Minimization of DNF Formulas Given a Truth Table

1. Minimization of DNF Formulas Given a Truth Table Lisa Hellerstein Polytechnic University Brooklyn, NY Joint with Eric Allender (Rutgers), Paul McCabe (Toronto), Toniann Pitassi (Toronto), and Michael Saks (Rutgers) [CCC 2006]

2. MinDNF Problem Given a truth table, find minimum DNF formula

3. Background • Many heuristics – old problem • Karnaugh maps • Software packages (Espresso) • Related problem • Given DNF formula, find smallest equivalent DNF formula • (Umans) Complete for

4. MinDNF • Decision Version: Given truth table and k > 0, does there exist DNF of size at most k? • In NP • Poly time = time polynomial in 2n • (Masek 1979) MinDNF is NP-complete • Gadget reduction from Circuit-SAT • Unpublished manuscript, cited in G & J • Recent versions Czort (1999), Umans et al. (2004)

5. [Umans, Villa, Sangiovanni-Vincentelli, 2004]

10. Main Results • New proof of NP-completeness of minDNF • Much simpler than Masek’s • Reduce from 3-partite set cover (3d-matching) • Theorem: If then for a fixed ε < 1, minDNF cannot be approximated to within an O(log ε N) factor of optimal

11. MinDNF as set covering problem Cover 1’s of f by prime implicants (minterms) of function

12. MinDNF as set covering problem Cover 1’s of f by prime implicants (minterms) of function

13. MinDNF as set covering problem Cover 1’s of f by prime implicants (minterms) of function

14. MinDNF as set covering problem Cover 1’s of f by prime implicants (minterms) of function

15. Approximation of minDNF • Can approximate minDNF using standard Greedy Set Cover Algorithm • Generate all prime implicants of f • “Cover” 1’s of f with prime implicants • Produces DNF that is within factor O(log N) of optimal • Very strong non-approximability result known for General Set Cover --- can’t do better than O(log N) approximation • Does this hold for minDNF also?

16. NP-completeness of minDNF • Two stage-reduction Stage 1: 3-partite set cover minDNF* Stage 2: min*DNF minDNF

17. Stage 1: 3-partite set cover minDNF* All sets in A x B x C B C A

18. MinDNF* Given a truth table with *’s, find minimum DNF formula

19. Idea [cf. Gimpel] 11111111 00000000

20. Idea [cf. Gimpel] 1

21. Idea [cf. Gimpel] 1 0

22. Idea [cf. Gimpel] 1 0 1 * 1 0

23. 11111111 Φ S 11101111 Φ b 11100010 Define Φ that maps each element b and set S to point in Boolean lattice s.t. b є S iff Φ(b) < Φ(S)

24. S1 S2 S3 S4 b1 b3 b2 b4 b5

25. S1 S2 S3 S4 1 1 1 1 1 b1 b3 b2 b4 b5

26. * S1 S2 S3 S4 * * * * 1 1 1 1 1 * * b1 b3 b2 b4 b5 * * * * * *

27. 0 0 0 0 0 0 0 0 0 * S1 S2 S3 S4 * * * * * 1 1 1 1 1 * * b1 b3 b2 b4 b5 * * * *

28. Size of min cover = Size of min DNF 0 0 0 0 * * 1 1 1 * 1 * * *

29. Define mapping φ A C B

30. Define mapping φ 00000000 00000000 00110101 A C B C A B

31. Define mapping φ 00000000 11001010 00000000 A C B C A B

32. Define mapping φ 11110000 00000000 00000000 A C B C A B

33. Define mapping φ 11110000 11001010 00110101 A C B Length is O(log n) C A B

34. NP-completeness of minDNF • Two stage-reduction Stage 1: 3-partite set cover minDNF* Stage 2: min*DNF minDNF size k size k + # of *’s

35. Non-approximability • Two stage-reduction Stage 1: Special case of set cover derived from Label Cover (Lund-Yannakakis) min*DNF Stage 2: min*DNF minDNF (make it approximation preserving)

36. Big Picture • Try to embed set cover instances with n elements into Boolean lattice on O(log n) variables • Closer relationship between set cover and minDNF • Apply techniques • Well known: Concrete instance of set cover on which greedy produces solution Ώ(log N) times worse than optimal • New: instance of minDNF on which greedy produces DNF of size Ω(logN) times worse than optimal

37. Open Questions • Improve non-approximability factor from O(logεN) to O(log N) • Independently, Feldman proved O(logεN) factor assuming P ≠ NP • Fixed parameter versions of minDNF • Solve minDNF when DNF-size is guaranteed to be less than k • Can do in poly time for k = • Do in poly time for k = ? (Related to learnability of log n term DNF)