Michael Heusch - IntCP 2006 Modeling and solving of a radio antennas deployment support application with discrete and interval constraints

Michael Heusch - IntCP 2006 Outline of the talk Presentation of the application Modeling with discrete and interval constraints Defining search heuristics Modeling the problem with the distn constraint Experimental results on solving the progressive deployment problem

Michael Heusch - IntCP 2006 Presentation of the LocRLFAP Informal description of the de radio antennas deployment problem : Constraints involved : Distance between frequencies depends on distance between antennas

Michael Heusch - IntCP 2006 Minimal and maximal distances between antennas Presentation of the LocRLFAP Informal description of the de radio antennas deployment problem : Difficulties : Hybrid combinatorial optimisation problem non-linear continuous constraints Constraints involved : Distance between frequencies depends on distance between antennas

Michael Heusch - IntCP 2006 Specification of the problem Formulation as a constrained optimisation problem: Data Fixed set of antennas (transmitter-receiver) Dispatched on n sites {P 1, …, P n } The links to establish is known in advance Variables of the problem: A solution associates one frequency to each antenna and a position to each site P i = (X i,Y i ): Position of a site f i,j : frequency allocated to the link from P i to P j Optimisation problem: Minimise the maximal frequency used

Michael Heusch - IntCP 2006 Constraints of the problem discrete constraints: Compatibility between antennas Forbidden frequencies continuous constraints Maximum distance between antennas (range) Minimum distance between the antennas (security, interference) mixed constraints Compatibility between the allocation and the deployment

Michael Heusch - IntCP 2006 Comparing the RLFAP/LocRLFAP with 5 sites RLFAPLocRLFAP

Michael Heusch - IntCP 2006 Comparing the RLFAP/LocRLFAP with 5 sites RLFAPLocRLFAP dist² (Si,Sj) = Σ i (Xi - Xj)²

Michael Heusch - IntCP 2006 Comparing the RLFAP/LocRLFAP

Michael Heusch - IntCP 2006 Hybrid solving with collaborating solvers Original approach Modeling with the finite domain constraint solver Eclair Full discretization of the problem Modeling three types of constraints Discrete constraints Continuous constraints Mixed constraints

Michael Heusch - IntCP 2006 Discrete constraints Co-site transmitter-receiver interference constraints: Duplex distance constraints for each bidirectional link Forbidden portions in the frequency range

Michael Heusch - IntCP 2006 Continuous and mixed constraints Elementary continuous constraints: dist²(P i,P j ) > m ij ², for all i<j dist²(P i,P j ) < M ij ², if there exists a radio link between P i and P j Mixed constraints: Compatibility constraints If dist(P i,P j )< d 1, great interference If d 1 <= dist(P i,P j )< d 2, limited interference Expression with elementary constraints { dist(P i,P j ) Δ 1 }, (i,j,k), i≠j, i≠k, j≠k { dist(P i,P j ) Δ 2 }, (i,j,k), i≠j, i≠k, j≠k d1d1 d2d2

Michael Heusch - IntCP 2006 Test set Full deployment of networks with 5 to 10 sites RLFAP LocRLFAP

Michael Heusch - IntCP 2006 Progressive deployment of networks with 9 and 10 sites P P P P P P P P P P

Michael Heusch - IntCP 2006 Solving with elementary constraints Full deployment in both models

Michael Heusch - IntCP 2006 Improvements to the search algorithm Usage of a naïve Branch & Bound with: Distinction of the type of variables The problem is under-constrained on positions Branch on disjunctions? Branch first on constraints entailing a strong interdistance? Variable selection heuristics minDomain min(dom/deg) minDomain+maxConstraints

Michael Heusch - IntCP 2006 Results with minDomain+maxConstraints 9 sites 10 sites 99% of the backtracks are performed on the continuous part of the search tree A bit less backtracks on the hybrid model Hybrid solving is 1 to 3 times slower Progressive deployment in both models

Michael Heusch - IntCP 2006 Introducing the distn global constraint distn ([P 1, …, P n ], V) P i = X i x Y i : Cartesian coordinates of the point p i V i,j : distance to maintain between P i and P j distn(p 1, …, p n ], v) satisfied if and only if dist(p i,p j ) = v i,j Filtering algorithm uses geometric approximation techniques

Michael Heusch - IntCP 2006 Applications of the constraint Molecular conformation Robotics Antennas deployment

Michael Heusch - IntCP 2006 Using distn in the model Second formulation of the problem with the global constraint: Simple continuous constraints Introduction of a matrix {V i,j } of distance variables: Domain(V i,j )=[m i,j, M i,j ] Expression of the set of min and max distance constraints: distn([P 1, …, P n ], V) Expression of the mixed « distant compatibility » disjunctions distn([P 1, …, P n ], V) { V ij Δ 1 }, (i,j,k), i≠j, i≠k, j≠k { V ij Δ 2 }, (i,j,k), i≠j, i≠k, j≠k

Michael Heusch - IntCP 2006 Results using distn (9 sites) Simple heuristicsAdvanced heuristics hybrid model / discrete model comparison: 1.8 times slower 1.5 times more backtracks Similar performance of both models wrt. simple model, distn divides by 2 the nb. of backtracks

Michael Heusch - IntCP 2006 Results using distn (10 sites) Simple heuristicsAdvanced heuristics hybrid model / discrete model comparison: 4 additional instances are solved Performance on the solved instances: 63% less backtracks All instances are solved

Michael Heusch - IntCP 2006 Quality of solutions 9 sites10 sites

Michael Heusch - IntCP 2006 Conclusion and perspectives We showed the relevance of coupling discrete and continuous constraints Obtain solution of greater quality Better performance when solving Independence w.r.t. the discretization step Validation on one industrial application Key points Definition of appropriate search heuristics Usage of the distn global constraint

Michael Heusch - IntCP 2006 Perspectives on the application Validation on instances of greater size Take forbidden zone constraints into account Provide deployment zones using polygons

Michael Heusch - IntCP 2006 Other approaches for solving the RLFAP Other approaches for solving the classical RLFAP Graph coloring Branch & Cut CP LDS [Walser – CP96] Russian Doll Search [Schiex et. al - CP97] Heuristics Tabou [Vasquez – ROADEF 2001] Simulated annealing, evolutionary algorithms… Motivations for an approach using CP Robustness wrt modification of the constraints of the problem

Michael Heusch - IntCP 2006 Sketch of distn’s filtering algorithm

Michael Heusch - IntCP 2006 Filtering algorithm on polygons Method using polygons for representing domains Theorem by K. Nurmela et P. Östergård (1999) M. Markót et T. Csendes: A New Verified Optimization Technique for the ``Packing Circles in a Unit Square'' Problems. SIAM Journal of Optimization, 2005 pi2pi2 p i k-1 pikpik pi1pi1 PiPi PjPj

Michael Heusch - IntCP 2006 Filtering algorithm on polygons P1 P2

Michael Heusch - IntCP 2006 Filtering algorithm on polygons P1 P2

Michael Heusch - IntCP 2006 Filtering algorithm on polygons P1 P2

Michael Heusch - IntCP 2006 Interval extension of the algorithm P1 P2

Michael Heusch - IntCP 2006 Filtering algorithm of distn P1 P2 Adjusting bounds of the distance variables