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Questions tagged [fine-grained]

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14
votes
1answer
319 views

The problem of deciding whether a monotone CNF implies a monotone DNF

Consider the following decision problem Input: A monotone CNF $\Phi$ and a monotone DNF $\Psi$. Question: Is $\Phi \to \Psi$ a tautology? Definitely you can solve this problem in $O(2^n \cdot \...
2
votes
0answers
160 views

Approximating the Radius of a (Dense) Graph

For a (dense) graph, computing its radius is as hard as computer "All Pairs Shortest Paths" (APSP) [1]. So we can focus on approximating the radius. A $(1+\epsilon)$-approximating of APSP for a ...
5
votes
1answer
120 views

Reference request: complexity of $k$-partite $k$-SAT

Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT: given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal ...
7
votes
0answers
246 views

Given a Boolean formula, does there exist a small circuit that computes a satisfying assignment?

The 3-SAT problem can be defined as follows: 3-SAT Input: A 3-CNF formula $\phi$ of size $m$ with $n$ variables. Question: Does there exist a variable assignment that satisfies $\phi$? ...
4
votes
1answer
140 views

Hardness of Subgraph isomorphism problem for sparse pattern graph

Subgraph isomorphism problem is a well studied problem: given graphs $G$ and $H$, one needs to answer if $H$ contains $G$ as a subgraph. It was proven that this problem requires $|H|^{\theta(|G|)}$ ...
4
votes
1answer
161 views

Algorithms in preprocessed universe [closed]

In celebrated paper Clustered integer 3SUM via additive combinatorics by TM Chan and M Lewenstein one of the provided algorithms is the one for preprocessed universe. They were able to provide an ...
5
votes
0answers
124 views

Online triangle counting

Please consider the following problem. It can (but probably shouldn't) be called offline version of online triangle detection on subgraphs. Given a graph $G$ and a collection $C$ of subsets of ...
6
votes
1answer
581 views

On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
7
votes
1answer
255 views

Two DFA intersection emptiness connections to SETH & L vs P

(re "fine grained complexity") Wehar has proved that Two DFA intersection emptiness in $O(n^{2-\epsilon})$ time → SETH false. does anyone see any particular key proof difficulty, challenge, ...
14
votes
0answers
549 views

Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
23
votes
3answers
876 views

What are the relationships between those hypotheses in Fine-Grained Complexity Theory?

Complexity theory, through such concepts as NP-completeness, distinguishes between computational problems that have relatively efficient solutions and those that are intractable. "Fine-grained" ...