Solution of Satisfiability Problems by Unconstrained Quadratic Programming Talk given by Gary Kochen

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Solution of Satisfiability ProblemsbyUnconstrain
ed Quadratic ProgrammingTalk given byGary
KochenbergerUniversity of ColoradoHearin
Center for Enterprise ScienceOther Team
MembersFred GloverBahram AlidaeeKeith
WomerHaitao Li
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Introduction sThe Tenet of Linearity in
Combinatorial Optimization sMetaheuristics
Methods call this devotion into
question sRecent work suggests that nonlinear
representations may offer a fruitful
alternative sIn this research we investigate
the use of such methods for satisfiability
problems (SAT)
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Definition of SATA SAT instance is a Boolean
CNF Formula with s A set of n variables s A
set of literals either a variable or its
negation s A set of Clauses
. Each Clause consists only of literals
connected by logical or connectives s
CNF . Where
is the logical and connective.Goal
Find an assignment of Truth values such that the
Boolean function evaluates to
True. For Max SAT, we want
an assignment that satisfies the maximum
number of clauses.
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Applications s Propositional Logic s Data
Base Design and Analysis s Automated
Reasoning s SAILOR 21 s Computer Aided
Design s Machine vision s Robotics s
Scheduling s Integrated Circuit Design
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Zero/One (IP) formulation of SAT and
Max-SAT subject to s
literals in clause m s
0/1 (logical) variables s whether or not
clause m is satisfied s
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Max 2 Sat Example (Hansen Jaumard) (n 4,
m 12) Clause Clause
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IP Model s 16 variables and 12
constraints s ,
all other variables equal to 1 s 11 of the 12
clauses are satisfied s (by setting equal
to 1 and equal to 0)
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Max 2-SAT Unconstrained Quadratic Binary
Programming s Quadratic Penalty Function s
Unconstrained Quadratic Binary Program s
2-SAT w Three possible clause types w Each
with linear constraint w Each with equivalent
exact penaltyThree possibilities are (a) No
negations Classical constraint
Exact Penalty (b) One negation
Classical constraint Exact Penalty (c)
Two negations Classical constraint
Exact penalty
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For our exampleClause Clause
Quadratic Penalty (penj) 1
2 3 4 5 6 7
8 9 10 11 12
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Form Penalty Function SAT (UQP) s penj
is the quadratic penalty associated with clause
j s In the form of xQx (i.e., UQP) s
Advances is solving UQPFor our example we
getor,where the matrix Q is given
bySolving SAT (UQP) gives at
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Remarks s This unconstrained quadratic model
in 4 variables is equivalent
to the 16 variable, 12 constraint IP s The
general SAT (UQP) model will be of the form
, where
c is a constant and s Model applies both to
the SAT and Max-SAT w If equals zero, we
have a SAT solution w Else, our solution is a
result for the Max-SAT problem s Eliminates the
need for the y variables w Problem with m
5,000 clauses and 200 variables w SAT (UQP)
model with a Q matrix of size 200 by 200
(very modest) w IP
model would have 5,200 variables and 5000
constraints
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• Random Problems.available from authors
• xQx times rounded up
• xQx code run for 50 SPAN cycles
• 2Sat 500d and 1000a too large for CPLEX
• CPLEX results from version 8.0

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• All times in seconds unless noted otherwise.
• Maxsat is an exact method developed by Borchers
  Furman
• Maxsat results obtained on IBM RS/6000-590
• xQx results obtained 1.6 MHZ PC.
• Each xQx run was for 50 SPAN cycles
• Problem 100_600 was previously unsolved.

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3-SAT Max 3-SAT s Four clause types s Each
with classical constraint penalty1. No
Negations Classical constraint Exact
Penalty2. One Negations Classical
constraint Exact Penalty3. Two
Negations Classical constraint Exact
Penalty4. Three Negations Classical
constraint Exact Penalty s As before,
minimize the sum of the penalty terms s Penalty
function is cubic s Convert to xQx via
procedure of Hammer and Boros w Replace the
product term xy by a new variable z w Add the
penalty term g penalty is zero only when g
else it is positive
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3-Sat Example(n 5, m 12)Clause
Clause 1 2
3 4 5 6 7 8
9 10 11 12
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Forming the penalty function s five cubic
terms s introduce two new variables w w s
with associated penaltiesWhich is of
the form
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Taking (arbitrarily) the penalty P to be 10, the
7-by-7 Q matrix isMinimizing xQx yields

for which . Thus,
implying that all 12 clauses are
satisfied at this solution.Remarks pertaining
to 3-SAT and Max 3-Sat s For 3-SAT, clauses
does matter s SAT (UQP) could be as large as (n
m) s But terms combine (and cancel) s And,
our ability to solve larger UQP continues to
advance s General SAT problems
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Solving xQx (An Overview) s Tabu Search (TS)
method is centered around the use of
strategic oscillation. s The method alternates
between constructive phases (add moves)
and destructive phases (drop moves). s To
control the underlying search process, we use a
memory structure that is updated at
critical events. s A parameter span is used to
indicate the amplitude of
oscillation about a critical event. s We begin
with span equal to 1 and gradually increase it to
some limiting value. s For
each value of span, a series of alternating
constructive and destructive phases is
executed before progressing to the next
span value. s At the limiting point,
span is gradually decreased, allowing
again for a series of alternating
constructive and destructive
phases. s When span reaches a value of 1, a
complete span cycle has been completed and
the next cycle is launched.
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Solving xQx (An Overview), continued s The
search process is typically allowed to run for a
pre-set number of span
cycles. s Information stored at critical events
is used to influence the
search process by penalizing potentially
attractive add moves and
inducing drop moves associated with assignments
of values to variables in
recent critical solutions. s Cumulative
critical event information is used to introduce a
subtle long-term bias into
the search process by means of
additional penalties and inducements similar to
those discussed above. s
Other standard elements of tabu search such as
short and long- term memory
structures are also included. s A complete
description of the method is given in Glover,
Kochenberger, Alidaee (Management
Science).
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Where we are headed s Further testing of max
2-SAT case w More test problems w State of
the Art comparisons s Extend to 3- SAT case w
Code and test generator w Converting cubic
penalty function to xQx w Test on standard
test bed w Assess performance/make
comparisons s Extend to General SAT case w
Convert back to xQx w Examine other solution
possibilities w Long term plans s Develop
improved heuristic method for solving xQx w
Enable larger instances to be solved w Useful
for other problem classes
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Wrap-up s Shown xQx to be a fruitful modeling
and solution framework for 2-SAT and Max
2-SAT problems. s Indicated how xQx could be
applied to more general SAT
problems. s Encouraged by our progress to date
and with the prospects for future
findings and contributions.