# The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

@article{Gopalan2006TheCO, title={The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies}, author={Parikshit Gopalan and Phokion G. Kolaitis and Elitza N. Maneva and Christos H. Papadimitriou}, journal={Electron. Colloquium Comput. Complex.}, year={2006}, volume={TR06} }

Given a Boolean formula, do its solutions form a connected subgraph of the hypercube? This and other related connectivity considerations underlie recent work on random Boolean satisfiability. We study connectivity properties of the space of solutions of Boolean formulas, and establish computational and structural dichotomies. Specifically, we first establish a dichotomy theorem for the complexity of the st-connectivity problem for Boolean formulas in Schaefer's framework. Our result asserts…

## 139 Citations

### The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

- Mathematics, Computer ScienceTheory of Computing Systems
- 2015

Connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp.

### Connectivity of Boolean satisfiability

- MathematicsArXiv
- 2015

A computational dichotomy is proved for the st-connectivity problem, asserting that it is either solvable in polynomial time or PSPACE-complete, and an aligned structural dichotomy for the connectivity problem is proved, asserting the maximal diameter of connected components is either linear in the number of variables, or can be exponential.

### The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants

- MathematicsArXiv
- 2014

For the diameter and the st-connectivity question, dichotomies analogous to those of Gopalan et al. are proved, and fragments are identified where it is in P, Where it is coNP-complete, and where it are PSPACE-complete in analogy to Gopala et al.'s trichotomy.

### A Computational Trichotomy for Connectivity of Boolean Satisfiability

- Mathematics, Computer ScienceJ. Satisf. Boolean Model. Comput.
- 2014

The trichotomy is proved: Connectivity is either in P, coNP-complete, or PSPACE-complete.

### A Dichotomy Theorem within Schaefer for the Boolean Connectivity Problem

- MathematicsElectron. Colloquium Comput. Complex.
- 2007

It is shown that the connectivity problem for bijunctive relations can be solved in O(min{n|φ|, T (n)}) time, where n denotes the number of variables, φ denotes the corresponding 2-CNF formula, and T ( n) denotes the time needed to compute the transitive closure of a directed graph of n vertices.

### On the Boolean connectivity problem for Horn relations

- MathematicsDiscret. Appl. Math.
- 2007

### An exact algorithm for the Boolean connectivity problem for k-CNF

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2010

### Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2017

This work studies the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment to another satisfying assignment of the Boolean formula and uses Birkhoff’s representation theorem on a set system that is shown to be a distributive lattice.

### Solution-Graphs of Boolean Formulas and Isomorphism

- MathematicsSAT
- 2016

The solution graph of a Boolean formula on n variables is the subgraph of the hypercube \(H_n\) induced by the satisfying assignments of the formula. The structure of solution graphs has been the…

### Finding Paths between Graph Colourings: Computational Complexity and Possible Distances

- MathematicsElectron. Notes Discret. Math.
- 2007

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