# On Problems as Hard as CNF-SAT

@article{Cygan2012OnPA, title={On Problems as Hard as CNF-SAT}, author={Marek Cygan and Holger Dell and Daniel Lokshtanov and D'niel Marx and Jesper Nederlof and Yoshio Okamoto and Ramamohan Paturi and Saket Saurabh and Magnus Wahlstrom}, journal={2012 IEEE 27th Conference on Computational Complexity}, year={2012}, pages={74-84} }

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 time algorithms have been found for a myriad of problems, including GRAPH COLORING, HAMILTONIAN PATH, DOMINATING SET and 3-CNF-SAT. In some instances, improving these algorithms further seems to be out of reach. The CNF-SAT problem is the canonical example of…

## 196 Citations

### On Problems as Hard as CNF-SATx

- Computer Science
- 2012

It is shown that, for every < 1, the problems HITTING SET, SET SPLITTing, and NAE-SAT cannot be computed in time O(2 ) unless SETH fails, and it is proved that the fastest known algorithms for STEINTER TREE, CONNECTED VERTEX COVER, SET PARTITIONING, and the pseudo-polynomial time algorithm for SUBSET SUM cannot be significantly improved.

### Polynomial formulations as a barrier for reduction-based hardness proofs

- Mathematics, Computer ScienceArXiv
- 2022

Evidence is provided that such ﬁne-grained reductions will be difﬁcult to derive Set Cover Conjecture from SETH, and that proving c n lower bound for any constant c > 1 (not just c = 2) under SETH for any of the problems above would imply new circuit lower bounds.

### Complexity of SAT Problems, Clone Theory and the Exponential Time Hypothesis

- Mathematics, Computer ScienceSODA
- 2013

A stronger version of Impagliazzo et al.'s sparsification lemma for k-SAT is proved; namely that all finite Boolean constraint languages S and S' such that SAT(·) is NP-complete can be sparsified into each other.

### Exact exponential algorithms for two poset problems

- Mathematics, Computer ScienceSWAT
- 2020

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute…

### Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

- Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

It is proved that sufficient progress would imply a breakthrough on one of five major open problems in the theory of algorithms, including dynamic versions of bipartite perfect matching, bipartites maximum weight matching, single source reachability, single sources shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems.

### SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

- Mathematics, Computer ScienceSODA
- 2019

A tight reduction is achieved from k -SAT to Subset Sum on dense instances, proving that Bellman’s 1962 pseudo-polynomial time algorithm cannot be improved to time T 1-ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.

### Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk)

- Computer Science, MathematicsIPEC
- 2015

Evidence is provided that a problem $A$ with a running time O(n^k) that has not been improved in decades, also requires n^{k-o(1)} time, thus explaining the lack of progress on the problem.

### Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs

- Computer Science, MathematicsSTACS
- 2013

This paper develops an algorithm running in O(2^{O((k log k)^{2/3})}n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph.

### Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

- Mathematics, Computer ScienceICALP
- 2021

This paper demonstrates the existence of maximally hard pattern graphs H that require time n tw( H )+1 − o (1) and proves the following asymptotic statement for large treewidth t: under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory.

### Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

- Computer Science, Mathematics2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

It is shown that the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing, and in several cases it is able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail.

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### On Problems as Hard as CNF-SATx

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It is shown that, for every < 1, the problems HITTING SET, SET SPLITTing, and NAE-SAT cannot be computed in time O(2 ) unless SETH fails, and it is proved that the fastest known algorithms for STEINTER TREE, CONNECTED VERTEX COVER, SET PARTITIONING, and the pseudo-polynomial time algorithm for SUBSET SUM cannot be significantly improved.

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### Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

- Computer Science, Mathematics2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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It is shown that the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing, and in several cases it is able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail.

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