A sublinear-time randomized approximation algorithm for matrix games
This paper presents a parallel randomized algorithm which computes a pair of ε-optimal strategies for a given ( m, n)-matrix game A = [ a ij ] ϵ [−1, 1] in O( ε −2log 2( n+ m)) expected time on an ( n+ m)/log( n+ m)-processor EREW PRAM. For any fixed accuracy ϵ > 0, the expected sequential runnin...
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| Published in: | Operations research letters Vol. 18; no. 2; pp. 53 - 58 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
01.09.1995
Elsevier |
| Subjects: | |
| ISSN: | 0167-6377, 1872-7468 |
| Online Access: | Get full text |
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| Abstract | This paper presents a parallel randomized algorithm which computes a pair of ε-optimal strategies for a given (
m,
n)-matrix game
A = [
a
ij
]
ϵ [−1, 1] in O(
ε
−2log
2(
n+
m)) expected time on an (
n+
m)/log(
n+
m)-processor EREW PRAM. For any fixed accuracy
ϵ > 0, the expected sequential running time of the suggested algorithm is O((
n +
m)log(
n +
m)), which is sublinear in mn, the number of input elements of
A. On the other hand, simple arguments are given to show that for
ε <
1
2
, any deterministic algorithm for computing a pair of ε-optimal strategies of an (
m,
n)-matrix game
A with ± 1 elements examines
Ω(mn) of its elements. In particular, for
m =
n the randomized algorithm achieves an almost quadratic expected speedup relative to any deterministic method. |
|---|---|
| AbstractList | This paper presents a parallel randomized algorithm which computes a pair of ε-optimal strategies for a given (
m,
n)-matrix game
A = [
a
ij
]
ϵ [−1, 1] in O(
ε
−2log
2(
n+
m)) expected time on an (
n+
m)/log(
n+
m)-processor EREW PRAM. For any fixed accuracy
ϵ > 0, the expected sequential running time of the suggested algorithm is O((
n +
m)log(
n +
m)), which is sublinear in mn, the number of input elements of
A. On the other hand, simple arguments are given to show that for
ε <
1
2
, any deterministic algorithm for computing a pair of ε-optimal strategies of an (
m,
n)-matrix game
A with ± 1 elements examines
Ω(mn) of its elements. In particular, for
m =
n the randomized algorithm achieves an almost quadratic expected speedup relative to any deterministic method. |
| Author | Khachiyan, Leonid G. Grigoriadis, Michael D. |
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| Cites_doi | 10.1016/0041-5553(77)90174-4 10.1002/cpa.3160110408 10.1137/0804004 10.2307/1969530 10.1287/moor.20.2.257 |
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| Issue | 2 |
| Keywords | Matrix games Approximation algorithms Linear programming Randomized algorithms Parallel algorithms Complexity Lower bound Algorithms Approximation Deterministic approach Computational complexity Matrix game Game theory |
| Language | English |
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| References | Khachiyan, Khachiyan (BIB6) 1977; 17 Grigoriadis, Khachiyan (BIB5) 1994; 4 Dantzig (BIB2) 1953 Grigoriadis, Khachiyan (BIB3) 1992 also available as TR91-73 from DIMACS, CoRE-Busch, Rutgers Univ., New Brunswick, NJ 08903. Brown (BIB1) 1951; Vol. 13 Plotkin, Shmoys, Tardos (BIB9) 1995; 20 Lipton, Young (BIB7) 1994 Luby, Nisan (BIB8) 1993 Shapiro (BIB11) 1958; 11 Robinson (BIB10) 1951; 54 Grigoriadis (10.1016/0167-6377(95)00032-0_BIB3) 1992 Grigoriadis (10.1016/0167-6377(95)00032-0_BIB5) 1994; 4 Khachiyan (10.1016/0167-6377(95)00032-0_BIB6_1) 1977; 17 Khachiyan (10.1016/0167-6377(95)00032-0_BIB6_2) 1978; 17 Shapiro (10.1016/0167-6377(95)00032-0_BIB11) 1958; 11 Lipton (10.1016/0167-6377(95)00032-0_BIB7) 1994 Robinson (10.1016/0167-6377(95)00032-0_BIB10) 1951; 54 Dantzig (10.1016/0167-6377(95)00032-0_BIB2) 1953 Brown (10.1016/0167-6377(95)00032-0_BIB1) 1951; Vol. 13 Luby (10.1016/0167-6377(95)00032-0_BIB8) 1993 Plotkin (10.1016/0167-6377(95)00032-0_BIB9) 1995; 20 10.1016/0167-6377(95)00032-0_BIB4 |
| References_xml | – reference: also available as TR91-73 from DIMACS, CoRE-Busch, Rutgers Univ., New Brunswick, NJ 08903. – start-page: 129 year: 1992 end-page: 136 ident: BIB3 article-title: Approximate solution of matrix games in parallel publication-title: Advances in Optimization and Parallel Computing – volume: Vol. 13 year: 1951 ident: BIB1 article-title: Iterative solution of games by fictitious play publication-title: Activity Analysis of Production and Allocation – start-page: 448 year: 1993 end-page: 457 ident: BIB8 article-title: A parallel approximation algorithm for positive linear programming publication-title: ACM Symp. on Theory of Computing – volume: 20 start-page: 257 year: 1995 end-page: 301 ident: BIB9 article-title: Fast approximation algorithms for fractional packing and covering problems publication-title: Math. Oper. Res. – start-page: 734 year: 1994 end-page: 740 ident: BIB7 article-title: Simple strategies for large zero-sum games with applications to complexity theory publication-title: ACM Symp. on Theory of Computing – volume: 11 start-page: 587 year: 1958 end-page: 593 ident: BIB11 article-title: Note on a computation method in the theory of games publication-title: Comm. Pure Appl. Math. – volume: 54 start-page: 296 year: 1951 end-page: 301 ident: BIB10 article-title: An iterative method of solving a game publication-title: Ann. Math. – year: 1953 ident: BIB2 article-title: Linear Programming and Extensions – volume: 4 start-page: 86 year: 1994 end-page: 107 ident: BIB5 article-title: Fast approximation schemes for convex programs with many blocks and coupling constraints publication-title: SIAM J. Optim. – volume: 17 start-page: 1421 year: 1977 end-page: 1431 ident: BIB6 article-title: Convergence rate of the game processes for solving matrix games publication-title: Zh. V – start-page: 448 year: 1993 ident: 10.1016/0167-6377(95)00032-0_BIB8 article-title: A parallel approximation algorithm for positive linear programming – start-page: 129 year: 1992 ident: 10.1016/0167-6377(95)00032-0_BIB3 article-title: Approximate solution of matrix games in parallel – volume: 17 start-page: 78 year: 1978 ident: 10.1016/0167-6377(95)00032-0_BIB6_2 publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(77)90174-4 – volume: 11 start-page: 587 year: 1958 ident: 10.1016/0167-6377(95)00032-0_BIB11 article-title: Note on a computation method in the theory of games publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.3160110408 – volume: 4 start-page: 86 year: 1994 ident: 10.1016/0167-6377(95)00032-0_BIB5 article-title: Fast approximation schemes for convex programs with many blocks and coupling constraints publication-title: SIAM J. Optim. doi: 10.1137/0804004 – volume: 54 start-page: 296 year: 1951 ident: 10.1016/0167-6377(95)00032-0_BIB10 article-title: An iterative method of solving a game publication-title: Ann. Math. doi: 10.2307/1969530 – volume: Vol. 13 year: 1951 ident: 10.1016/0167-6377(95)00032-0_BIB1 article-title: Iterative solution of games by fictitious play – ident: 10.1016/0167-6377(95)00032-0_BIB4 – volume: 17 start-page: 1421 year: 1977 ident: 10.1016/0167-6377(95)00032-0_BIB6_1 article-title: Convergence rate of the game processes for solving matrix games publication-title: Zh. Vychisl. Mat. Mat. Fiz. – volume: 20 start-page: 257 year: 1995 ident: 10.1016/0167-6377(95)00032-0_BIB9 article-title: Fast approximation algorithms for fractional packing and covering problems publication-title: Math. Oper. Res. doi: 10.1287/moor.20.2.257 – start-page: 734 year: 1994 ident: 10.1016/0167-6377(95)00032-0_BIB7 article-title: Simple strategies for large zero-sum games with applications to complexity theory publication-title: ACM Symp. on Theory of Computing – year: 1953 ident: 10.1016/0167-6377(95)00032-0_BIB2 |
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| Snippet | This paper presents a parallel randomized algorithm which computes a pair of ε-optimal strategies for a given (
m,
n)-matrix game
A = [
a
ij
]
ϵ [−1, 1] in O(... |
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| StartPage | 53 |
| SubjectTerms | Applied sciences Approximation algorithms Complexity Exact sciences and technology Game theory Linear programming Matrix games Operational research and scientific management Operational research. Management science Parallel algorithms Randomized algorithms |
| Title | A sublinear-time randomized approximation algorithm for matrix games |
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