CP-AI-OR-02 Gomes & Shmoys 1 The Promise of LP to Boost CSP Techniques for Combinatorial Problems Carla P. Gomes David Shmoys Department of Computer Science School of Operations Research and Industrial Engineering Cornell University CP-AI-OR 2002

CP-AI-OR-02 Gomes & Shmoys 2 Motivation Increasing interest in combining Constraint Satisfaction Problem (CSP) formulations and Linear Programming (LP) based techniques for solving hard computational problems. Successful results for solving problems that are a mixture of linear constraints – where LP excels – and combinatorial constraints – where CSP excels. However, surprisingly difficult to successfully integrate LP and CSP based techniques in a purely combinatorial setting. Example: Satisfiability

CP-AI-OR-02 Gomes & Shmoys 3 Power of Randomization Randomization is magic --- we have some intuitions why it works.

CP-AI-OR-02 Gomes & Shmoys 4 Challenge Can the LP provide information that is not easily uncovered by CSP propagation and inference techniques with a much smaller computational cost? Key Issue:

CP-AI-OR-02 Gomes & Shmoys 5 Outline of Talk A purely combinatorial problem domain Problem formulations CSP formulation LP formulations –Assignment formulation –Packing Formulation Randomization Heavy-tailed behavior in combinatorial search Approximation Algorithms for QCP A Hybrid Complete CSP/LP Randomized Rounding Backtrack Search Approach Empirical Results Conclusions

CP-AI-OR-02 Gomes & Shmoys 6 A purely combinatorial problem domain

CP-AI-OR-02 Gomes & Shmoys 7 Quasigroups or Latin Squares: An Abstraction for Real World Applications Gomes and Selman 97 Quasigroup or Latin Square (Order 4) A Quasigroup or Latin Square is an n- by-n matrix such that each row and column is a permutation of the same n colors 68% holes The Quasigroup or Latin Square Completion Problem (QCP):

CP-AI-OR-02 Gomes & Shmoys 8 Complexity Better characterization beyond worst case? Critically constrained area 42%50%20% Complexity of Latin Square Completion EASY AREA 35%42%50% Time: QCP is NP-Complete

CP-AI-OR-02 Gomes & Shmoys 9 Problem Formulations

CP-AI-OR-02 Gomes & Shmoys 10 QCP as a CSP Variables - Constraints - row column

CP-AI-OR-02 Gomes & Shmoys 11 Pure CSP approaches solve QCP instances up to order 33 relatively well. Higher orders (e.g.,critically constrained area) are beyond the reach of CSP solvers.

CP-AI-OR-02 Gomes & Shmoys 12 LP Formulations

CP-AI-OR-02 Gomes & Shmoys 13 Assignment Formulation Cubic representation of QCP Columns Rows Colors

CP-AI-OR-02 Gomes & Shmoys 14 QCP Assignment Formulation Row/color line Column/color line Row/column line Max number of colored cells

CP-AI-OR-02 Gomes & Shmoys 15 Packing formulation Max number of colored cells in the selected patterns s.t. one pattern per family a cell is covered at most by one pattern Families of patterns (partial patterns are not shown)

CP-AI-OR-02 Gomes & Shmoys 16 Packing Formulation Definitions: Compatible matching for color k – any extension of a partial solution with respect to color k. family of all compatible matchings for color k - variable denoting each compatible matching M in |M| number of colored cells in a compatible matching

CP-AI-OR-02 Gomes & Shmoys 17 QCP Packing Formulation one pattern per color at most one pattern covering each cell Max number of colored cells

CP-AI-OR-02 Gomes & Shmoys 18 Any feasible solution to the packing LP relaxation is also a solution to the assignment LP relaxation  The value of the assignment relaxation is at least the bound implied by the packing formulation => the packing formulation provides a tighter upper bound than the assignment formulation  Limitation – size of formulation is exponential in n. (one may apply column generation techniques)

CP-AI-OR-02 Gomes & Shmoys 19 Randomization

CP-AI-OR-02 Gomes & Shmoys 20 Background Stochastic strategies have been very successful in the area of local search. Simulated annealing Genetic algorithms Tabu Search Walksat and variants. Limitation: inherent incomplete nature of local search methods.

CP-AI-OR-02 Gomes & Shmoys 21 Randomized variable and/or value selection – lots of different ways. Example: randomly breaking ties in variable and/or value selection. Compare with standard lexicographic tie-breaking. Note: No problem maintaining the completeness of the algorithm! Randomized backtrack search

CP-AI-OR-02 Gomes & Shmoys 22 Sample mean Erratic Behavior of Mean Number runs Empirical Evidence of Heavy-Tails (*) no solution found - reached cutoff: 2000 Time:(*)3011(*)7 Easy instance – 15 % preassigned cells Gomes et al Median = 1 !

CP-AI-OR-02 Gomes & Shmoys 23 Decay of Distributions Standard Exponential Decay e.g. Normal: Heavy-Tailed Power Law Decay e.g. Pareto-Levy: Power Law Decay Standard Distribution (finite mean & variance) Exponential Decay Infinite variance, infinite mean

CP-AI-OR-02 Gomes & Shmoys 24 Exploiting Heavy-Tailed Behavior Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc. Consequence for algorithm design: Use restarts or parallel / interleaved runs to exploit the extreme variance performance. Restarts eliminate heavy-tailed behavior 70% unsolved 1-F(x) Unsolved fraction Number backtracks (log) 250 (62 restarts) 0.001% unsolved

CP-AI-OR-02 Gomes & Shmoys 25 Randomized backtrack search – active research area -> very effective when combined with no-good learning! solved open problems different variants of randomization/restarts, e.g., biased probability function for variable/value selection, “jumping” to different points in the search tree State-of-the-art Sat Solvers incorporate randomized restarts: ChaffRelsat GraspGoldberg’s Solver QuestSatZ, SATO, … used to verify 1/7 of a Alpha chip (Pentium IV)

CP-AI-OR-02 Gomes & Shmoys 26 Randomized Rounding

CP-AI-OR-02 Gomes & Shmoys 27 Randomized Rounding Solve a relaxation of combinatorial problem; Use randomization to go from the relaxed version to the original problem;

CP-AI-OR-02 Gomes & Shmoys 28 Randomized Rounding of a 0-1 Integer Programming Solve the LP relaxation; Interpret the resulting fractional solution as providing the probability distribution over which to set the variables to 1. Note: The resulting solution is not guaranteed to be feasible. Nevertheless, good intuition of why randomized rounding is a powerful tool.

CP-AI-OR-02 Gomes & Shmoys 29 LP Based Approximations

CP-AI-OR-02 Gomes & Shmoys 30 Approximation Algorithm Assumption: Maximization problem the value of the objective function delivered by algorithm A for input instance I. the optimal value of the objective function for input instance I. The performance ratio of an algorithm A is the infimum (supremum, for min) over all I of the ratio A is an - approximation algorithm if it has performance ratio at least (at most, for min)

CP-AI-OR-02 Gomes & Shmoys 31 Approximation Algorithm For randomized algorithms we replace by in the definition of performance ratio. (expectation is taken over the random choices performed by the algorithm). Note: the only randomness in the performance guarantee stems from the randomization of the algorithm itself, and not due to any probabilistic assumptions on the instance. In general, the term approximation algorithm will denote a polynomial-time algorithm.

CP-AI-OR-02 Gomes & Shmoys 32 QCP Assignment Formulation Row/color line Column/color line Row/column line Max number of colored cells

CP-AI-OR-02 Gomes & Shmoys 33 Approximations Based on Assignment Formulation Kumar et. al 99  Algorithm1 - at each iteration, the algorithm solves the LP relaxation and sets to 1 the variable closest to 1. This is an 1/3 approximation algorithm. Algorithm 2 – at each iteration, the algorithm selects a compatible matching for a color, for which the LP relaxation places the greatest total weight. This is an 1/2 approximation algorithm. Experimental evaluation -> problems up to order 9.

CP-AI-OR-02 Gomes & Shmoys 34 QCP Packing Formulation one compatible matching per color at most one compatible matching covering each cell Max number of colored cells

CP-AI-OR-02 Gomes & Shmoys 35 Approximation Based on Packing Formulation Randomization scheme: for each color K choose a pattern with probability (so that some matching is selected for each color) As a result we have a pattern per color. Problem: some patterns may overlap, even though in expectation, the constraints imply that the number of matchings in which a cell is involved is 1.

CP-AI-OR-02 Gomes & Shmoys 36 Packing formulation Max number of colored cells in the selected patterns s.t. one pattern per family a cell is covered at most by one pattern

CP-AI-OR-02 Gomes & Shmoys 37 (1-1/e)- Approximation Based on Packing Formulation Let’s assume that the PLS is completable Z*=h What is the expected number of cells uncolored by our randomized procedure due to overlapping conflicts? From we can compute So, the desired probability corresponds to the probability of a cell not be colored with any color, i.e.:

CP-AI-OR-02 Gomes & Shmoys 38 (1-1/e)- Approximation Based on Packing Formulation This expression is maximized when all the are equal therefore: So the expected number of uncolored cells is at most  at least holes are expected to be filled by this technique.

CP-AI-OR-02 Gomes & Shmoys 39 Putting all the pieces together

CP-AI-OR-02 Gomes & Shmoys 40 CSP Model LP Model + LP Randomized Rounding Heavy-tails We want to maintain completeness How do we put all the pieces together? A HYBRID COMPLETE CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH

CP-AI-OR-02 Gomes & Shmoys 41 HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH Central features of algorithm: Complete Backtrack search algorithm It maintains two formulations CSP model Relaxed LP model LP Randomized rounding  for setting values at the top of the tree CSP + LP inference

CP-AI-OR-02 Gomes & Shmoys 42 Variable setting controlled by LP Randomized Rounding CSP & LP Inference Search & Inference controlled by CSP %LP Interleave-LP HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH Populate CSP Model Perform propagation Populate LP solver Solve LP Adaptive CUTOFF

CP-AI-OR-02 Gomes & Shmoys 43 1.Initialize CSP model and perform propagation of constraints (Ilog Solver); 2.Solve LP model (Ilog Cplex Barrier) LP provides good heuristic guidance and pruning information for the search. However solving the LP is relatively expensive. 3.Two parameters control the LP effort %LP – this parameter controls the percentage of variables set based on the LP rounding (%LP=0  pure CSP strategy) Interleave-LP – sets the frequency in which we re- solve the LP. 4.Randomized rounding scheme: rank variables according to the LP value. Select the highest ranked variable and set its value to 1 with probability p given by its LP value. With probability (1-p), randomly select a color form the colors allowed in the CSP model. 5.Perform propagation CSP propagation after each variable setting. (A total of Interleave-LP variables is assigned this way without resolving the LP) 6.Use a cutoff value to restart the sercah (keep increasing it to maintain completeness) HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH

CP-AI-OR-02 Gomes & Shmoys 44 Empirical Results

CP-AI-OR-02 Gomes & Shmoys 45 Time Performance

CP-AI-OR-02 Gomes & Shmoys 46 Performance in Backtracks

CP-AI-OR-02 Gomes & Shmoys 47 Performance With the hybrid strategy we also solve instances of order 40 in critically constrained area – out of reach for pure CSP; We even solved a few balanced instances of order 50 in the critically constrained order! more systematic experimentation is required to better understand limitations and strengths of approach.

CP-AI-OR-02 Gomes & Shmoys 48 Conclusions

CP-AI-OR-02 Gomes & Shmoys 49 Conclusions Approximations based on LP randomized rounding (variable/value setting) + CSP propagation --- very powerful. Combating heavy-tails of backtrack search through randomization --- very effective. Consequence: New ways of designing algorithms - aim for strategies which have highly asymmetric distributions that can be exploited using restarts, portfolios of algorithms, and interleaved/parallel runs. General approach  holds promise for a range of combinatorial problems Final TAKE HOME MESSAGE Randomization does not  incomplete search !!!

CP-AI-OR-02 Gomes & Shmoys Check also: Check also: Demos, papers, etc.

CP-AI-OR-02 Gomes & Shmoys 51 Eighth International Conference on the Principles and Practice of Constraint Programming September 7-13 Cornell, Ithaca NY CP 2002