Upper and Lower Bounds for Weak Backdoor Set Detection

TL;DR

This paper establishes new upper and lower exponential time bounds for detecting weak backdoor sets in 3CNF formulas, improving previous algorithms and proving tight bounds under the Strong Exponential Time Hypothesis.

Contribution

It introduces faster algorithms for detecting weak backdoor sets into Horn and Krom classes and proves matching lower bounds assuming SETH.

Findings

4.54^k algorithm for Horn backdoor detection

2.27^k algorithm for Krom backdoor detection

2^k lower bound under SETH

Abstract

We obtain upper and lower bounds for running times of exponential time algorithms for the detection of weak backdoor sets of 3CNF formulas, considering various base classes. These results include (omitting polynomial factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Krom formulas. These bounds improve an earlier known bound of 6^k. We also prove a 2^k lower bound for these problems, subject to the Strong Exponential Time Hypothesis.