1)}$ time on an $n$-node graph. Our main result is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real $3$SUM, the Real APSP, and the OV hypotheses is true. Combined with slight modifications of prior reductions, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic Max Flow, now under the new weaker hypothesis. Our main result is built on the following two lines of reductions. * Real APSP and Real $3$SUM hardness for the All-Edges Sparse Triangle problem. Prior reductions only worked from the integer variants of these problems. * Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem. Along the way we show that Triangle Collection is equivalent to a simpler restricted version of the problem, simplifying prior work. Our techniques also have other interesting implications, such as a super-linear lower bound of Integer All-Numbers $3$SUM based on the Real $3$SUM hypothesis, and a tight lower bound for a string matching problem based on the OV hypothesis.

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arXiv2022

All-Pairs Shortest Path Distances with Differential Privacy: Improved Algorithms for Bounded and Unbounded Weights

We revisit the problem of privately releasing the all-pairs shortest path distances of a weighted undirected graph up to low additive error, which was first studied by Sealfon [Sea16]. In this paper, we improve significantly on Sealfon's results, both for arbitrary weighted graphs and for bounded-weight graphs on $n$ nodes. Specifically, we provide an approximate-DP algorithm that outputs all-pairs shortest path distances up to maximum additive error $\tilde{O}(\sqrt{n})$, and a pure-DP algorithm that outputs all pairs shortest path distances up to maximum additive error $\tilde{O}(n^{2/3})$ (where we ignore dependencies on $\varepsilon, δ$). This improves over the previous best result of $\tilde{O}(n)$ additive error for both approximate-DP and pure-DP [Sea16], and partially resolves an open question posed by Sealfon [Sea16, Sea20]. We also show that if the graph is promised to have reasonably bounded weights, one can improve the error further to roughly $n^{\sqrt{2}-1+o(1)}$ in the approximate-DP setting and roughly $n^{(\sqrt{17}-3)/2 + o(1)}$ in the pure-DP setting. Previously, it was only known how to obtain $\tilde{O}(n^{1/2})$ additive error in the approximate-DP setting and $\tilde{O}(n^{2/3})$ additive error in the pure-DP setting for bounded-weight graphs [Sea16].

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arXiv2022

Tight Dynamic Problem Lower Bounds from Generalized BMM and OMv

The main theme of this paper is using $k$-dimensional generalizations of the combinatorial Boolean Matrix Multiplication (BMM) hypothesis and the closely-related Online Matrix Vector Multiplication (OMv) hypothesis to prove new tight conditional lower bounds for dynamic problems. The combinatorial $k$-Clique hypothesis, which is a standard hypothesis in the literature, naturally generalizes the combinatorial BMM hypothesis. In this paper, we prove tight lower bounds for several dynamic problems under the combinatorial $k$-Clique hypothesis. For instance, we show that: * The Dynamic Range Mode problem has no combinatorial algorithms with $\mathrm{poly}(n)$ pre-processing time, $O(n^{2/3-ε})$ update time and $O(n^{2/3-ε})$ query time for any $ε> 0$, matching the known upper bounds for this problem. Previous lower bounds only ruled out algorithms with $O(n^{1/2-ε})$ update and query time under the OMv hypothesis. Other examples include tight combinatorial lower bounds for Dynamic Subgraph Connectivity, Dynamic 2D Orthogonal Range Color Counting, Dynamic 2-Pattern Document Retrieval, and Dynamic Range Mode in higher dimensions. Furthermore, we propose the OuMv$_k$ hypothesis as a natural generalization of the OMv hypothesis. Under this hypothesis, we prove tight lower bounds for various dynamic problems. For instance, we show that: * The Dynamic Skyline Points Counting problem in $(2k-1)$-dimensional space has no algorithm with $\mathrm{poly}(n)$ pre-processing time and $O(n^{1-1/k-ε})$ update and query time for $ε> 0$, even if the updates are semi-online. Other examples include tight conditional lower bounds for (semi-online) Dynamic Klee's measure for unit cubes, and high-dimensional generalizations of Erickson's problem and Langerman's problem.

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arXiv2021

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph (WMG) of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates. When the input is the WMG, to our knowledge, the fastest algorithm for computing all winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths problem and runs in $O(m^{2.69})$ time, where $m$ is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all winners. This running time is essentially optimal. If the input to the Schulze winners problem is not the WMG but the preference profile, then constructing the WMG is a bottleneck that increases the running time significantly; in the special case when there are $m$ candidates and $n=O(m)$ voters, the running time is $O(m^{2.69})$, or $O(m^{2.5})$ if there is a nearly-linear time algorithm for multiplying dense square matrices. To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the WMG. We prove a similar connection between the so called Dominating Pairs problem and the problem of finding a winner in the Schulze method. Our paper is the first to bring fine-grained complexity into the field of computational social choice. Using it we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic.

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arXiv2021

Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in $n^{3-o(1)}$ time, and it is widely open whether it has an $O(n^{3-ε})$ time algorithm for $ε> 0$. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product. A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an $n$-vertex graph with potentially negative edge weights in $\{-M, \ldots, M\}$. SSRP with positive integer edge weights bounded by $M$ can be solved in $\tilde{O}(Mn^ω)$ time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in $O(M^{0.7519} n^{2.5286})$ time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved $O(M^{0.8043} n^{2.4957})$ time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require $n^{2.5-o(1)}$ time. Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an $\tilde{O}(M n^ω)$ time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.

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arXiv2021

Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths

APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an $\tilde{O}(n^ω)$ time algorithm, where $ω<2.373$ is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be $Ω(n^{2.5})$ even if $ω=2$ [Zwick'02]. To understand this $n^{2.5}$ bottleneck, we build a web of reductions around directed unweighted APSP. We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on the value of $ω$. As a consequence, we establish an equivalence between APSP in directed unweighted graphs, APSP in directed graphs with small $(\tilde{O}(1))$ integer weights, All-Pairs Longest Paths in DAGs with small weights, approximate APSP with additive error $c$ in directed graphs with small weights, for $c\le \tilde{O}(1)$ and several other graph problems. We also provide fine-grained reductions from directed unweighted APSP to All-Pairs Shortest Lightest Paths (APSLP) in undirected graphs with $\{0,1\}$ weights and $\#_{\text{mod}\ c}$APSP in directed unweighted graphs (computing counts mod $c$). We complement our hardness results with new algorithms. We improve the known algorithms for APSLP in directed graphs with small integer weights and for approximate APSP with sublinear additive error in directed unweighted graphs. Our algorithm for approximate APSP with sublinear additive error is optimal, when viewed as a reduction to Min-Plus product. We also give new algorithms for variants of #APSP in unweighted graphs, as well as a near-optimal $\tilde{O}(n^3)$-time algorithm for the original #APSP problem in unweighted graphs. Our techniques also lead to a simpler alternative for the original APSP problem in undirected graphs with small integer weights.

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arXiv2020

Monochromatic Triangles, Triangle Listing and APSP

One of the main hypotheses in fine-grained complexity is that All-Pairs Shortest Paths (APSP) for $n$-node graphs requires $n^{3-o(1)}$ time. Another famous hypothesis is that the $3$SUM problem for $n$ integers requires $n^{2-o(1)}$ time. Although there are no direct reductions between $3$SUM and APSP, it is known that they are related: there is a problem, $(\min,+)$-convolution that reduces in a fine-grained way to both, and a problem Exact Triangle that both fine-grained reduce to. In this paper we find more relationships between these two problems and other basic problems. Pătraşcu had shown that under the $3$SUM hypothesis the All-Edges Sparse Triangle problem in $m$-edge graphs requires $m^{4/3-o(1)}$ time. The latter problem asks to determine for every edge $e$, whether $e$ is in a triangle. It is equivalent to the problem of listing $m$ triangles in an $m$-edge graph where $m=\tilde{O}(n^{1.5})$, and can be solved in $O(m^{1.41})$ time [Alon et al.'97] with the current matrix multiplication bounds, and in $\tilde{O}(m^{4/3})$ time if $ω=2$. We show that one can reduce Exact Triangle to All-Edges Sparse Triangle, showing that All-Edges Sparse Triangle (and hence Triangle Listing) requires $m^{4/3-o(1)}$ time also assuming the APSP hypothesis. This allows us to provide APSP-hardness for many dynamic problems that were previously known to be hard under the $3$SUM hypothesis. We also consider the previously studied All-Edges Monochromatic Triangle problem. Via work of [Lincoln et al.'20], our result on All-Edges Sparse Triangle implies that if the All-Edges Monochromatic Triangle problem has an $O(n^{2.5-ε})$ time algorithm for $ε>0$, then both the APSP and $3$SUM hypotheses are false. We also connect the problem to other ``intermediate'' problems, whose runtimes are between $O(n^ω)$ and $O(n^3)$, such as the Max-Min product problem.

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