1. The Connectivity of Boolean Satisfiability: Structural and Computational Dichotomies Elitza Maneva (UC Berkeley) Joint work with Parikshit Gopalan, Phokion Kolaitis and Christos Papadimitriou

2. Features of our dichotomy Refers to the structure of the entire space of solutions The dichotomy cuts across Boolean clones Motivated by recent heuristics for random input CSP.

3. Space of solutions 11111 00000 n-dimensional hypercube

4. Space of solutions 11111 00000

5. 11111 00000

6. Connectivity of graph of solutions? 11111 00000 11111 00000

7. Our dichotomy Computational problems –CONN: Is the solution graph connected? –st-CONN: Are two solutions connected? Structural property –Possible diameter of components PSPACE-complete exponential NP-complete CONN st-CONN diameter SAT Tight CSP Non-tight CSP in co-NP in P linear P and NP-complete

8. Motivation for our study Heuristics for random CSP are influenced by the structure of the solution space Random 3-SAT with parameter  : n variables,  n clauses are chosen at random 4.154.27 0  EasyHard Unsat

9. Motivation for our study Heuristics for random CSP are influenced by the structure of the solution space Survey propagation algorithm [Mezard, Parisi, Zecchina ‘02] designed to work for clustered random problems very successful for such random instances based on statistical physics analysis

10. Clustering in random CSP What is known? 2-SAT: a single cluster up to the satisfiability threshold 3-SAT to 7-SAT: not known, but conjectured to have clusters before the satisfiability threshold 8-SAT and above: exponential number of clusters [Achlioptas, Ricci-Tersenghi `06] [Mezard, Mora, Zecchina `05]

11. Our dichotomy PSPACE-complete exponential CONN st-CONN diameter Tight CSP Non-tight CSP in coNP in P linear

12. OR-free CSPs 11111 00000 NAND-free CSPs Distance preserving CSPs 11111 00000 11111 00000 Graph distance = Hamming distance Tight

13. 11111 00000 OR / NAND-free CSP Set of relations neither of which can express OR by substituting constants Includes Horn Includes some NP-complete CSP, e.g. POS-1-in-k SAT

14. 11111 00000 Graph distance = Hamming distance Distance preserving CSP Set of relations, for which every component is a 2-SAT formula (component-wise bijunctive) Includes bijunctive Includes some NP-complete CSP

15. Proof for the hard side of the dichotomy Proof for 3-SAT Expressibility theorem like Schaefer’s

16. Schaefer expressibility A relation  is expressible from set of relations S if there is a CNF(S) formula , s.t. :  (x 1, …, x n ) =  w 1, …,w t  (x 1, …, x n, w 1, …, w t )

17. Faithful expressibility A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. :  (x 1, …, x n ) =  w 1, …,w t  (x 1, …, x n, w 1, …, w t )

18. Faithful expressibility A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. :  (x 1, …, x n ) =  w 1, …,w t  (x 1, …, x n, w 1, …, w t )

19. Faithful expressibility A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t.:  (x 1, …, x n ) =  w 1, …,w t  (x 1, …, x n, w 1, …, w t ) and (1) For every a  {0,1} n with  ( a )=1, the graph of solutions of  ( a, w) is connected. (2) For every a, b  {0,1} n with  ( a )=  ( b )=1, | a-b |=1, there exists w s.t.  ( a, w )=  ( b, w )=1

20. Lemma: For 3-SAT (a) Exist formulas with exponential diameter (b) CONN and st-CONN are PSPACE-complete Lemma: Faithful expressibility: (a) preserves diameter up to a polynomial factor (b) Is a poly time reduction for CONN and st-CONN Faithful Expressibility Theorem: If S is not tight, every relation is faithfully expressible from S. Proof for the hard side of the dichotomy

21. Theorem : If S is not tight, every relation is faithfully expressible from S. Proof in 4 steps. Step 0: Express 2-SAT clauses. Some relation can express OR (NAND). Other 2-SAT clauses by resolution: (x 1  x 2 ) =  w (x 1  w)  (w  x 2 ) Faithful Expressibility Theorem ___

22.  (x 1  x 3 ) = Faithful Expressibility Theorem Theorem : If S is not tight, every relation is faithfully expressible from S. Proof in 4 steps. Step 1 : Express a relation where some distance expands. Use R which is not component-wise bijunctive.

23. Faithful Expressibility Theorem Theorem : If S is not tight, every relation is faithfully expressible from S. Proof in 4 steps. Step 2 : Express a path of length 4 between vertices at distance 2.

24. Theorem : If S is not tight, every relation is faithfully expressible from S. Proof in 4 steps. Step 3: Express all 3-SAT clauses from such paths. [Demaine-Hearne ‘02] Faithful Expressibility Theorem

25. Theorem : If S is not tight, every relation is faithfully expressible from S. Proof in 4 steps. Step 4: Express all relations from 3-SAT clauses. Faithful Expressibility Theorem

26. Open questions Trichotomy for CONN?Trichotomy for CONN? –P for component-wise bijunctive –coNP-complete for non-Schaefer tight relations –open for Horn/dual-Horn Which Boolean CSPs have a clustered phase?Which Boolean CSPs have a clustered phase?

27. Thank you