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SSG’s, MPG’s, DPG’s and PG’s Are All LP-type Problems

SSG’s, MPG’s, DPG’s and PG’s Are All LP-type Problems. Nir Halman, MIT. 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. PG  MPG [Pu95]. MPG  DPG [ZP96].

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SSG’s, MPG’s, DPG’s and PG’s Are All LP-type Problems

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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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