The Connectivity of Boolean Satisfiability: Structural and Computational Dichotomies

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. Space of solutions 11111 00000

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

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

8. 0  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 Unsat Easy Hard 4.15 4.27

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 CSPWhat 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 Tight CSP Non-tight CSP CONN st-CONN diameter in coNP in P linear PSPACE-complete PSPACE-complete exponential

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

13. OR / NAND-free CSP 11111 00000 • 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. Distance preserving CSP 11111 Graph distance = Hamming distance 00000 • 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. :  (x1, …, xn) =  w1, … ,wt(x1, …, xn, w1, …, wt)

17. Faithful expressibility A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. :  (x1, …, xn) =  w1, … ,wt(x1, …, xn, w1, …, wt)

18. Faithful expressibility A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. :  (x1, …, xn) =  w1, … ,wt(x1, …, xn, w1, …, wt)

19. Faithful expressibility • A relation  is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t.: • (x1, …, xn) =  w1, … ,wt(x1, …, xn, w1, …, wt) 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 ws.t. (a, w)=(b, w)=1

20. Proof for the hard side of the dichotomy 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.

21. Faithful Expressibility Theorem • 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: • (x1 x2) =  w (x1 w)  (w  x2) _ _ _

22. 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. (x1 x3) =

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. Faithful Expressibility Theorem 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]

25. Faithful Expressibility Theorem 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.

26. Open questions • 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?

27. Thank you