1 Consistencies for Ultra-Weak Solutions in Minimax Weighted CSPs Using the Duality Principle Arnaud...

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Consistencies for Ultra-Weak Solutions inMinimax Weighted CSPs Using the Duality Principle

Arnaud Lallouet1, Jimmy H.M. Lee2, and Terrence W.K. Mak2

1Université de Caen, GREYC, Caen, FranceArnaud.Lallouet@unicaen.fr

2The Chinese University of Hong Kong, Shatin, N.T., Hong Kong{jlee,wkmak}@cse.cuhk.edu.hk

IntroductionIntroduction• Motivation

• Minimax Weighted CSPs– Ultra-weakly solved, weakly solved, and strongly solved

• Consistency Techniques1. Lower Bound formulations2. Upper Bounds using duality principle3. Strengthening lower and upper bounds by adopting

WCSP consistencies

• Performance Evaluations

• Conclusion & Future Work

Radio Link Frequency Assignment ProblemsRadio Link Frequency Assignment Problems

• Soft Constrained Problems– Model: Weighted CSPs/COPs

• CELAR Problem [Cabon et al., 1999]:– Given a set S of radio links located

between pairs of sites

– Assign frequencies to S:• Prevent/Minimize interferences

– Involves two types of constraints

CELAR Problem: http://www.inra.fr/mia/T/schiex/Doc/CELAR.shtml

Communication from A to B

Communication from B to A

Technological constraints|fAB - fBA| = constant

between two sites

Constraints to prevent interferences:e.g. |fAB - fBC| > threshold

between links close to each otherfBA

Sometimes the problem is unfeasible…

Soft constraints to minimize interferences:e.g. max(0, threshold - |fAB - fBC|)

between links close to each otherfBA

insecure regionSubject to

control by adversaries

Minimize interferences?

• Nature of the problem:– Optimization: Minimizing interferences– Adversaries: Controlling parts of the links

• We can solve:1. Many COPs/WCSPs

• Each perform optimization on one combination of adversary’s frequency adjustment

2. Multiple QCSPs• Reducing into a decision problem

• Viewing in game theory:– Two-person zero-sum turn-based game

• Allis [1994] proposes three solving levels:– Ultra-weakly solved

• Best-worst case for a player

– Weakly solved• Strategies for a player to achieve his/her best against

all possible moves by his/her opponent

– Strongly solved• Strategies for a player to achieve his/her best against

all legal moves

Stronger

Our work

insecure region

Minimize interferences a

priori?

Assume worst case adversary

Finding frequency assignments for the worst possible case!

posteriori?

Soft Constraints

Minimax Weighted CSPsMinimax Weighted CSPs

Minimax Weighted CSPs

Weighted CSPs Quantified CSPs+

CSPs+Min/MaxQuantifiers

To avoid multiple sub-problems, we propose:

Minimax Weighted CSPsMinimax Weighted CSPs

• Minimax Weighted CSP [Lee et al., 2011]

– Variables:• x1, x2, x3

– Domains:• D1=D3 ={a,b,c}, D2 = {a,b}

– Soft Constraints:– Global Upper Bound k: 11– Valuation structure:

• ([0..k] , ⊕, ≤ )– Quantifier Sequence:

• Q1 = max,• Q2 = min,• Q3 = max

x1 Cost

x2 Cost

x2 x3 Cost

Soft constraintsx3 Cost

x1 x2 Cost

Unary constraint

Binary constraint

max x1

10 5 4 11 8 6 7 2 1 7 4 2 6 1 0 8 5 3

c a b c a b c a b ca b c a b

min x2

max x3

A-Cost for Sub-problemsA-Cost for Sub-problemsx1 Cost

x2 Cost

x2 x3 Cost

x3 Cost

x1 x2 Cost

4 0 5 1 0 = ⊕ ⊕ ⊕ ⊕ 10

max(10,5,4) = 10 11 7 7 6 8

min(10,11) = 10 7 6

max(10,7,6)=10

max x1

10 5 4 11 8 6 7 2 1 7 4 2 6 1 0 8 5 3

min x2

max x3

max x1

10 5 4 11 8 6 7 2 1 7 4 2 6 1 0 8 5 3

min x2

max x3

max(10,5,4) = 10 11 7 7 6 8

min(10,11) = 10 7 6

max(10,7,6)=10

A-Cost for Sub-problemsA-Cost for Sub-problems

Best-worst case (ultra-weak solution): {x1 = a, x2 = a, x3 = a}

A-cost for the problem: 10

Algorithms for Ultra-Weak Sol.Algorithms for Ultra-Weak Sol.Previous Work [Lee et al., 2011]:1. Alpha-beta prunings

– Maintains two bounds• Alpha lb: Best costs for max players• Beta ub: Best costs for min players

2. Suggest Two sufficient conditions to perform prunings and backtracks

Theorem:

For the set S of sub-problems P ’, where vi is assigned to xi:

∀P ’ ∈ S, A-cost(P ’) ≥ ub (Condition 1), or

∀P ’ ∈ S, A-cost(P ’) ≤ lb (Condition 2)

We can prune or backtrack according to the table:

A-cost(P ’) ≥ ub ≤ lbQi = min prune vi backtrac

Qi = max backtrack

prune vi

Computing the exact A-cost is hard! (NP-hard)

Sufficient Conditions for PruningsSufficient Conditions for Prunings

Corollary:

For the set of sub-problems P ’ obtained from P, where vi is assigned to xi:

A-cost(P ’) ≥ lbaf(P ,xi = vi) ≥ ub (Condition 1), or

A-cost(P ’) ≤ ubaf(P ,xi = vi) ≤ lb (Condition 2)

We can prune or backtrack according to the table below:

lbaf(P ,xi = vi) ≥ ub ubaf(P ,xi = vi) ≤ lb

Qi = min prune vi backtrack

Qi = max backtrack prune vi

How to compute efficiently?

ConsistenciesConsistencies• Local consistency enforcement

– Make implicit costs information explicit• E.g. bounds, prunings/backtracks

• Consistencies composes of 3 parts: 1. Lower bound estimation: lbaf(P ,xi = vi)

– NC & AC version

2. Upper bound estimation: ubaf(P ,xi = vi) – Two dualities: DC & DQ

3. Strengthening lower & upper estimation by projections/extensions– Adopt WCSP consistencies: NC*, AC*, FDAC*

– Naming convention:– DC-NC[proj-NC*], DQ-AC[proj-FDAC*]

Lower Bound EstimationLower Bound Estimation

• Lower bound estimation: lbaf(P ,xi = vi)

• Consider a simplified problem:– Only unary constraints, i.e. no binary

Lemma:

The A-cost of an MWCSP P with only unary constraints is equal to:

Q1C1 ⊕ Q2C2 ⊕ … ⊕ QnCn

x1 Cost

x2 Cost

x3 Cost

Q1 = max Q2 = min Q3 = max

⊕ ⊕ = 8

• Lower bound (NC version): nclb(P ,xi = vi)

• Example:– nclb(P ,x1 = b)

– nclb(P ,x2 = a)

x1 Cost

x2 Cost

x3 Cost

x1 Cost

x2 Cost

x3 Cost

Q1 = max Q2 = min Q3 = maxFor all sub-problems

where x2 = a

CØ ⊕ (⊕j<i min Cj ) ⊕ Ci(vi) ⊕ (⊕i<j Qj Cj )

• Lower bound (AC version): aclb[Cij](P ,xi = vi)

– nclb(P ,xi = vi) + a binary constraint Cij

• Example:– aclb(P ,x1 = b)

x1 Cost

x2 Cost

x3 Cost

x1 x2 Cost

a a 17

a b 13

b a 11

b b 16

Upper Bound EstimationUpper Bound Estimation• Upper bound ubaf(): Duality of Constraints

Definition of Dual Problem:Given an MWCSP P = (X,D,C,Q,k).

The dual problem of P is P Τ = (X,D,C Τ,Q Τ,k) where:

1. Quantifier: Qi = max → Q Τi= min & Qi = min → Q Τ

i= max

2. Cost: For a complete assignment l, cost(l) = -1*costΤ(l)

Construction Method:

x1 Cost

b 1x1 x2 Cost

a a -7

a b -3

b a -1

b b -6x1 Cost

x1 x2 Cost

Q1 = max Q2 = min

Q2 = maxQ1 = min

Upper Bound EstimationUpper Bound Estimation• Upper bound: Duality of Constraints (DC)

– Corollary:

A lbaf(P Τ,xi = vi) on the dual multiply by -1 is an ubaf(P ,xi = vi) for the original problem

max x1

c a b c a b c a b ca b

min x2

max x3

2 10 1 11

0 1 0 0 11 100 2 1 0 10 10

Upper bound ub : 10Lower bound lb : 1

min x1

c a b c a b c a b ca b

max x2

min x3

-2 -10 -1 -11

0 -1 0 0 -11 100 -2 -1 0 -10 -10

Upper bound ub : -1Lower bound lb : -10

lbaf(P Τ,x2 = b) ≤ -11→ -1 * lbaf(P Τ,x2 = b) ≥ 11

Upper Bound EstimationUpper Bound Estimation• Following the corollary:

• We implement ubaf(P ,xi = vi) by:

– NC version: nclb(P Τ ,xi = vi)

– AC version :aclb[Cij] (P Τ ,xi = vi)

• Advantage for Duality of Constraints (DC)– Reuse the same lbaf()– New lbaf() can be used as ubaf()

• Upper bound: Duality of Quantifiers (DQ)

• Creating/Writing new ubaf() via:• Flipping quantifiers of existing lbaf()

• Example:– nclb(P ,x2 = a)

– ncub(P ,x2 = a)

Upper Bound EstimationUpper Bound Estimation

x1 Cost

x2 Cost

x3 Cost

x1 Cost

x2 Cost

x3 Cost

For all sub-problems where x2 = a,

guarantee a lower bound

For all sub-problems where x2 = a,

guarantee an upper bound

• Upper bound: Duality of Quantifiers (DQ)• Creating/Writing new ubaf() via:

• Flipping quantifiers of existing lbaf()

• Immediate attempt:

• Problem: Binary constraints add costs!

CØ ⊕ (⊕j<i min Cj ) ⊕ Ci(vi) ⊕ (⊕i<j Qj Cj )

min to max

CØ ⊕ (⊕j<i max Cj ) ⊕ Ci(vi) ⊕ (⊕i<j Qj Cj )

• Upper bound: Duality of Quantifiers (DQ)• Creating/Writing new ubaf() via:

• Flipping quantifiers of existing lbaf()

• To fix:

• Further add maximum costs for constraints which are not covered in the function

• For implementation: 1. We pre-compute and add these maximum costs

before search2. We maintain the added sum during search

CØ ⊕ (⊕j<i max Cj ) ⊕ Ci(vi) ⊕ (⊕i<j Qj Cj )

ConsistenciesConsistencies

• We have methods to compute:– lbaf(): NC & AC version

• Standard approximation analysis

– ubaf(): Two dualities• Inspired from QCSP consistencies and algorithms• [Bordeaux and Monfroy, 2002]• [Gent et al., 2005]

ConsistenciesConsistencies• Can we further strengthen both estimation

functions?

• Utilize projections & extensions conditions– WCSP consistencies: NC*, AC*, and FDAC* [Cooper et al., 2010]

• For Duality of Constraints (DC) consistencies– Conditions are enforced in both the original and

dual problem

Performance EvaluationPerformance Evaluation• Compare and study different consistency notions

– DQ-NC[proj-NC*], DQ-AC[proj-AC*], DQ-AC[proj-FDAC*]– DC-NC[proj-NC*], DC-AC[proj-AC*], DC-AC[proj-FDAC*]

• Benchmarks:1. Randomly Generated Problems2. Graph Coloring Game3. Generalized Radio Link Frequency Assignment Problem

• Each set of parameters:– 20 instances & taking average result– If there are unsolved instances, we state the #solved

besides runtime

• Compare our results against:– Alpha-beta pruning– QeCode: A solver for solving QCOP+

Randomly Generated Problems [Lee et al.,2011]– (n,d,p): (# of vars, domain size, constraint density)– Integer costs of a binary constraint

• Generated uniformly in [0…30] for each tuple of assignments– Probability of 50%: a min (max resp.) quantifier– Time limit: 900s

Performance EvaluationPerformance Evaluation

Stronger projection/extensionWe may:•Strengthening lbaf() (ubaf() resp.)•Weakening ubaf() (lbaf() resp.)

Duality of ConstraintsExtracts costs from two different copies of constraints (original and dual) and resolve the issue

ConclusionConclusion

• Define and implement various consistency notions for MWCSPs1. Lower bound by costs estimations2. Upper bound by duality principle3. Strengthening lower & upper bound

estimation functions: • Adopting projection/extension conditions in

WCSP consistencies

• Discussions on our solving techniques on the two other stronger solutions

Related WorkRelated Work

• Related CSP frameworks tackling adversaries:– Stochastic CSPs [Walsh, 2002]– Adversarial CSPs [Brown et al., 2004]– QCSP+/QCOP+ [Benedetti et al., 2007]

[Benedetti et. al, 2008]

• Other related frameworks:– Bi-level Programming– Plausibility-Feasibility-Utility framework

[Pralet et al., 2009]

Future WorkFuture Work

• Consistency algorithms:– High-arity Soft Table Constraints, and– Global Soft Constraints

• Theoretical comparisons on different consistency notions

• Algorithms tackling stronger solutions• Online & Distributed Algorithms• Value ordering heuristics

– ICTAI 2012

Q & AQ & A

Graph Coloring Game [Lee et al.,2011]– Two player zero-sum games

• Writing numbers of nodes– (v,c,d): (# of vertices, # allowed numbers, edge density)– Turns:

• Odd/Even numbered turns - Player 1/Player 2 → A series of alternating quantifiers

– Time limit: 900s

1 Consistencies for Ultra-Weak Solutions in Minimax Weighted CSPs Using the Duality Principle Arnaud...

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