Randomized Rounding

Randomized Rounding

CS6999 Probabilistic Methods in Integer Programming Randomized Rounding Andrew D. Smith April 2003 Overview 2 Background Integer Programming The Probabilistic Method Randomization and Approximation The Covering Radius of a Code Randomized Rounding The Algorithm Probabilistic Analysis Handling Feasibility Covering and Packing Problems Scaling Variables Derandomization Conditional Probabilities Pessimistic Estimators Advanced Techniques Positive Correlation Limited Dependence Randomized Rounding – Overview Version 2.0 – April 2003 Andrew D. Smith Integer and Linear Programming 3 maximizing or minimizing a linear function over a polyhedron m n given matrix A R × find a vector x optimizing 2 vector b Rm 2 max cx Ax b vector c Rn f j ≤ g 2 min cx x 0; Ax b f j ≥ ≥ g Alexander Schrijver (1986) “Theory of linear and integer programming” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Integer and Linear Programming 4 maximizing or minimizing a linear function over a polyhedron m n given matrix A R × find a vector x optimizing 2 vector b Rm 2 max cx Ax b vector c Rn f j ≤ g 2 min cx x 0; Ax b f j ≥ ≥ g x is the solution vector max cx or min cx are objective functions Ax b, Ax b, x 0 are constraints on the solution ≤ ≥ ≥ Alexander Schrijver (1986) “Theory of linear and integer programming” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Integer and Linear Programming 5 maximizing or minimizing a linear function over a polyhedron m n given matrix A R × find a vector x optimizing 2 vector b Rm 2 max cx Ax b vector c Rn f j ≤ g 2 min cx x 0; Ax b f j ≥ ≥ g x is the solution vector max cx or min cx are objective functions Ax b, Ax b, x 0 are constraints on the solution ≤ ≥ ≥ linear programming x Rn polynomial time 2 integer programming x Zn NP-complete (decision version) 2 Alexander Schrijver (1986) “Theory of linear and integer programming” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Integer Programming 6 solving optimization problems over discrete structures famous integer programs some exact solution techniques Facility Location Branch and Bound • • Knapsack Problem Dynamic Programming • • Job Shop Scheduling Cutting Planes • • Travelling Salesperson Ellispoid/Separation • • all NP-hard! exact algorithms fail eventually! Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Integer Programming 7 solving optimization problems over discrete structures famous integer programs some exact solution techniques Facility Location Branch and Bound • • Knapsack Problem Dynamic Programming • • Job Shop Scheduling Cutting Planes • • Travelling Salesperson Ellispoid/Separation • • all NP-hard! exact algorithms fail eventually! what happens when no exact algorithm is fast enough but some solution must be found? Find an approximately optimal solution more on this shortly : : : Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Linear Programming Relaxations 8 relaxation: remove constraints geometrically: expand (and simplify) the polyhedron translation: allow more vectors to be solutions Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Linear Programming Relaxations 9 relaxation: remove constraints geometrically: expand (and simplify) the polyhedron translation: allow more vectors to be solutions fractional relaxation: remove integrality constraints very important technique in solving integer programs n n x Z x¯ R 2 7−! 2 recall that linear programs can be solved in polynomial time • (e.g. primal dual interior point) fractional relaxations for NP-hard problems can be very easy • (e.g. fractional knapsack reduces to sorting) provide intuition about (geometric?) structure of problem • (important concept) integrality gap = cx=cx¯ • Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Linear Programming Relaxations 10 relaxation: remove constraints geometrically: expand (and simplify) the polyhedron translation: allow more vectors to be solutions fractional relaxation: remove integrality constraints very important technique in solving integer programs n n x Z x¯ R 2 7−! 2 recall that linear programs can be solved in polynomial time • (e.g. primal dual interior point) fractional relaxations for NP-hard problems can be very easy • (e.g. fractional knapsack reduces to sorting) provide intuition about (geometric?) structure of problem • (important concept) integrality gap = cx=cx¯ • true understanding of an NP-complete compinatorial optimization problem requires understand its fractional relaxation Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Randomized and Approximation Algorithms 11 Randomized Algorithms any algorithm with behavior depending on random bits (we assume a source of random bits) quality measures time complexity number of random bits required probability of success Motwani & Raghavan (1995) “Randomized algorithms” D. Hochbaum (1995) “Approximation algorithms for NP-hard problems” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Randomized and Approximation Algorithms 12 Randomized Algorithms any algorithm with behavior depending on random bits (we assume a source of random bits) quality measures time complexity number of random bits required or probability of success Approximation Algorithms polynomial time algorithm for an NP-hard optimization problem optimal approximate performance ratio approximate (maximization) optimal (minimization) performance ratios are better when closer to 1 and cannot be < 1 quality measures time complexity performance ratio Motwani & Raghavan (1995) “Randomized algorithms” D. Hochbaum (1995) “Approximation algorithms for NP-hard problems” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith Randomized and Approximation Algorithms 13 Randomized Algorithms any algorithm with behavior depending on random bits (we assume a source of random bits) quality measures time complexity number of random bits required or probability of success Approximation Algorithms polynomial time algorithm for an NP-hard optimization problem optimal approximate performance ratio approximate (maximization) optimal (minimization) performance ratios are better when closer to 1 and cannot be < 1 quality measures time complexity performance ratio combine the two Randomized Approximation Algorithms Motwani & Raghavan (1995) “Randomized algorithms” D. Hochbaum (1995) “Approximation algorithms for NP-hard problems” Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith The Probabilistic Method 14 Trying to prove that a structure with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in this space with positive probability. — Alon & Spencer Alon & Spencer (2000) “The probabilistic method” 2nd edition Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith The Probabilistic Method 15 Trying to prove that a structure with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in this space with positive probability. — Alon & Spencer importance for algorithms can be used to show a solution exists • and to show that a solution exists with particular qualities • and to show that a solution exists where an algorithm will find it • essential tool in design and analysis of randomized algorithms • Alon & Spencer (2000) “The probabilistic method” 2nd edition Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith The Probabilistic Method 16 Trying to prove that a structure with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in this space with positive probability. — Alon & Spencer importance for algorithms can be used to show a solution exists • and to show that a solution exists with particular qualities • and to show that a solution exists where an algorithm will find it • essential tool in design and analysis of randomized algorithms • caveat non-constructive by definition Alon & Spencer (2000) “The probabilistic method” 2nd edition Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith A Quick Illustration 17 K5 is the complete graph on 5 vertices Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith A Quick Illustration 18 K5 is the complete graph on 5 vertices A 5 player tournament is an orientation of K5 Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith A Quick Illustration 19 K5 is the complete graph on 5 vertices A 5 player tournament is an orientation of K5 A Hamiltonian path touches each vertex exactly once Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith A Quick Illustration 20 Theorem There is a tournament with n players and at least (1 n) n!2 − Hamiltonian paths. Randomized Rounding – Background Version 2.0 – April 2003 Andrew D. Smith A Quick Illustration 21 Theorem There is a tournament with n players and at least (1 n) n!2 − Hamiltonian paths. Proof Let T be a random tournament on n players and let X be the number of Hamiltonian paths in T . For each permutation σ let Xσ be the indicator random variable for σ giving a Hamiltonian path - i.e., satisfying (σ(i); σ(i + 1)) T for 1 i n. 2 ≤ ≤ Then X = Xσ and (1 n) P E[X] = E[Xσ] = n!2 − : Thus some tournament has at leastX E[X] Hamiltonian

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