arXiv subjects

Nick Fischer

Publications and source records attributed to Nick Fischer.

At least 37 records · Page 2Linked to original sources

arXiv2024

Sumsets, 3SUM, Subset Sum: Now for Real!

We study a broad class of algorithmic problems with an "additive flavor" such as computing sumsets, 3SUM, Subset Sum and geometric pattern matching. Our starting point is that these problems can often be solved efficiently for integers, owed to the rich available tool set including bit-tricks, linear hashing, and the Fast Fourier Transform. However, for real numbers these tools are not available, leading to significant gaps in the best-known running times for integer inputs versus for real inputs. In this work our goal is to close this gap. As our key contribution we design a new technique for computing real sumsets. It is based on a surprising blend of algebraic ideas (like Prony's method and coprime factorizations) with combinatorial tricks. We then apply our new algorithm to the aforementioned problems and successfully obtain, in all cases, equally fast algorithms for real inputs. Specifically, we replicate the running times of the following landmark results by randomized algorithms in the standard real RAM model: - Sumsets: Given two sets $A,B$, their sumset $A+B=\{a+b:a\in A,b\in B\}$ can be computed in time $\tilde O(|A+B|)$ [Cole, Hariharan; STOC'02]. - Geometric pattern matching: Given two sets $A,B$, we can test whether there is some shift such that $A+s\subseteq B$ in time $\tilde O(|A|+|B|)$ [Cardoze, Schulman; FOCS'98]. - 3SUM with preprocessing: We can preprocess three size-$n$ sets $A,B,C$ in time $\tilde O(n^2)$ such that upon query of sets $A'\subseteq A,B'\subseteq B,C'\subseteq C$, the 3SUM instance $(A',B',C')$ can be decided in time $\tilde O(n^{13/7})$ [Chan, Lewenstein; STOC'15]. - Output-sensitive Subset Sum: Given a size-$n$ (multi-)set $X$ and a target $t$, we can compute the set of subset sums $\{Σ(X'):X'\subseteq X,Σ(X')\leq t\}$ in output-sensitive time $\tilde O(n+\mathrm{out}^{4/3})$ [Bringmann, Nakos; STOC'20].

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arXiv2024

New Applications of 3SUM-Counting in Fine-Grained Complexity and Pattern Matching

The 3SUM problem is one of the cornerstones of fine-grained complexity. Its study has led to countless lower bounds, but as has been sporadically observed before -- and as we will demonstrate again -- insights on 3SUM can also lead to algorithmic applications. The starting point of our work is that we spend a lot of technical effort to develop new algorithms for 3SUM-type problems such as approximate 3SUM-counting, small-doubling 3SUM-counting, and a deterministic subquadratic-time algorithm for the celebrated Balog-Szemerédi-Gowers theorem from additive combinatorics. As consequences of these tools, we derive diverse new results in fine-grained complexity and pattern matching algorithms, answering open questions from many unrelated research areas. Specifically: - A recent line of research on the "short cycle removal" technique culminated in tight 3SUM-based lower bounds for various graph problems via randomized fine-grained reductions [Abboud, Bringmann, Fischer; STOC '23] [Jin, Xu; STOC '23]. In this paper we derandomize the reduction to the important 4-Cycle Listing problem. - We establish that \#3SUM and 3SUM are fine-grained equivalent under deterministic reductions. - We give a deterministic algorithm for the $(1+ε)$-approximate Text-to-Pattern Hamming Distances problem in time $n^{1+o(1)} \cdot ε^{-1}$. - In the $k$-Mismatch Constellation problem the input consists of two integer sets $A, B \subseteq [N]$, and the goal is to test whether there is a shift $c$ such that $|(c + B) \setminus A| \leq k$ (i.e., whether $B$ shifted by $c$ matches $A$ up to $k$ mismatches). For moderately small $k$ the previously best running time was $\tilde O(|A| \cdot k)$ [Cardoze, Schulman; FOCS '98] [Fischer; SODA '24]. We give a faster $|A| \cdot k^{2/3} \cdot N^{o(1)}$-time algorithm in the regime where $|B| = Θ(|A|)$.

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arXiv2024

A Simple Parallel Algorithm with Near-Linear Work for Negative-Weight Single-Source Shortest Paths

We give the first parallel algorithm with optimal $\tilde{O}(m)$ work for the classical problem of computing Single-Source Shortest Paths in general graphs with negative-weight edges. In graphs without negative edges, Dijkstra's algorithm solves the Single-Source Shortest Paths (SSSP) problem with optimal $\tilde O(m)$ work, but is inherently sequential. A recent breakthrough by Bernstein, Nanongkai, Wulff-Nilsen; FOCS '22 achieves the same for general graphs. Parallel shortest path algorithms are more difficult and have been intensely studied for decades. Only very recently, multiple lines of research culminated in parallel algorithms with optimal work $\tilde O(m)$ for various restricted settings, such as approximate or exact algorithms for directed or undirected graphs without negative edges. For general graphs, the best known algorithm by [shvinkumar, Bernstein, Cao, Grunau, Haeupler, Jiang, Nanongkai, Su; ESA '24 still requires $m^{1+o(1)}$ work. This paper presents a randomized parallel algorithm for SSSP in general graphs with near-linear work $\tilde O(m)$ and state-of-the-art span $n^{1/2 + o(1)}$. We follow a novel bottom-up approach leading to a particularly clean and simple algorithm. Our algorithm can be seen as a \emph{near-optimal parallel black-box reduction} from SSSP in general graphs to graphs without negative edges. In contrast to prior works, the reduction in this paper is both parallel and essentially without overhead, only affecting work and span by polylogarithmic factors.

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arXiv2024

Recognizing Sumsets is NP-Complete

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm that takes exponential time in the size of the given set, but can we do polynomial or even linear time? This basic computational question is indirectly asking a fundamental structural question: do the special characteristics of sumsets allow them to be efficiently recognizable? In this paper, we answer this question negatively by proving that the problem is NP-complete. Specifically, our results hold for integer sets and over any finite field. Assuming the Exponential Time Hypothesis, our lower bound becomes $2^{Ω(n^{1/4})}$.

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arXiv2024

Faster Combinatorial k-Clique Algorithms

Detecting if a graph contains a $k$-Clique is one of the most fundamental problems in computer science. The asymptotically fastest algorithm runs in time $O(n^{ωk/3})$, where $ω$ is the exponent of Boolean matrix multiplication. To date, this is the only technique capable of beating the trivial $O(n^k)$ bound by a polynomial factor. Due to this technique's various limitations, much effort has gone into designing "combinatorial" algorithms that improve over exhaustive search via other techniques. The first contribution of this work is a faster combinatorial algorithm for $k$-Clique, improving Vassilevska's bound of $O(n^{k}/\log^{k-1}{n})$ by two log factors. Technically, our main result is a new reduction from $k$-Clique to Triangle detection that exploits the same divide-and-conquer at the core of recent combinatorial algorithms by Chan (SODA'15) and Yu (ICALP'15). Our second contribution is exploiting combinatorial techniques to improve the state-of-the-art (even of non-combinatorial algorithms) for generalizations of the $k$-Clique problem. In particular, we give the first $o(n^k)$ algorithm for $k$-clique in hypergraphs and an $O(n^3/\log^{2.25}{n} + t)$ algorithm for listing $t$ triangles in a graph.

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arXiv2024

New Graph Decompositions and Combinatorial Boolean Matrix Multiplication Algorithms

We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic $O(n^ω)$ time, where $ω<3$; much work has gone into bringing $ω$ closer to $2$. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The naive $O(n^3)$-time algorithm was initially improved by a $\log^2{n}$ factor [Arlazarov et al.; RAS'70], then by $\log^{2.25}{n}$ [Bansal and Williams; FOCS'09], then by $\log^3{n}$ [Chan; SODA'15], and finally by $\log^4{n}$ [Yu; ICALP'15]. We design a combinatorial algorithm for BMM running in time $n^3 / 2^{Ω(\sqrt[7]{\log n})}$ -- a speed-up over cubic time that is stronger than any poly-log factor. This comes tantalizingly close to refuting the conjecture from the 90s that truly subcubic combinatorial algorithms for BMM are impossible. This popular conjecture is the basis for dozens of fine-grained hardness results. Our main technical contribution is a new regularity decomposition theorem for Boolean matrices (or equivalently, bipartite graphs) under a notion of regularity that was recently introduced and analyzed analytically in the context of communication complexity [Kelley, Lovett, Meka; arXiv'23], and is related to a similar notion from the recent work on $3$-term arithmetic progression free sets [Kelley, Meka; FOCS'23].

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arXiv2023

The Effect of Sparsity on $k$-Dominating Set and Related First-Order Graph Properties

We revisit $k$-Dominating Set, one of the first problems for which a tight $n^k-o(1)$ conditional lower bound (for $k\ge 3$), based on SETH, was shown (Pătraşcu and Williams, SODA 2007). However, the underlying reduction creates dense graphs, raising the question: how much does the sparsity of the graph affect its fine-grained complexity? We first settle the fine-grained complexity of $k$-Dominating Set in terms of both the number of nodes $n$ and number of edges $m$. Specifically, we show an $mn^{k-2-o(1)}$ lower bound based on SETH, for any dependence of $m$ on $n$. This is complemented by an $mn^{k-2+o(1)}$-time algorithm for all $k\ge 3$. For the $k=2$ case, we give a randomized algorithm that employs a Bloom-filter inspired hashing to improve the state of the art of $n^{ω+o(1)}$ to $m^{ω/2+o(1)}$. If $ω=2$, this yields a conditionally tight bound for all $k\ge 2$. To study if $k$-Dominating Set is special in its sensitivity to sparsity, we consider a class of very related problems. The $k$-Dominating Set problem belongs to a type of first-order definable graph properties that we call monochromatic basic problems. These problems are the natural monochromatic variants of the basic problems that were proven complete for the class FOP of first-order definable properties (Gao, Impagliazzo, Kolokolova, and Williams, TALG 2019). We show that among these problems, $k$-Dominating Set is the only one whose fine-grained complexity decreases in sparse graphs. Only for the special case of reflexive properties, is there an additional basic problem that can be solved faster than $n^{k\pm o(1)}$ on sparse graphs. For the natural variant of distance-$r$ $k$-dominating set, we obtain a hardness of $n^{k-o(1)}$ under SETH for every $r\ge 2$ already on sparse graphs, which is tight for sufficiently large $k$.

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arXiv2023

Faster Sublinear-Time Edit Distance

We study the fundamental problem of approximating the edit distance of two strings. After an extensive line of research led to the development of a constant-factor approximation algorithm in almost-linear time, recent years have witnessed a notable shift in focus towards sublinear-time algorithms. Here, the task is typically formalized as the $(k, K)$-gap edit distance problem: Distinguish whether the edit distance of two strings is at most $k$ or more than $K$. Surprisingly, it is still possible to compute meaningful approximations in this challenging regime. Nevertheless, in almost all previous work, truly sublinear running time of $O(n^{1-\varepsilon})$ (for a constant $\varepsilon > 0$) comes at the price of at least polynomial gap $K \ge k \cdot n^{Ω(\varepsilon)}$. Only recently, [Bringmann, Cassis, Fischer, and Nakos; STOC'22] broke through this barrier and solved the sub-polynomial $(k, k^{1+o(1)})$-gap edit distance problem in time $O(n/k + k^{4+o(1)})$, which is truly sublinear if $n^{Ω(1)} \le k \le n^{\frac14-Ω(1)}$.The $n/k$ term is inevitable (already for Hamming distance), but it remains an important task to optimize the $\mathrm{poly}(k)$ term and, in general, solve the $(k, k^{1+o(1)})$-gap edit distance problem in sublinear-time for larger values of $k$. In this work, we design an improved algorithm for the $(k, k^{1+o(1)})$-gap edit distance problem in sublinear time $O(n/k + k^{2+o(1)})$, yielding a significant quadratic speed-up over the previous $O(n/k + k^{4+o(1)})$-time algorithm. Notably, our algorithm is unconditionally almost-optimal (up to subpolynomial factors) in the regime where $k \leq n^{\frac13}$ and improves upon the state of the art for $k \leq n^{\frac12-o(1)}$.

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arXiv2023

Deterministic 3SUM-Hardness

As one of the three main pillars of fine-grained complexity theory, the 3SUM problem explains the hardness of many diverse polynomial-time problems via fine-grained reductions. Many of these reductions are either directly based on or heavily inspired by Pătraşcu's framework involving additive hashing and are thus randomized. Some selected reductions were derandomized in previous work [Chan, He; SOSA'20], but the current techniques are limited and a major fraction of the reductions remains randomized. In this work we gather a toolkit aimed to derandomize reductions based on additive hashing. Using this toolkit, we manage to derandomize almost all known 3SUM-hardness reductions. As technical highlights we derandomize the hardness reductions to (offline) Set Disjointness, (offline) Set Intersection and Triangle Listing -- these questions were explicitly left open in previous work [Kopelowitz, Pettie, Porat; SODA'16]. The few exceptions to our work fall into a special category of recent reductions based on structure-versus-randomness dichotomies. We expect that our toolkit can be readily applied to derandomize future reductions as well. As a conceptual innovation, our work thereby promotes the theory of deterministic 3SUM-hardness. As our second contribution, we prove that there is a deterministic universe reduction for 3SUM. Specifically, using additive hashing it is a standard trick to assume that the numbers in 3SUM have size at most $n^3$. We prove that this assumption is similarly valid for deterministic algorithms.

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arXiv2023

Dynamic Dynamic Time Warping

The Dynamic Time Warping (DTW) distance is a popular similarity measure for polygonal curves (i.e., sequences of points). It finds many theoretical and practical applications, especially for temporal data, and is known to be a robust, outlier-insensitive alternative to the \frechet distance. For static curves of at most $n$ points, the DTW distance can be computed in $O(n^2)$ time in constant dimension. This tightly matches a SETH-based lower bound, even for curves in $\mathbb{R}^1$. In this work, we study \emph{dynamic} algorithms for the DTW distance. Here, the goal is to design a data structure that can be efficiently updated to accommodate local changes to one or both curves, such as inserting or deleting vertices and, after each operation, reports the updated DTW distance. We give such a data structure with update and query time $O(n^{1.5} \log n)$, where $n$ is the maximum length of the curves. As our main result, we prove that our data structure is conditionally \emph{optimal}, up to subpolynomial factors. More precisely, we prove that, already for curves in $\mathbb{R}^1$, there is no dynamic algorithm to maintain the DTW distance with update and query time~\makebox{$O(n^{1.5 - δ})$} for any constant $δ> 0$, unless the Negative-$k$-Clique Hypothesis fails. In fact, we give matching upper and lower bounds for various trade-offs between update and query time, even in cases where the lengths of the curves differ.

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arXiv2023

Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

The "short cycle removal" technique was recently introduced by Abboud, Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of approximation. Its main technical result is that listing all triangles in an $n^{1/2}$-regular graph is $n^{2-o(1)}$-hard under the 3-SUM conjecture even when the number of short cycles is small; namely, when the number of $k$-cycles is $O(n^{k/2+γ})$ for $γ<1/2$. Abboud et al. achieve $γ\geq 1/4$ by applying structure vs. randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve the best possible $γ=0$ and the following lower bounds under the 3-SUM conjecture: * Approximate distance oracles: The seminal Thorup-Zwick distance oracles achieve stretch $2k\pm O(1)$ after preprocessing a graph in $O(m n^{1/k})$ time. For the same stretch, and assuming the query time is $n^{o(1)}$ Abboud et al. proved an $Ω(m^{1+\frac{1}{12.7552 \cdot k}})$ lower bound on the preprocessing time; we improve it to $Ω(m^{1+\frac1{2k}})$ which is only a factor 2 away from the upper bound. We also obtain tight bounds for stretch $2+o(1)$ and $3-ε$ and higher lower bounds for dynamic shortest paths. * Listing 4-cycles: Abboud et al. proved the first super-linear lower bound for listing all 4-cycles in a graph, ruling out $(m^{1.1927}+t)^{1+o(1)}$ time algorithms where $t$ is the number of 4-cycles. We settle the complexity of this basic problem by showing that the $\widetilde{O}(\min(m^{4/3},n^2) +t)$ upper bound is tight up to $n^{o(1)}$ factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog-Szemerédi-Gowers theorem and Rusza's covering lemma. A key ingredient that may be of independent interest is a subquadratic algorithm for 3-SUM if one of the sets has small doubling.

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arXiv2023

Deterministic Sparse Pattern Matching via the Baur-Strassen Theorem

How fast can you test whether a constellation of stars appears in the night sky? This question can be modeled as the computational problem of testing whether a set of points $P$ can be moved into (or close to) another set $Q$ under some prescribed group of transformations. Consider, as a simple representative, the following problem: Given two sets of at most $n$ integers $P,Q\subseteq[N]$, determine whether there is some shift $s$ such that $P$ shifted by $s$ is a subset of $Q$, i.e., $P+s=\{p+s:p\in P\}\subseteq Q$. This problem, to which we refer as the Constellation problem, can be solved in near-linear time $O(n\log n)$ by a Monte Carlo randomized algorithm [Cardoze, Schulman; FOCS'98] and time $O(n\log^2 N)$ by a Las Vegas randomized algorithm [Cole, Hariharan; STOC'02]. Moreover, there is a deterministic algorithm running in time $n\cdot2^{O(\sqrt{\log n\log\log N})}$ [Chan, Lewenstein; STOC'15]. An interesting question left open by these previous works is whether Constellation is in deterministic near-linear time (i.e., with only polylogarithmic overhead). We answer this question positively by giving an $n\cdot(\log N)^{O(1)}$-time deterministic algorithm for the Constellation problem. Our algorithm extends to various more complex Point Pattern Matching problems in higher dimensions, under translations and rigid motions, and possibly with mismatches, and also to a near-linear-time derandomization of the Sparse Wildcard Matching problem on strings. We find it particularly interesting how we obtain our deterministic algorithm. All previous algorithms are based on the same baseline idea, using additive hashing and the Fast Fourier Transform. In contrast, our algorithms are based on new ideas, involving a surprising blend of combinatorial and algebraic techniques. At the heart lies an innovative application of the Baur-Strassen theorem from algebraic complexity theory.

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arXiv2023

The Time Complexity of Fully Sparse Matrix Multiplication

What is the time complexity of matrix multiplication of sparse integer matrices with $m_{in}$ nonzeros in the input and $m_{out}$ nonzeros in the output? This paper provides improved upper bounds for this question for almost any choice of $m_{in}$ vs. $m_{out}$, and provides evidence that these new bounds might be optimal up to further progress on fast matrix multiplication. Our main contribution is a new algorithm that reduces sparse matrix multiplication to dense (but smaller) rectangular matrix multiplication. Our running time thus depends on the optimal exponent $ω(a,b,c)$ of multiplying dense $n^a\times n^b$ by $n^b\times n^c$ matrices. We discover that when $m_{out}=Θ(m_{in}^r)$ the time complexity of sparse matrix multiplication is $O(m_{in}^{σ+ε})$, for all $ε> 0$, where $σ$ is the solution to the equation $ω(σ-1,2-σ,1+r-σ)=σ$. No matter what $ω(\cdot,\cdot,\cdot)$ turns out to be, and for all $r\in(0,2)$, the new bound beats the state of the art, and we provide evidence that it is optimal based on the complexity of the all-edge triangle problem. In particular, in terms of the input plus output size $m = m_{in} + m_{out}$ our algorithm runs in time $O(m^{1.3459})$. Even for Boolean matrices, this improves over the previous $m^{\frac{2ω}{ω+1}+ε}=O(m^{1.4071})$ bound [Amossen, Pagh; 2009], which was a natural barrier since it coincides with the longstanding bound of all-edge triangle in sparse graphs [Alon, Yuster, Zwick; 1994]. We find it interesting that matrix multiplication can be solved faster than triangle detection in this natural setting. In fact, we establish an equivalence to a special case of the all-edge triangle problem.

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arXiv2023

Can You Solve Closest String Faster than Exhaustive Search?

We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X \subseteq Σ^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $\bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $Σ^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-ε) d} poly(nd))$, for any $ε> 0$, unless the Strong Exponential Time Hypothesis fails. $\bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 \pm o(1)}$ whenever the dimension is $ω(\log n) < d < n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$.

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arXiv2023

Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!

In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's algorithm), while others are significantly more involved (e.g. to find negative cycles we design an algorithm reminiscent of noisy binary search and analyze it with drift analysis). As side results, we obtain an algorithm to compute the minimum cycle mean in the same running time as well as a new construction for computing Low-Diameter Decompositions in directed graphs.

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arXiv2023

Almost-Optimal Sublinear-Time Edit Distance in the Low Distance Regime

We revisit the task of computing the edit distance in sublinear time. In the $(k,K)$-gap edit distance problem the task is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. It has been established by Goldenberg, Krauthgamer and Saha (FOCS '19), with improvements by Kociumaka and Saha (FOCS '20), that the $(k,k^2)$-gap problem can be solved in time $\widetilde O(n/k+\operatorname{poly}(k))$. One of the most natural questions in this line of research is whether the $(k,k^2)$-gap is best-possible for the running time $\widetilde O(n/k+\operatorname{poly}(k))$. In this work we answer this question by significantly improving the gap. Specifically, we show that in time $O(n/k+\operatorname{poly}(k))$ we can even solve the $(k,k^{1+o(1)})$-gap problem. This is the first algorithm that breaks the $(k,k^2)$-gap in this running time. Our algorithm is almost optimal in the following sense: In the low distance regime ($k\le n^{0.19}$) our running time becomes $O(n/k)$, which matches a known $n/k^{1+o(1)}$ lower bound for the $(k,k^{1+o(1)})$-gap problem up to lower order factors. Our result also reveals a surprising similarity of Hamming distance and edit distance in the low distance regime: For both, the $(k,k^{1+o(1)})$-gap problem has time complexity $n/k^{1\pm o(1)}$ for small $k$. In contrast to previous work, which employed a subsampled variant of the Landau-Vishkin algorithm, we instead build upon the algorithm of Andoni, Krauthgamer and Onak (FOCS '10). We first simplify their approach and then show how to to effectively prune their computation tree in order to obtain a sublinear-time algorithm in the given time bound. Towards that, we use a variety of structural insights on the (local and global) patterns that can emerge during this process and design appropriate property testers to effectively detect these patterns.

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arXiv2022

Improved Sublinear-Time Edit Distance for Preprocessed Strings

We study the problem of approximating the edit distance of two strings in sublinear time, in a setting where one or both string(s) are preprocessed, as initiated by Goldenberg, Rubinstein, Saha (STOC '20). Specifically, in the $(k, K)$-gap edit distance problem, the goal is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. We obtain the following results: * After preprocessing one string in time $n^{1+o(1)}$, we can solve $(k, k \cdot n^{o(1)})$-gap edit distance in time $(n/k + k) \cdot n^{o(1)}$. * After preprocessing both strings separately in time $n^{1+o(1)}$, we can solve $(k, k \cdot n^{o(1)})$-gap edit distance in time $k \cdot n^{o(1)}$. Both results improve upon some previously best known result, with respect to either the gap or the query time or the preprocessing time. Our algorithms build on the framework by Andoni, Krauthgamer and Onak (FOCS '10) and the recent sublinear-time algorithm by Bringmann, Cassis, Fischer and Nakos (STOC '22). We replace many complicated parts in their algorithm by faster and simpler solutions which exploit the preprocessing.

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arXiv2022

A Structural Investigation of the Approximability of Polynomial-Time Problems

We initiate the systematic study of a recently introduced polynomial-time analogue of MaxSNP, which includes a large number of well-studied problems (including Nearest and Furthest Neighbor in the Hamming metric, Maximum Inner Product, optimization variants of $k$-XOR and Maximum $k$-Cover). Specifically, MaxSP$_k$ denotes the class of $O(m^k)$-time problems of the form $\max_{x_1,\dots, x_k} \#\{y:ϕ(x_1,\dots,x_k,y)\}$ where $ϕ$ is a quantifier-free first-order property and $m$ denotes the size of the relational structure. Assuming central hypotheses about clique detection in hypergraphs and MAX3SAT, we show that for any MaxSP$_k$ problem definable by a quantifier-free $m$-edge graph formula $ϕ$, the best possible approximation guarantee in faster-than-exhaustive-search time $O(m^{k-δ})$ falls into one of four categories: * optimizable to exactness in time $O(m^{k-δ})$, * an (inefficient) approximation scheme, i.e., a $(1+ε)$-approximation in time $O(m^{k-f(ε)})$, * a (fixed) constant-factor approximation in time $O(m^{k-δ})$, or * an $m^ε$-approximation in time $O(m^{k-f(ε)})$. We obtain an almost complete characterization of these regimes, for MaxSP$_k$ as well as for an analogously defined minimization class MinSP$_k$. As our main technical contribution, we rule out approximation schemes for a large class of problems admitting constant-factor approximations, under the Sparse MAX3SAT hypothesis posed by (Alman, Vassilevska Williams'20). As general trends for the problems we consider, we find: (1) Exact optimizability has a simple algebraic characterization, (2) only few maximization problems do not admit a constant-factor approximation; these do not even have a subpolynomial-factor approximation, and (3) constant-factor approximation of minimization problems is equivalent to deciding whether the optimum is equal to 0.

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