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The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be…
The Strong Exponential Time Hypothesis (SETH) is a standard assumption in (fine-grained) parameterized complexity and many tight lower bounds are based on it. We consider a number of reasonable weakenings of the SETH, with sources from (i)…
We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $\epsilon>0$, compute a $(1+\epsilon)$-approximation for ED and a…
The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential…
The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its…
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight…
A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the $O(n^2)$ dynamic programming solution to the Longest…
The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of…
Subset-Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. One of the most intriguing open problems in this area is…
The edit distance (a.k.a. the Levenshtein distance) between two strings is defined as the minimum number of insertions, deletions or substitutions of symbols needed to transform one string into another. The problem of computing the edit…
We prove several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any $\varepsilon>0$, assuming the Strong Exponential Time Hypothesis (SETH), there are no near-linear time…
The CNF formula satisfiability problem (CNF-SAT) has been reduced to many fundamental problems in P to prove tight lower bounds under the Strong Exponential Time Hypothesis (SETH). Recently, the works of Abboud, Hansen, Vassilevska W. and…
We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in $\text{Quasi-NP} = \text{NTIME}[n^{(\log n)^{O(1)}}]$ and…
The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original…
Subgraph similarity search, one of the core problems in graph search, concerns whether a target graph approximately contains a query graph. The problem is recently touched by neural methods. However, current neural methods do not consider…
We study approximation algorithms for the following three string measures that are widely used in practice: edit distance (ED), longest common subsequence (LCS), and longest increasing sequence (LIS). All three problems can be solved…
Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this…
We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent,…
Classic similarity measures of strings are longest common subsequence and Levenshtein distance (i.e., the classic edit distance). A classic similarity measure of curves is dynamic time warping. These measures can be computed by simple…
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut…