1. Optimal Solutions for Generalized Covering Problems
Matthias F. Stallmann, 2006/02/07, NCSU IE/OR Joint Seminar
Xiao Yu Li
Seattle, WA, USA
Franc Brglez
Raleigh, NC, USA
Joint work with:
Our paper addresses the unate covering and binate covering problems. They are both special cases of the Minimum-Cost Satisfiability Problem. Most work discussed in this presentation was done while I was a Ph.D. student at North Carolina State University. I am now working as an optimization engineer at Amazon.com.
11/20/2018

2. Outline Unate and binate cover problems Basics of branch and bound
Lower bounds make a lot of difference
All-integer dual
Some preliminary results and conclusions
Stochastic search for better upper bounds
Final thoughts
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3. Background Work Unate (Set) Cover Problem (UCP)
1987: espresso, Rudell et al
1993: espresso-signature, McGeer at al ….
Binate Cover Problem (BCP)
1996: scherzo, Coudert
2002: bsolo, Manquinho et al …
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4. Native MinCostSat Problems
Covering
Non-Covering
Minimum-Length
Plan
ATPG
Binate Covering
Unate Covering
Logic Minimization
Crew Scheduling
Vehicle Routing
Many optimization problems can be formulated by the minimum-cost assignment problem. They can be roughly categorized as Covering and non-covering. On the covering side, there are the unate covering problem and the binate covering problem. Set covering has applications in two-level logic minimization and application in OR that include crew scheduling, and vehicle routing. Binate covering’s main applications are in the logic synthesis domain that include technology mapping, finite state machine minimization and boolean Relations. On the non-covering side, MinCostSat can formulated problems in automated test pattern generation, minimum length plan. And it also has applications in expert systems.
Technology Mapping
FSM Minimization
Boolean Relations
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5. Unate (Set) Covering Three backup positions to fill.
Five players to choose from.
Want to minimize the number of players.
Duncan
Francis
Malone
Stockton
Yao
Before we get too carried away …
Guard 1 1
Forward 1 1
Center 1 1 1
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6. Unate Covering as CNF Sat
Conjunctive Normal Form
Guard F  S
Forward D  M
Center D  M  Y
Goal: minimize F + S + D + M + Y
Min-cost solutions: F =1, D =1, S = M = Y = 0
M =1, S =1, D = F = Y = 0
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7. Binate Covering Problem
Additional constraint 1:
Malone and Stockton
are together ( M  S)
M  S M  S
Additional constraint 2:
Duncan and Stockton
can’t be signed together ( D  S )
D  S
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8. Binate Covering Problem (cont.)
constraints
covering matrix D F M S Y 1 -1
F  S
D  M
D  M  Y
M  S
D  S
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9. MinCostSat Problems Revisited
Covering
Non-Covering
Minimum-Length
Plan
ATPG
Binate
Unate
Technology Mapping
FSM Minimization
Boolean Relations
Logic Minimization
Crew Scheduling
Vehicle Routing
Most rows have both 1’s and -1’s
Covering Matrix
No -1’s
Most rows have no -1’s
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10. Solve MinCostSat as ILP
MinCostSat is a special case of 0-1
integer linear programming
For each constraint in the MinCostSat instance...
x1  x2  x3  x4
...replace xi with (1 - xi):
MinCostSAT can be transformed in ILP easily. We show an example here.
(1 - x1) + x2 + (1 - x3) + x4  1
- x1 + x2 - x3 + x4  -1
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11. MinCostSat as ILP: an example
-x1 + x2 - x3 + x4  -1
-x1 - x2 - x  -2
x1 + x2 + x  1
x1 v x2 v x3 v x4
x1 v x2 v x4
x1 v x2 v x3
Assume unit cost, minimize: x1 + x2 + x3 + x4
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12. Branch and Bound Solvers
1. Upper/lower bound calculation
(for a unate example)
2. Branching (variable assignment)
3. One iteration of the main algorithm
4. Effect of better (global) upper bounds and (local) lower bounds.
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13. An Example (Steiner 9) UB = 6 find “cover” to get UB LB = 3 based on1
UB = 6
find “cover”
to get UB
LB = 3
based on
max. ind. set
(MIS)
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14. After assignment of a variable
st09
UB = 4 (+ 1 = 5)
X1 = 0
X1 = 1 1 X9 X8 X7 X6 X5 X4 X3 X2
LB = 2 (+ 1 = 3)
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15. The algorithm: one iteration
2. select a variable
1. dequeue “best”
node in priority queue x
x=0
x=1
UB=5
LB=4
UB=6
LB=3
3. create two
new nodes
UB=5
LB=4
UB=5
LB=3
UB=5
LB=3
4. do reduction and
compute bounds
5. enqueue both new nodes
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16. Lower and Upper Bounds UB= 7 6 5 x5=0 x5=1 x9=0 x9=1 legend 7/3 7/5
on queue
UB/LB
UB= 7 6 5
7/3
x5=0
x5=1
7/5
6/4
x9=0
x9=1
dead end
7/6
5/4
dead end
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17. Branch-and-bound solver: eclipse
The algorithm
Initialize root; Q.enqueue(root); globalUB=UB(root)
while ( Q is not empty ) {
node = Q.dequeue();
if ( LB(node) < globalUB ) {
select branching variable x_i
for ( b = 0 to 1 ) {
n_b = (node for x_i = b)
do reduction and bounds for n_b
if ( UB(n_b) < globalUB ) { globalUB = UB(n_b) }
if ( LB(n_b) < globalUB ) { Q.enqueue( n_b) }
}}}
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18. Eclipse Performance Factors
Seven Performance Factors
Lower Bounding (ILP techniques)
Upper Bounding (stochastic search)
Search-Tree Exploration Strategies
Branching Variable Selection
Search Pruning
Reductions
Data Structures
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19. Factor 1: Lower Bounding
Four lower-bounding strategies:
Maximal Independent Set (MIS) of rows (e.g., maximal matching for vertex cover)
Linear Programming Relaxation (LPR)
Cutting Planes (CP)
All Integer Dual Simplex
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20. Lower Bounding: LP Relaxation
Relax integer constraints (LPR):
Feasible region x2
Integer
optimum = 2
(1, 1)
The Gomery cuts never exclude integer feasible solutions from an ILP problem, but has the potential of eliminating optimal non-integer solution for the LP problem; therefore, increases the lower bound. In this case, (0.5, 0.5) is excluded by the Gomery cuts.
(0.5, 0.5)
LP cost = 1.0 x1
LP optimum is a lower bound on integer optimum
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21. Lower Bounding: Cutting Planes (CP)x2
Feasible region
(0.7, 1.1)
The Gomery cuts never exclude integer feasible solutions from an ILP problem, but has the potential of eliminating optimal non-integer solution for the LP problem; therefore, increases the lower bound. In this case, (0.5, 0.5) is excluded by the Gomery cuts.
Cut
(0.5, 0.5) x1
CP cuts off fractional solution but not integer solutions
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22. Lower Bound Methods (rot.b)
are needed to see this picture.
Graphics decompressor 90 95
100
105
110
115
QuickTimeｪ and a CP
LPR
MIS2
MIS 1 2 3 4
OPT 5
(MIS2 is MIS with extra tricks)
Most effective by far: CP
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23. Experimental Results The solvers Scherzo [Coudert, DAC1996]
Cplex (State of the art IP/ILP Solver)
Eclipse-lpr (LP Relaxation)
Eclipse-cp (Cutting Planes)
The benchmarks
Logic minimization (unate covering)
FSM minimization (binate covering)
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24. Unate Results (nodes visited)
“harder”
(timed out)
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25. Typical Results: Unate (runtime)
timeout
= 1 hour
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26. Large binate benchmarks
We visit fewer nodes...
... but spend more time
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27. Hardest Binate Benchmarks
Solver ->
Benchmark
cplex (time)
cplex (nodes)
eclipse (time)
eclipse (nodes)
rot.b
R:2984 C:1452 L:40755
12.49
70.65::52.88
308
3967::3706
time out
alu4
R:1838 C:808 L:36250
e64.b
R:1053 C:608 L:8200
(mean::stdev)
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28. All-Integer Dual Simplex
vertex cover of a triangle x2 x3 1 -1 -1 1
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29. example (continued) -1 1 -1 1 -2-1 1 -2
At this point, when we divide by the pivot,
we get fractions. The LP solution is
x1 = ½, x2 = ½, x3 = ½
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30. Maintaining an integer tableaux3 e1 e3 ct -1 1 -2 -2 1 -1 x1 x1 e2 e2 x2 x2 ct x3
Negative of the solution
is shown in green circle.
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31. Features of All-Integer Dual
Terminates with optimum solution (but may take a long time).
Upper left tableau entry (negated) can always be used as a lower bound for (primal) solution.
With clever choice of pivot row, can be encouraged to increase the LB rapidly.
Can choose pivot row so that a greedy maximal independent set of rows is chosen first.
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32. Some Preliminary Results
32 runs, Intel® Xeon™ CPU 3.20GHz, 2MB cache, default(4096 it.,64K flips)
Benchmark
Runtime
Nodes
rot.b
( 8192 it.)
25.4::18.8
24.7::22.1
1.8::1.4
1.1::0.3
alu4
4.3::4.3
4.8::6.0
1.2::0.6
1.0::0.0
e64. b
( it.)
416.0::835.0
21.6::27.0
72.1::149.0
1.1::0.7
Large variance in number of iterations needed to match UB
and in whether optimum UB is found at root
Large variance in when optimal UB is first found
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33. Preliminary Conclusions
All-integer dual, when used in conjunction with stochastic search, is a powerful tool (even with no B&B).
Different instance types need radically different solution techniques:
Unate problems require more effort for upper bounds – simpler, faster, LB’s are best (e.g., ordinary LP)
Binate problems – all-int dual + stoch search
Non-covering – satisfiability techniques (feasibility is hard)
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34. Still MIA: the unate benchmark test4.pi
variables
constraints
literals
time/cplex
test4.pi
6139
1437
109318
???
ex5
2428
831
41085
65.6
exam.pi
4676
509
25694
21.0
rot.b
1451
2932
40755
62.0
apex4.a
4316
11912
57595
9.9
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35. Simple Stochastic Search
Each step flips the value of one variable
type of step
how
often
details
satisfy a constraint
0.95
pick a random unsatisfied constraint and flip its ‘best’ variable
OR flip a random variable to decrease cost
aim for lower cost
0.0499
pick ‘best’ variable to decrease cost (lowest number of clauses left unsatisfied)
completely random
0.0001
pick a variable at random -- need this to get out of a rut
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36. Anytime Algorithms
Can be terminated any time based on time budget or number of iterations
Will report a feasible solution if one has been found
Realistic method for judging algorithms: want to know how good a solution is available given a time budget
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37. Anytime Algorithms: cplex vs. umbra
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38. Summary and Open Problems
Need a lot of different approaches to deal with hard problems
No one method wins all the time
Order of presentation matters (sometimes a lot)
Benchmark “profiling”
Better theory (e.g., more sophisticated branching and bounding)
Better knowledge of machine architecture (parallelism and pipelining, distributed computing)
11/20/2018