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Outline A Structured Benchmark Domain Randomization Conclusions Gomes, Hogg, Walsh, Zhang 117 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Outline A Structured Benchmark Domain Randomization Conclusions Gomes, Hogg, Walsh, Zhang 118 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroups or Latin Squares: An Abstraction for Real World Applications Given an N X N matrix, and given N colors, a quasigroup of order N is a a colored matrix, such that: -all cells are colored. - each color occurs exactly once in each row. - each color occurs exactly once in each column. Quasigroup or Latin Square (Order 4) Gomes, Hogg, Walsh, Zhang 119 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroup Completion Problem (QCP) Given a partial assignment of colors (10 colors in this case), can the partial quasigroup (latin square) be completed so we obtain a full quasigroup? Example: 32% preassignment Gomes, Hogg, Walsh, Zhang& (Gomes Selman 97) 120 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroup Completion Problem A Framework for Studying Search NP-Complete. Has a structure not found in random instances, such as random K-SAT. Leads to interesting search problems when structure is perturbed (more about it later). Good abstraction for several real world problems: scheduling and timetabling, routing in fiber optics, coding, etc (Anderson 85, Colbourn 83, 84, Denes & Keedwell 94, Fujita et al. 93, Gent et al. 99, Gomes & Selman 97, Gomes et al. 98, Meseguer & Walsh 98, Stergiou and Walsh 99, Shaw et al. 98, Stickel 99, Walsh 99 ) Gomes, Hogg, Walsh, Zhang 121 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Fiber Optic Networks Nodes connect point to point fiber optic links Gomes, Hogg, Walsh, Zhang 122 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Fiber Optic Networks Nodes connect point to point fiber optic links Each fiber optic link supports a large number of wavelengths Nodes are capable of photonic switching --dynamic wavelength routing -which involves the setting of the wavelengths. Gomes, Hogg, Walsh, Zhang 123 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Routing in Fiber Optic Networks preassigned channels Input Ports 1 Output Ports 1 2 2 3 3 4 4 Routing Node How can we achieve conflict-free routing in each node of the network? IJCAI-01 Tutorial on Phase Dynamic wavelength routing is a NP-hard problem. Gomes, Hogg, Walsh, Zhang 124 Transitions and Structures-SA4 QCP Example Use: Routers in Fiber Optic Networks Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Quasigroup Completion Problem. •each channel cannot be repeated in the same input port (row constraints); • each channel cannot be repeated in the same output port (column constraints); 1 2 3 4 Output Port 1 2 3 4 Input ports Input Port Output ports CONFLICT FREE LATIN ROUTER (Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99) Gomes, Hogg, Walsh, Zhang 125 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 NP-completeness is a worstcase notion – what about average complexity? Structural differences between instances of the same NP- complete problem (QCP) Are all the Quasigroup Instances (of same size) Equally Difficult? Time performance: 150 1820 165 IJCAI-01 Tutorial on Phase What is Walsh, the Zhang fundamental difference between instances? Gomes, Hogg, 127 Transitions and Structures-SA4 Are all the Quasigroup Instances Equally Difficult? Time performance: 150 Fraction of preassignment: 1820 165 Gomes, Hogg, Walsh, Zhang 40% IJCAI-01 Tutorial on Phase 50% Transitions and Structures-SA4 35% 128 Median Runtime (log scale) Complexity of Quasigroup Completion Critically constrained area Underconstrained area 20% Gomes, Hogg, Walsh, Zhang Overconstrained area 42% 50% Fraction of pre-assignment 129 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Complexity Graph Phase Transition Fraction of unsolvable cases Phase transition from almost all solvable to almost all unsolvable Gomes, Hogg, Walsh, Zhang Almost all solvable area Almost all unsolvable area Fraction of pre-assignment 130 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 These results for the QCP - a structured domain, nicely complement previous results on phase transition and computational complexity for random instances such as SAT, Graph Coloring, etc. (Broder et al. 93; Clearwater and Hogg 96, Cheeseman et al. 91, Cook and Mitchell 98, Crawford and Auton 93, Crawford and Baker 94, Dubois 90, Frank et al. 98, Frost and Dechter 1994, Gent and Walsh 95, Hogg, et al. 96, Mitchell et al. 1992, Kirkpatrick and Selman 94, Monasson et 99, Motwani et al. 1994, Pemberton and Zhang 96, Prosser 96, Schrag and Crawford 96, Selman and Kirkpatrick 97, Smith and Grant 1994, Smith and Dyer 96, Zhang and Korf 96, and more) Gomes, Hogg, Walsh, Zhang 131 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 QCP Different Representations / Encodings Rows Colors Columns Cubic representation of QCP Gomes, Hogg, Walsh, Zhang 133 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 QCP as a MIP O(n3) cell i, j has color k; i, j,k =1, 2, ...,n. • Variables - x ijk x ∈ {0,1} ijk • Constraints - O(n2) Row/color line ∀ x ≤ 1 i, j,k =1, 2, ...,n. ∑ j,k ijk i Column/color line ∀ ∑ x ≤ 1 i, j,k =1, 2, ...,n. i,k ijk j Row/column line ∀ , ∑ x ≤ 1 i, j,k =1, 2, ...,n. i, j ijk IJCAI-01 Tutorial on Phase k Gomes, Hogg, Walsh, Zhang Transitions and Structures-SA4 134 QCP as a CSP • Variables - O(n2) [ vs. O(n3) for MIP] x color of cell i, j; i, j =1, 2, ...,n. i, j x ∈ {1, 2, ...,n} i, j • Constraints - O(n) [ vs. O(n2) for MIP] alldiff (x , x ,..., x ); i =1, 2, ...,n. i,n i,1 i,2 row alldiff (x , x ,..., x ); j =1, 2, ...,n. column n, j 1, j 2, j Gomes, Hogg, Walsh, Zhang 135 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Exploiting Structure for Domain Reduction • A very successful strategy for domain reduction in CSP is to exploit the structure of groups of constraints and treat them as global constraints. Example using Network Flow Algorithms: • All-different constraints (Caseau and Laburthe 94, Focacci, Lodi, & Milano 99, Nuijten & Aarts 95, Ottososon & Thorsteinsson 00, Refalo 99, Regin 94 ) Gomes, Hogg, Walsh, Zhang 136 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Exploiting Structure in QCP ALLDIFF as Global Constraint Matching on Two solutions: a Bipartite graph All-different constraint we can update the domains of the column variables (Berge 70, Regin 94, Shaw and Walsh 98 ) Gomes, Hogg, Walsh, Zhang 137 Analogously, we can update the domains of the other variables IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Exploiting Structure Arc Consistency vs. All Diff Arc Consistency AllDiff Solves up to order 20 Size search space 20400 Solves up to order 40 Size search space 401600 Gomes, Hogg, Walsh, Zhang 138 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroup as Satisfiability Two different encodings for SAT: 2D encoding (or minimal encoding); 3D encoding (or full encoding); Gomes, Hogg, Walsh, Zhang 139 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 2D Encoding or Minimal Encoding 3 Variables: n x cell i, j has color k; i, j,k =1, 2, ...,n. ijk x ∈ {0,1} ijk Each variables represents a color assigned to a cell. Clauses: O(n4) • Some color must be assigned to each cell (clause of length n); ∀ (x ∨ x x ) ij ij1 ij2 ijn • No color is repeated in the same row (sets of negative binary clauses); ∀ (¬x ∨ ¬x ) ∧ (¬x ∨ ¬x ) (¬x ∨ ¬x ) ik i1k i2k i1k i3k i1k ink • No color is repeated in the same column (sets of negative binary clauses); ∀ (¬x ∨ ¬x ) ∧ (¬x ∨ ¬x ) (¬x ∨ ¬x ) jk 1 jk 2 jk 1 jk 3 jk 1 jk njk Gomes, Hogg, Walsh, Zhang 140 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 3D Encoding or Full Encoding This encoding is based on the cubic representation of the quasigroup: each line of the cube contains exactly one true variable; Variables: Same as 2D encoding. O(n4) Clauses: • Same as the 2 D encoding plus: • Each color must appear at least once in each row; • Each color must appear at least once in each column; • No two colors are assigned to the sameIJCAI-01 cell;Tutorial on Phase Gomes, Hogg, Walsh, Zhang 141 Transitions and Structures-SA4 Capturing Structure Performance of SAT Solvers State of the art backtrack and local search and complete SAT solvers using 3D encoding are very competitive with specialized CSP algorithms. In contrast SAT solvers perform very poorly on 2D encodings (SATZ or SATO); In contrast local search solvers (Walksat) perform well on 2D encodings; Gomes, Hogg, Walsh, Zhang 142 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 SATZ on 2D encoding (Order 20 -28) Order 28 1,000,000 Order 20 SATZ and SATO can only solve up to order 28 when using 2D encoding; When using 3D encoding problems of the same size take only 0 or 1 backtrack and much higher orders can be solved; IJCAI-01 Tutorial on Phase Gomes, Hogg, Walsh, Zhang 143 Transitions and Structures-SA4 Walksat on 2D and 3D encoding (Order 30-33) 1,000,000 3D order 33 2D order 33 Walksat shows an unsual pattern the 2D encodings are somewhat easier than the 3D encoding at the peak and harder in the undereconstrained region; Gomes, Hogg, Walsh, Zhang 144 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroup - Satisfiability Encoding the quasigroup using only Boolean variables in clausal form using the 3D encoding is very competitive. Very fast solvers - SATZ, GRASP, SATO,WALKSAT; Gomes, Hogg, Walsh, Zhang 145 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Structural features of instances provide insights into their hardness namely: Backbone Inherent Structure and Balance Gomes, Hogg, Walsh, Zhang 146 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Backbone Backbone is the shared structure of all the solutions to a given instance. This instance has 4 solutions: Backbone Gomes, Hogg, Walsh, Zhang Total number of backbone variables: IJCAI-01 2 Tutorial on Phase 147 Transitions and Structures-SA4 Phase Transition in the Backbone • We have observed a transition in the backbone from a phase where the size of the backbone is around 0% to a phase with backbone of size close to 100%. • The phase transition in the backbone is sudden and it coincides with the hardest problem instances. (Achlioptas, Gomes, Kautz, Selman 00, Monasson et al. 99) Gomes, Hogg, Walsh, Zhang 148 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 New Phase Transition in Backbone QCP (satisfiable instances only) % of Backbone % Backbone Sudden phase transition in Backbone Computational cost Fraction of preassigned cells Gomes, Hogg, Walsh, Zhang 149 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Inherent Structure and Balance Gomes, Hogg, Walsh, Zhang 150 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Quasigroup Patterns and Problems Hardness Rectangular Pattern Aligned Pattern Tractable Balanced Pattern Very hard (Kautz, Ruan, Achlioptas, Gomes, Selman 2001) Gomes, Hogg, Walsh, Zhang 151 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 SATZ Balanced QCP Rectangular QCP QCP QWH Aligned QCP Gomes, Hogg, Walsh, Zhang 152 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Walksat Balanced filtered QCP Balance QWH QCP QWH aligned rectangular We observe the same ordering in hardness when using Walksat, SATZ, and SATO – Balacing makes instances harder Gomes, Hogg, Walsh, Zhang 153 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Phase Transitions, Backbone, Balance Summary The understanding of the structural properties of problem instances based on notions such as phase transitions, backbone, and balance provides new insights into the practical complexity of many computational tasks. Active research area with fruitful interactions between computer science, physics (approaches from statistical mechanics), and mathematics (combinatorics / random structures). Gomes, Hogg, Walsh, Zhang 154 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Outline A Structured Benchmark Domain Randomization Conclusions Gomes, Hogg, Walsh, Zhang 155 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Local Search Stochastic strategies have been very successful in the area of local search. Simulated annealing Genetic algorithms Tabu Search Gsat and variants. Limitation: inherent incomplete nature of local search methods. Gomes, Hogg, Walsh, Zhang 156 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Randomized Backtrack Search Goal: explore the addition of a stochastic element to a systematic search procedure without losing completeness. We introduce randomness in a backtrack search method by randomly breaking ties in variable and/or value selection. Compare with standard lexicographic tiebreaking. Gomes, Hogg, Walsh, Zhang 157 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Distributions of Randomized Backtrack Search Key Properties: I Erratic behavior of mean II Distributions have “heavy tails”. Gomes, Hogg, Walsh, Zhang 158 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Erratic Behavior of Search Cost Quasigroup Completion Problem 3500! sample mean 2000 Median = 1! 500 Gomes, Hogg, Walsh, Zhang number of runs 159 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Heavy-Tailed Distributions … infinite variance … infinite mean Introduced by Pareto in the 1920’s --- “probabilistic curiosity.” Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena. Examples: stock-market, earthquakes, weather,... Gomes, Hogg, Walsh, Zhang 160 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Decay of Distributions Standard --- Exponential Decay e.g. Normal: Pr[ X > x]≈ Ce − x 2, for some C > 0, x >1 Heavy-Tailed --- Power Law Decay e.g. Pareto-Levy: Pr[ X > x ] = C x − α , x > 0 Gomes, Hogg, Walsh, Zhang 161 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Power Law Decay Exponential Decay Standard Distribution (finite mean & variance) Gomes, Hogg, Walsh, Zhang 162 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 How to Check for “Heavy Tails”? Log-Log plot of tail of distribution should be approximately linear. Slope gives value of α α <1 infinite mean and infinite variance 1≤α < 2 infinite variance Gomes, Hogg, Walsh, Zhang 163 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 (1-F(x))(log) Unsolved fraction Heavy-Tailed Behavior in QCP Domain α = 0.153 α = 0.319 18% unsolved α = 0.466 α <1 => Infinite mean 0.002% unsolved Number backtracks (log) Gomes, Hogg, Walsh, Zhang 164 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Small-World Vs. Heavy-Tailed Behavior Does a Small-World topology (Watts & Strogatz) induce heavy-tail behavior? The constraint graph of a quasigroup exhibits a small-world topology (Walsh 99) Gomes, Hogg, Walsh, Zhang 165 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 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 provably eliminate heavy-tailed behavior. (Gomes et al. 97, Hoos 99, Horvitz 99, Huberman, Lukose and Hogg 97, Karp et al 96, Luby et al. 93, Rish et al. 97, Wlash 99) Gomes, Hogg, Walsh, Zhang 166 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Restarts 1-F(x) Unsolved fraction no restarts 70% unsolved restart every 4 backtracks 0.001% unsolved 250 (62 restarts) Number backtracks (log) Gomes, Hogg, Walsh, Zhang 167 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Sketch of proof of elimination of heavy tails X − number of backtracks to solve the problem Let’s truncate the search procedure after m backtracks. Probability of solving problem with truncated version: pm = Pr[ X ≤ m] Run the truncated procedure and restart it repeatedly. Gomes, Hogg, Walsh, Zhang 168 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Y − total number backtracks with restarts N u m b e r o f R e s ta r ts = Y / m ~ G e o m e tr ic ( pm ) F = P r[Y > y ] ≤ (1 − pm ) Y /m ≈ c1 e − c2 y Y - does not have Heavy Tails Gomes, Hogg, Walsh, Zhang 169 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Decoding in Communication Systems Voice waveform, binary digits from a cd, output of a set of sensors in a space probe, etc. Telephone line, a storage medium, a space communication link, etc. usually subject to NOISE Source Encoder Channel Processing prior to transmission, e.g., insertion of redundancy to combat the channel noise. Decoder Destination Processing of the channel output with the objective of producing at the destination an acceptable replica of the source output. Decoding in communication systems is NP-hard. (Berlekamp, McEliece, and van Tilborg 1978, Barg 1998) Gomes, Hogg, Walsh, Zhang 170 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Retransmissions in Sequential Decoding 1-F(x) Unsolved fraction without retransmissions with retransmissions Number backtracks (log) Gomes et al. 2000 / 20001 Gomes, Hogg, Walsh, Zhang 171 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Paramedic Crew Assignment (Austin, Texas) Paramedic crew assignment is the problem of assigning paramedic crews IJCAI-01constraints. Tutorial on Phase from different stations Gomes, Hogg, Walsh, Zhang to cover a given region, given several resource 172 Transitions and Structures-SA4 Deterministic Search Gomes, Hogg, Walsh, Zhang Austin, Texas 173 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Restarts Gomes, Hogg, Walsh, Zhang Austin, Texas 174 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Results on Effectiveness of Restarts Deterministic Logistics Planning Scheduling 14 108 mins. 411 sec Scheduling 16 Scheduling 18 Circuit Synthesis 1 Circuit Synthesis 2 ---(*) ---(*) ---(*) ---(*) 3 R 95 sec. 250 sec 1.4 hours ~18 hrs 165sec. 17min. (*) not found after 2 days Gomes, Hogg, Walsh, Zhang 175 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Portfolio of Algorithms A portfolio of algorithms is a collection of algorithms running interleaved or on different processors. Goal: to improve the performance of the different algorithms in terms of: expected runtime “risk” (variance) Efficient Set or Pareto set: set of portfolios that are best in terms of expected value and risk. (Gomes and Selman 97, Huberman, Lukose, Hogg 97 ) Gomes, Hogg, Walsh, Zhang 176 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Cumulative Frequencies Brandh & Bound for MIP Depth-first vs. Best-bound Optimal strategy: Best Bound Best-Bound: Average-1400 nodes; St. Dev.- 1300 Depth-first 45% 30% Best bound Depth-First: Average - 18000;St. Dev. 30000 Gomes, Hogg, Walsh, Zhang Number of nodes 177 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Heavy-tailed behavior of Depth-first Gomes, Hogg, Walsh, Zhang 178 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Expected run time of portfolios Portfolio for 6 processors 0 DF / 6 BB 3 DF / 3 BB Efficient set 4 DF / 2 BB 6 DF / 0BB 5 DF / 1BB Standard deviation of run time of portfolios IJCAI-01 Tutorial on Phase Gomes, Hogg, Walsh, Zhang 179 Transitions and Structures-SA4 Expected run time of portfolios Portfolio for 20 processors 0 DF / 20 BB The optimal strategy is to run Depth First on the 20 processors! Optimal collective behavior emerges from suboptimal individual behavior. 20 DF / 0 BB Standard Gomes, Hogg, Walsh, Zhang deviation of run time of portfolios IJCAI-01 Tutorial on Phase 180 Transitions and Structures-SA4 Compute Clusters and Distributed Agents With the increasing popularity of compute clusters and distributed problem solving / agent paradigms, portfolios of algorithms --- and flexible computation in general --- are rapidly expanding research areas. (Baptista and Marques da Silva 00, Boddy & Dean 95, Bayardo 99, Davenport 00, Hogg 00, Horvitz 96, Matsuo 00, Steinberg 00, Russell 95, Santos 99, Welman 99. Zilberstein 99) Gomes, Hogg, Walsh, Zhang 181 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Bayesian Model Structure Learning Learning to infer predictive models from data and to identify key variables ==> restarts, cutoffs and other adaptive behavior of search algorithms. (Horvitz, Ruan, Gomes, Kautz, Selman, Chickering 2001) Gomes, Hogg, Walsh, Zhang 182 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 4XDVLJURXS 2UGHU&63 Variance in number of uncolored cells across rows and columns Min depth Avg Depth Number uncolored cells per column Max number of uncolored cells across rows and columns Green - long runs Gray - short runs Model accuracy 96.8% vs 48% for the marginal model Gomes, Hogg, Walsh, Zhang 183 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Analysis of different solver features and problem features Gomes, Hogg, Walsh, Zhang 184 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Outline A Structured Benchmark Domain Randomization Conclusions Gomes, Hogg, Walsh, Zhang 185 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Summary Finding optimal solution is harder than deciding solubility Phase transitions can be used to characterize complex problems and their behavior Understanding phase transitions can help to design and develop more efficient search algorithms Gomes, Hogg, Walsh, Zhang 186 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Summary Phase transitions are seen in other complexity classes P, PSPACE, … Structure is important to phase transition behavior backbones, small world topology, .... Gomes, Hogg, Walsh, Zhang 187 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Summary The understanding of the structural properties of problem instances based on notions such as phase transitions, backbone, and balance provides new insights into the practical complexity of many computational tasks. Active research area with fruitful interactions between computer science, physics (approaches from statistical mechanics), and mathematics (combinatorics / random structures). Gomes, Hogg, Walsh, Zhang 188 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4 Summary Stochastic search methods (complete and incomplete) have been shown very effective. Restart strategies and portfolio approaches can lead to substantial improvements in the expected runtime and variance, especially in the presence of heavy-tailed phenomena. Randomization is therefore a tool to improve algorithmic performance and robustness. Machine Learning techniques can be used to learn predicitive models. IJCAI-01 Tutorial on Phase Gomes, Hogg, Walsh, Zhang 189 Transitions and Structures-SA4 Bridging the Gap General Solution Methods Exploiting Structure: Tractable Components Transition Aware Systems (phase transition constrainedness backbone resources) Randomization Exploits variance to improve robustness and performance Real World Problems Gomes, Hogg, Walsh, Zhang 190 IJCAI-01 Tutorial on Phase Transitions and Structures-SA4