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1 Consistencies for Ultra-Weak Solutions in Minimax Weighted CSPs Using the Duality Principle Arnaud Lallouet 1, Jimmy H.M. Lee 2, and Terrence W.K. Mak 2 1 Université de Caen, GREYC, Caen, France 2 The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

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2 Introduction Motivation Minimax Weighted CSPs –Ultra-weakly solved, weakly solved, and strongly solved Consistency Techniques 1.Lower Bound formulations 2.Upper Bounds using duality principle 3.Strengthening lower and upper bounds by adopting WCSP consistencies Performance Evaluations Conclusion & Future Work

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3 Radio 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:

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4 Radio Link Frequency Assignment Problems A BC D Communication from A to B Communication from B to A

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5 Radio Link Frequency Assignment Problems A BC D Technological constraints |f AB - f BA | = constant f AB f BA between two sites

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6 Radio Link Frequency Assignment Problems A BC D Constraints to prevent interferences: e.g. |f AB - f BC | > threshold f AB f BC between links close to each other f BA f CB Sometimes the problem is unfeasible…

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7 Radio Link Frequency Assignment Problems A BC D Soft constraints to minimize interferences: e.g. max(0, threshold - |f AB - f BC |) f AB f BC between links close to each other f BA f CB

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8 Radio Link Frequency Assignment Problems A BC D f BD f DB insecure region Subject to control by adversaries Minimize interferences?

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9 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 Radio Link Frequency Assignment Problems

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10 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 Radio Link Frequency Assignment Problems Our work

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11 Radio Link Frequency Assignment Problems A BC D f BD f DB insecure region Minimize interferences a priori? Assume worst case adversary Finding frequency assignments for the worst possible case! Minimize interferences a posteriori?

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12 Soft Constraints Minimax Weighted CSPs ≈ Weighted CSPs Quantified CSPs + = CSPs+ Min/Max Quantifiers + To avoid multiple sub-problems, we propose:

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13 Minimax Weighted CSPs Minimax Weighted CSP [Lee et al., 2011] –Variables: x 1, x 2, x 3 –Domains: D 1 = D 3 ={ a, b, c }, D 2 = { a, b } –Soft Constraints: –Global Upper Bound k : 11 –Valuation structure: ([0.. k ], ⊕, ≤ ) –Quantifier Sequence: Q 1 = max, Q 2 = min, Q 3 = max x1x1 Cost a 4 b 0 c 0 x2x2 a 0 b 2 x2x2 x3x3 aa 1 ab 1 ac 0 ba 0 bb 2 bc 0 Soft constraints x3x3 Cost a 5 b 0 c 0 x1x1 x2x2 aa 0 ab 0 ba 1 bb 0 ca 0 cb 1 Unary constraint Binary constraint

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14 A-Cost for Sub-problems x1x1 Cost a 4 b 0 c 0 x2x2 a 0 b 2 x2x2 x3x3 aa 1 ab 1 ac 0 ba 0 bb 2 bc 0 x3x3 a 5 b 0 c 0 x1x1 x2x2 aa 0 ab 0 ba 1 bb 0 ca 0 cb 1 4 ⊕ 0 ⊕ 5 ⊕ 1 ⊕ 0 = 10

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16 A-Cost for Sub-problems Best-worst case (ultra-weak solution): { x 1 = a, x 2 = a, x 3 = a} A-cost for the problem: 10

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17 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 v i is assigned to x i : ∀ 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 ≤ lb Q i = minprune v i backtrack Q i = maxbacktrackprune v i Computing the exact A-cost is hard! (NP-hard)

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18 Sufficient Conditions for Prunings Corollary: For the set of sub-problems P ’ obtained from P, where v i is assigned to x i : A-cost( P ’) ≥ lbaf ( P, x i = v i ) ≥ ub (Condition 1), or A-cost( P ’) ≤ ubaf ( P, x i = v i ) ≤ lb (Condition 2) We can prune or backtrack according to the table below: lbaf ( P, x i = v i ) ≥ ububaf ( P, x i = v i ) ≤ lb Q i = minprune v i backtrack Q i = maxbacktrackprune v i How to compute efficiently?

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19 Consistencies Local consistency enforcement –Make implicit costs information explicit E.g. bounds, prunings/backtracks Consistencies composes of 3 parts: 1.Lower bound estimation: lbaf ( P, x i = v i ) –NC & AC version 2.Upper bound estimation: ubaf ( P, x i = v i ) –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*]

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20 Lower Bound Estimation Lower bound estimation: lbaf ( P, x i = v i ) 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: Q 1 C 1 ⊕ Q 2 C 2 ⊕ … ⊕ Q n C n x1x1 Cost a 4 b 1 c 2 x2x2 a 8 b 6 c 1 x3x3 a 1 b 3 Q 1 = max Q 2 = min Q 3 = max ⊕⊕ = 8

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21 Lower bound (NC version): nc lb ( P, x i = v i ) Example: – nc lb ( P, x 1 = b ) – nc lb ( P, x 2 = a ) Lower Bound Estimation x1x1 Cost a 4 b 1 c 2 x2x2 a 8 b 6 c 1 x3x3 a 1 b 3 Q 1 = max Q 2 = min Q 3 = max x1x1 Cost a 4 b 1 c 2 x2x2 a 8 b 6 c 1 x3x3 a 1 b 3 Q 1 = max Q 2 = min Q 3 = max For all sub-problems where x 2 = a C Ø ⊕ ( ⊕ j < i min C j ) ⊕ C i ( v i ) ⊕ ( ⊕ i < j Q j C j )

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22 Lower Bound Estimation Lower bound (AC version): ac lb [ C ij ]( P, x i = v i ) – nc lb ( P, x i = v i ) + a binary constraint C ij Example: – ac lb ( P, x 1 = b ) x1x1 Cost a 4 b 1 x2x2 a 8 b 6 Q 1 = max Q 2 = min Q 3 = max x3x3 Cost a 4 b 1 c 2 x1x1 x2x2 aa 5 ab 3 ba 2 bb 9 x1x1 x2x2 aa 17 ab 13 ba 11 bb 16

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23 Upper 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: Q i = max → Q Τ i = min & Q i = min → Q Τ i = max 2.Cost: For a complete assignment l, cost(l) = -1*cost Τ (l) Construction Method: x1x1 Cost a 4 b 1 x1x1 x2x2 aa -7 ab -3 ba bb -6 x1x1 Cost a -4 b x1x1 x2x2 Cost aa 7 ab 3 ba 1 bb 6 Q 1 = max Q 2 = min Q 2 = max Q 1 = min

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24 Upper Bound Estimation Upper bound: Duality of Constraints (DC) –Corollary: A lbaf ( P Τ, x i = v i ) on the dual multiply by -1 is an ubaf ( P, x i = v i ) for the original problem lbaf ( P Τ, x 2 = b ) ≤ -11 → -1 * lbaf ( P Τ, x 2 = b ) ≥ 11

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25 Upper Bound Estimation Following the corollary: We implement ubaf ( P, x i = v i ) by: –NC version: nc lb ( P Τ, x i = v i ) –AC version : ac lb [ C ij ] ( P Τ, x i = v i ) Advantage for Duality of Constraints (DC) –Reuse the same lbaf () –New lbaf () can be used as ubaf ()

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26 Upper bound: Duality of Quantifiers (DQ) Creating/Writing new ubaf() via: Flipping quantifiers of existing lbaf () Example: – nc lb ( P, x 2 = a ) – nc ub ( P, x 2 = a ) Upper Bound Estimation x1x1 Cost a 4 b 1 c 2 x2x2 a 8 b 6 c 1 x3x3 a 1 b 3 Q 1 = max Q 2 = min Q 3 = max x1x1 Cost a 4 b 1 c 2 x2x2 a 8 b 6 c 1 x3x3 a 1 b 3 Q 1 = max Q 2 = min Q 3 = max For all sub-problems where x 2 = a, guarantee a lower bound For all sub-problems where x 2 = a, guarantee an upper bound

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27 Upper bound: Duality of Quantifiers (DQ) Creating/Writing new ubaf() via: Flipping quantifiers of existing lbaf () Immediate attempt: Problem: Binary constraints add costs! Upper Bound Estimation C Ø ⊕ ( ⊕ j < i min C j ) ⊕ C i ( v i ) ⊕ ( ⊕ i < j Q j C j ) min to max C Ø ⊕ ( ⊕ j < i max C j ) ⊕ C i ( v i ) ⊕ ( ⊕ i < j Q j C j )

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28 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 search 2.We maintain the added sum during search Upper Bound Estimation C Ø ⊕ ( ⊕ j < i max C j ) ⊕ C i ( v i ) ⊕ ( ⊕ i < j Q j C j )

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29 Consistencies 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]

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30 Consistencies 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

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31 Performance 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 Problems 2.Graph Coloring Game 3.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+

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32 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 Evaluation Stronger projection/extension We may: Strengthening lbaf () ( ubaf () resp.) Weakening ubaf () ( lbaf () resp.) Duality of Constraints Extracts costs from two different copies of constraints (original and dual) and resolve the issue

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33 Conclusion Define and implement various consistency notions for MWCSPs 1.Lower bound by costs estimations 2.Upper bound by duality principle 3.Strengthening lower & upper bound estimation functions: Adopting projection/extension conditions in WCSP consistencies Discussions on our solving techniques on the two other stronger solutions

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34 Related 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]

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35 Future 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

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36 Q & A

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37 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 Performance Evaluation Similar results

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38 Generalized Radio Link Frequency Assignment –Designed according to two CELAR sub-instances –Minimize interference beforehand –( i, n, d, r ): (CELAR sub-instance index, # of links, # of allowed frequencies, ratio of adversary links) –Time limit: 7200s Performance Evaluation Projection/extension in FDAC* Slightly improves the search only Quantifier info. not considered

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39 Algorithms for Stronger Sol. Solution Size –Ultra-weak: O( n ) –Weak: O(( n - m ) d m ) –Strong: O( d n ) Where: –# of variables: n –# of adversary variables: m –Maximum domain size: d Ultra-weak solutions are linear

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40 Algorithms for Stronger Sol. Pruning Conditions –A sound pruning condition when solving a weaker solution may not hold in stronger ones Reason: –Removal of the assumption of optimal/perfect plays Theorem: For the set S of sub-problems P ’, where v i is assigned to x i : ∀ 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 ≤ lb Q i = minprune v i backtrack Q i = maxbacktrackprune v i Invalid: When finding weak solutions Adversary min player Invalid: When finding weak solutions Adversary max player

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41 Relations with complexity classes Weighted CSPs: –NP-hard Quantified CSPs: –PSPACE-complete Theorem: –Finding the truthfulness of QCSPs can be reduced (by Karp reduction) to finding the A-Cost of MWCSPs → MWCSPs: –PSPACE-hard Assumption: P ≠ PSPACE

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42 Transforming MWCSP to QCOP Theorem: –An MWCSP P can be transformed into a QCOP P ’. The A-cost of P can be found by solving the optimal strategy of P ’. Proof (Sketch): –Using ‘ Soft As Hard ’ approach [Petit et. al, 2001] Transform soft constraints into hard constraints

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43 Graph Coloring Game (GCG) Maximize costs Player A Player B Minimize costs Owned by A Owned by B How do they play the game?

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44 Graph Coloring Game 1/A 2/B 3/A4/B 5/B6/A 7/B 8/A Player A Player B Write number 3 on node 1 Write number 6 on node Game Cost: |3 - 6| = 3

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45 Graph Coloring Game 1/A 2/B 3/A4/B 5/B6/A 7/B 8/A Player A 3 6 so on… Maximize costs What should I do ? Place 0 Gain a cost of 3 Place 3 No cost gain

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46 Graph Coloring Game 1/A 2/B 3/A4/B 5/B6/A 7/B 8/A Final Game Cost: When the game terminates… What we want to study…

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47 1/A 2/B 3/A4/B 5/B6/A 7/B 8/A 1/A 3/A 6/A 8/A /A 3/A 6/A 8/A so on… Modeled and solved by COP/ Weighted CSP 1/A 3/A 6/A 8/A Modeled and solved by COP/ Weighted CSP Approach 1:

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48 Modeling GCG 1/A 2/B 3/A4/B 5/B 6/A 7/B8/A 1.Guess a threshold: 56 2.Generate a Quantified CSP [Bordeaux and Monfroy, 2002] which asks: –Can player A finds numbers against player B ’ s moves –s.t. Player A gets costs < 56? Approach 2:

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49 Modeling GCG Approach 1: –Number of COPs/ Weighted CSPs constructed is exponential to the possible numbers player B can write Approach 2: –Generate Quantified CSPs based on the objective function

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50 x1x1 Cost a 4 b 1 Q 1 = max x2x2 Cost a 8 b 6 Q 2 = min Q 3 = max x3x3 Cost a 4 b 1 c 2 x1x1 x2x2 aa 5 ab 3 ba 10 bb 9 x1x1 x2x2 Cost aa 17 ab 13 ba 19 bb 16 Q 1 = max Q 2 = min Q 3 = max x3x3 Cost a 4 b 1 c 2 NC AC x1x1 Cost a 4 b 1 x2x2 a 8 b 6 x1x1 x2x2 aa 5 ab 3 ba 10 bb 9 Merge

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51 x1x1 Cost a 4 b 1 x1x1 x2x2 aa 7 ab 3 ba 1 bb 6 DC Original ProblemDual Problem Q 1 = max Q 2 = min Q 2 = max Q 1 = min x1x1 Cost a -4 b x1x1 x2x2 Cost aa -7 ab -3 ba bb -6

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