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LP-Type Problems in Stochastic Games: Algorithms and Applications

Explore LP-Type Problems within stochastic games, including strategies, values, and optimal solutions. Learn about related games such as Parity Games and Discounted Payoff Games, and discover an algorithmic approach for solving them efficiently. Find out how to formulate Stochastic Games as LP-Type Problems and solve them in linear time.

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LP-Type Problems in Stochastic Games: Algorithms and Applications

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  1. SSG’s, MPG’s, DPG’s and PG’s Are All LP-type Problems Nir Halman, MIT

  2. av. 2 1-sink 0.1 0.1 0.6 0-sink min 0.2 0.7 0.3 max av. 1 Simple Stochastic Games Notations: strategies, values, optimal strategy, binary SSG’s

  3. PG  MPG [Pu95] MPG  DPG [ZP96] DPG  SSG [ZP96] Related Games All games defined on graphs with 2 kinds of nodes and no sink nodes (  infinite games) •Parity Games • Min Payoff Games • Discounted Payoff Games

  4. Previous vrs. Our Results One unifying algorithm for all 5 games

  5. LP-type Problems [SW92] Def: a pair (H, ), H:constraints :objective function satisfying: •monotonicity (FGH,(F)(G)) •locality Goal: calculate (H)

  6. r An Example Smallest Enclosing Ball (SEB) H: points(H): radius of the smallest enclosing ball of H Wanted: •The optimal value r =(H) •A basis of H combinatorial dimension:d+1 (fixed) Two algorithmic primitives: Violation test & Basis calculation

  7. LP-type Algorithms Linear time (randomized) algorithms when the dimension is fixed [Cl88], [Ka92],[SW92] tv=time for violation test; tb=time for basis calculation; Alg of [SW92] runs in O(e d log d n(tv +tb)) Corollary: SEB is solvable in linear time Runs in time sub-exponential in dimension d

  8. Usage of LP-type Framework In computational geometry / facilitylocation : • distance between polytopes • smallest enclosing ball/ellipsoid • largest ball/ellipsoid in polytope • angle-optimal placement of point in polygon • line transversal of translates •p-center on the line/in the plane with rectilinear norm • convex Hausdorff distance •p-recovery points on the line and on directed trees• etc.

  9. Formulating a SSG as an LP-type Problem Def: G=(V,E=E0 E1 Ea). E’ E1, G(E’) =(V,E0E’Ea) is sub-game of G Def: E’ E1,(E’),(E’) be a pair of optimal strategies in G(E’): (E’)=vVv(E’),(E’)vV1outgoing edge in E’, -otherwise. Thm 1: G=(V,E=E0 E1 Ea). (E1,) is a |V1|-dimensional LP-type problem A basis = set of edges in an optimal strategy

  10.  (E’)=vVv(E’),(E’)vV1 outgoing edge in (E’),-otherwise. Cont’ Def: G=(V,E=E0 E1 Ea). E’ E0,(E’),(E’) be a pair of optimal strategies in G(E’): (E’)=-vVv(E’),(E’)vV0 an outgoing edge in E’,-otherwise. Thm 2: G=(V,E=E0 E1 Ea). (E0, ) is a |V0|-dimensional LP-type problem

  11. (v’) v3 V’ e’ v4 Solving SSG in eO( n log n)time Thm 3: SSGG with out-degree 1 for all max nodes. (E0,) is solvable in eO( n log n)time Proof idea:violation test B  {e’ } in constant time. Basis calculation of B  {e’ } = comparing 2 games with trivial max and min strategies. (E’)=-vVv(E’),(E’)vV0 an outgoing edge in (E’)-otherwise.

  12. Cont’ Thm 4: SSGis solvable in eO( n log n)time Proof idea:consider corresponding LP-type (E1,). violation test in constant time. Basis calculation of B  {e’ } = comparing 2 games with out-degree 1 for all max nodes. Corollary: PG, MPG and DPG are all solvable in eO( n log n)time (E’)=vVv(E’),(E’)vV1 outgoing edge in (E’),-otherwise.

  13. Future Research • Solve any of the mentioned games in polynomial time • Solve 2-person bi-matrix games in sub-exponential time • Find more sub-exponential applications of LP-type model

  14. Thank you !

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