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

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

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:

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

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

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…

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

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

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

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

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?

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

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

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

15

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

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)

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?

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*]

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

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 )

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

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

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

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 ()

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

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 )

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 )

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]

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

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+

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

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

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]

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

36 Q & A

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

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

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

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

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

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

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?

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

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

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…

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:

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:

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

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

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