---
title: "Phase Transitions of Bounded Satisfiability Problems"
authors:
  - "Delbert D. Bailey"
  - "Phokion G. Kolaitis"
doi: ""
url: "https://ijcai.org/proceedings/2003/169"
pdf: "https://ijcai.org/proceedings/2003/0169.pdf"
venue: "Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence"
year: 2003
pages: "1187-1192"
track: ""
keywords:
---

# Phase Transitions of Bounded Satisfiability Problems

Delbert D. Bailey, Phokion G. Kolaitis

*Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence*, pages 1187-1192.

## Abstract

We investigate phase transitions for the family of bounded satisfiability problems 3SAT(b), recently introduced by Zhang, that ask: given a 3CNFformula, is there a truth assignment that violates no more than b of its clauses. Zhang's results were experimental and for a fixed number of variables (n = 25), and suggested that the locations of the phase transitions for 3SAT(b) are separated and move significantly as b increases. Analysis of these locations was posed as an open question . We analytically show that the phase transitions of all 3SAT(b) problems must occur within a narrow region, regardless of how large the value of b is. Moreover, our experiments reveal that the phase transitions for these problems occur in a remarkable way. Specifically, unlike 3SAT, the probability curves for 3SAT(b) do not have a quasi-common intersection point about which they rotate as they become steeper with increasing n. Instead, they move rapidly to the left toward the narrow region that the analysis predicts.