Resolution based Rules for the Weighted MaxSAT problem

1
Resolution based Rules for the Weighted Max-SAT
problem

• Federico Heras and Javier Larrosa

2
Overview

• Max-SAT Given a set of weighted clauses, find an
  assignment of the variables such that the sum of
  the weights of the violated clauses is minimized.
• Complexity NP-Hard.

3
Overview

• Applications
• Recent Algorithms

• Max-Cut
• Max-Clique
• Bayesian Networks

• Combinatorial Auctions
• Binate Covering Problem
• ...

• de Givry et al. CP03
• Shen and Zhang AAAI04
• Xing and Zang CP04

• de Givry et al. IJCAI05
• Alsinet et al SAT05
• Li et al CP05

4
Previous Work

• Our previous work
• Larrosa Heras IJCAI 05
• Extension of DPLL to Max-SAT.
• Extension of Resolution to Max-SAT.
• Heras Larrosa AAAI06
• Inference rules based on Resolution Rule for
  Max-SAT. Their combination produces an efficient
  Max-SAT solver.

5
New Contributions

• We generalize previous rules Heras Larrosa
  AAAI06.
• We proppose new ones envolving hard and soft
  clauses.

6
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

7
Max-SAT Notation

• x, y, z, boolean variables
• l, literal (positive or negative var.)
• A,B,C, clauses (set of literals)
• ? empty clause
• (C,u), weighted clauses
• (?,w) empty clause (Lower Bound)
• T maximum weight (Upper Bound)
• Negation ((l v C),w) ? (l v C,w), (C,w)

8
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

9
Max-SAT Search

• Function Max-DPLL(F, T) nat
• FInference(F)
• if (?, T)? F then ret T
• if FØ then ret 0
• if F(?, w) then ret w
• l SelectLiteral(F)
• vMax-DPLL(Fl, T)
• vMax-DPLL(Fl,v)
• ret v

Larrosa Heras IJCAI 05
10
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

11
Weighted Resolution (Max-RES)
Clashing clauses
(A ? B,m),(x ? A,u-m), (x ? B, w-m), (x ? A ?
B,m), (x ? A ? B,m)
(x ? A,u), (x ? B,w)
?
(where mminu,w)
12
Weighted Resolution (Max-RES)
Clashing clauses
(A ? B,m),(x ? A,u-m), (x ? B, w-m), (x ? A ?
B,m), (x ? A ? B,m)
Resolvent
(x ? A,u), (x ? B,w)
Posterior clashing clauses
?
Compensation clauses
(where mminu,w)
13
Weighted Resolution (Max-RES)
Clashing clauses
(A ? B,m),(x ? A,u-m), (x ? B, w-m), (x ? A ?
B,m), (x ? A ? B,m)
Resolvent
(x ? A,u), (x ? B,w)
Posterior clashing clauses
?
Compensation clauses
(where mminu,w)
Introduced in Larrosa Heras IJCAI 05
Completeness proved in Bonet Levy Manyà SAT
06
14
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

15
Inference Rules

• Simplification rules
• 1 Step of resolution NRES and DRES.
• Hyper-Resolution 2-RES and Chain-RES and
  Hard-RESa and Hard-RESb.
• They simplify the problem
• Reduce the clauses arity.
• Increase the lower bound.
• Create new unit clauses.

16
Neighborhood Resolution
(A,m),(x ? A,u-m), (x ? A, w-m)
(x ? A,u), (x ? A,w) ?
(where mminu,w)
17
Neighborhood Resolution
(A,m),(x ? A,u-m), (x ? A, w-m)
(x ? A,u), (x ? A,w) ?
(where mminu,w)
Example
(x ? y ? z,1), (x ? y ? z,1)
(y ? z,1)
?
18
Directional Resolution DRES
(x,u-m),(x ? y,w-m),(x v y, m),(y,m)
(x,u),(x ? y,w)
?
(where mminu,w) , x lt y
19
Directional Resolution DRES
(x,u-m),(x ? y,w-m),(x v y, m),(y,m)
(x,u),(x ? y,w)
?
(where mminu,w) , x lt y
Example
(x ? y,1), (x,1)
(x ? y,1), (y,1)
?
20
Hyper-Resolution 2-RES
(x ? y,u-m),(x ? z,w-m),(y ?
z,v-m),(x,m),(x v y v z,m),(x v y v z,m)
(x ? y,u),(x ? z,w),(y v z,v)
?
21
Hyper-Resolution 2-RES
(x ? y,u-m),(x ? z,w-m),(y ?
z,v-m),(x,m),(x v y v z,m),(x v y v z,m)
(x ? y,u),(x ? z,w),(y v z,v)
?
Example
(x ? y,1), (x v z,1), (y v z,1)
(x,1), (x v y v z,1), (x v y v z,1)
(x v z,1), (x v z,1),(x v y v z,1)
?
?
22
Hyper-Resolution Chain-RES
(l0,w0 - m), (l1,w2 - m), (l0 v l1,w1 - m), (l0
v l1,m), (?,m)
(l0,w0), (l0 v l1,w1), (l1 ,w2)
?
(where mminw0,...,w2) and 2 steps of resolution
23
Hyper-Resolution Chain-RES
(l0,w0 - m), (l1,w2 - m), (l0 v l1,w1 - m), (l0
v l1,m), (?,m)
(l0,w0), (l0 v l1,w1), (l1 ,w2)
?
(where mminw0,...,w2) and 2 steps of resolution
Example
(x ? y,1), (x,1),(y,1)
(x ? y,1), (y,1),(y,1)
(x ? y,1), (?,1)
?
?
24
Hyper-Resolution Chain-RES
(l0,w0 - m), (l2,w3 - m), (l0 v l1,w1 - m),
(l1 v l2,w2 - m), (l0 v l1,m), (l1 v
l2,m), (?,m)
(l0,w0), (l0 v l1,w1), (l1 v l2,w2), (l2 ,w3)
?
(where mminw0,...,w3) and 3 steps of resolution
25
Hyper-Resolution Chain-RES
(l0,w0 - m), (l3,w4 - m), (l0 v l1,w1 - m),
(l1 v l2,w2 - m), (l2 v l3,w3 - m), (l0 v
l1,m), (l1 v l2,m), (l2 v l3,m), (?,m)
(l0,w0), (l0 v l1,w1), (l1 v l2,w2), (l2 v
l3,w3), (l3 ,w4)
?
(where mminw0,...,w3) and 4 steps of resolution
26
Hyper-Resolution Chain-RES
(l0,w0 - m), (li-1 v li,wi - m) 1ltiltk (lk
,wk1 - m), (li-1 v li,m) 1ltiltk (?,m)
(l0,w0), (li-1 v li,wi) 1ltiltk (lk ,wk1)
?
(where mminw0,...,wk) and k1 steps of
resolution
Extension of the 3-RES presented in Heras
Larrosa AAAI06
27
Example
(?,1), (x,2), (x v y,2), (y v z,3),(z v w,2)
(w,2)
28
Example
(?,1), (x,2), (x v y,2), (y v z,3),(z v w,2)
(w,2)
4 steps of resolution
(?,3), (x v y,2), (y v z,2), (y v z,1), (z v
w,2)
29
Resolution with hard clauses

• Compensation clauses subsumed by hard clauses
  Hard-RESa and Hard-RESb.
• Bin(l1,l2,,lk) specifies that at most one
  literal in l1,l2,,lk is set to false.

(l1 ? l2,T), (l1 ? l3,T),, (l1 ?
lk,T), (l2 ? l3,T), (l2 ? l4,T),, (l2 ?
lk,T), , (lk-1 ? lk,T)
Bin(l1,l2,..,lk)
30
Hyper-Resolution Hard-RESa
(l1 v l2 v v lk,T), Bin(l1,l2,,lk), (l1,u1-m),
(l2,u2-m), (lk,uk,-m) (?,m)
(l1 v l2 v v lk,T), Bin(l1,l2,,lk), (l1,u1), (l
2,u2), (lk,uk)
?
(where mminu1,...,uk) and k steps of resolution
It increases the lower bound
31
Hyper-Resolution Hard-RESb
(l1 v l2 v v lk,T), Bin(l1,l2,,lk), (h v
l1,u1-m), (h v l2,u2-m), (h v lk,uk,-m) (h,m)
(l1 v l2 v v lk,T), Bin(l1,l2,,lk), (h v
l1,u1), (h v l2,u2), (h v lk,uk)
?
(where mminu1,...,uk) and k steps of resolution
It creates new unit clauses
32
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

33
Experimental Results

• Benchmarks
• State-of-the-art solvers

• Max-CUT
• Max-2-SAT
• Max-CSP (direct encoding)

• Shen and Zhang AAAI04
• Xing and Zang CP04

• de Givry et al. IJCAI05
• Li et al CP05

34
Experimental Results
MAX-CUT 2-RES
35
Experimental Results
MAX-2-SAT 2-RES and Chain-RES
36
Experimental Results
WCSP Hard-RESa and Hard-RESb.
37
Outline

• Preliminaries
• Max-SAT Search
• Max-SAT Inference
• Inference Rules
• Experimental Results
• Conclusions and Future Work

38
Conclusions and Future Work

• We have propposed new inference rules for the
  Weighted Max-SAT problem.
• We plan to Extend SAT techniques to Max-SAT
• Clause learning.
• Restarts.