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Shannon-like games are difficult
Discrete Mathematics North-Holland
481
111 (1993) 481-483
Shannon-like
games are difficult
Y6hei Y amasaki College of Biomedical Received
Technology,
Osaka University.
Toyonaka,
Osaka 560, Japan
22 July 1991
Abstract Yamasaki,
Y., Shannon-like
games are difficult, Discrete
Mathematics
111 (1993) 481483.
We deal with an analogy of the Shannon switching game, called n-pair games, played on a graph with n pairs of terminals (+lr -1), (+2, -2), . . . . (+., -.). H ere these 2n terminals are not necessarily distinct and a coincidence of terminals is indicated by =, Each player at his turn labels an edge unlabeled as yet. Short intends to make a path consisting of the edges labeled by him between one of the given n pairs of terminals and Cut intends to obstruct it. The classical strategy theory can be generalized for the so-called double-matroid games, including 2-pair games [cf. Yamasaki (1978, 1989)]. However, certain particular 3-pair games are ‘difficult’ (cf. Yamasaki, to appear).
The simplest strategy one may imagine is the matching strategy: each move depends only on the preceding move of the opponent. It cannot be applied even for l-pair games, i.e. classical Shannon games. Fig. 1 gives a situation in a l-pair game where no matching strategy exists for Short although he is winning, that is, he can win even playing second. The next strategy is more global. A winning player-Short, in Fig. l-takes a pair of disjoint subsets of edges of the same size. He chooses an element of the subset different from the one in which his opponent has chosen the preceding move. This strategy is applied generally to the double-matroid games, but not to the n-pair games. Short in Fig. 2 and Cut in Fig. 3 are winning, but neither of them has such a strategy. How can we obtain an improved strategy? While our strategy requires a sort of correspondence, we would expect in any position where a player rc is winning that the strategy for 7c is as follows: There is an even size subset S of edges. If the opponent the corresponding move can be found in it.
of 7cplays in S, then
Even this sort of strategy exists for neither of the winning player in Figs. 2 and 3, because he is no more winning when an edge is labeled by his opponent, although there are odd number of edges in the given graph. Correspondence Machikaneyama-cho, 0012-365X/93/$06.00
to: Yohei Yamasaki, College Toyonaka, 560, Japan. 0
1993-Elsevier
of Biomedical
Science Publishers
Technology,
Osaka
B.V. All rights reserved
University,
l-l,
482
Y. Yamasaki
Fig. 1. l-pair
+z
+I
=
=
game (n= 1).
-4
+3
-3
=
-5
0
A
+s =
-2
+4
=
--I
Fig. 2. 5-pair game (n=5).
+,
-1
=
+z
-2
=
+3
-3
Fig. 3. 3-pair game (n = 3).
Table 1 Game
2-Pair 3-Pair Hex
game game
Correspondence strategy
Complexity
Exists Does not exist Does not exist
Polynomial ????? PSPACE-complete
(cf. C71)
Table 1 lists the complexity and the existence of the correspondence strategy of several classes of games. Here hex is a class of Shannon switching games on vertices whose complexity is PSPACE-complete [4] and has no correspondence strategy. The 2-pair game is polynomial because of its correspondence strategy. How is the ?????-part? To prove that it is polynomial, one needs a quite different idea from the correspondence strategy. On the other hand, if one states some hardness of it, then his proof must distinguish the 2-pair game from the 3-pair game, provided PSPACE-complete is not polynomial.
Shannon-like games are dijficult
483
References [l] [2]
C. Berge, Graphs E.R. Berlekamph,
[3]
S. Even and R.E. Tarjan, A combinatorial problem which is complete in polynomial space, Proc. 7th ACM Symp. on the Theory of Computing, May 1975. S. Reisch, Hex ist PSPACE-vollstlndig, Acta Inform. 15 (1981). Y. Yamasaki, Theory of division games, Publ. Res. Inst. Math. Sci. 17 (1978) 337-358. Y. Yamasaki, Shannon switching games without terminals II, Graphs and Combin. 5 (1989) 275-282. Y. Yamasaki, Combinatorial Games: Back and Front (in Japanese) (Springer, Tokyo, 1989). Y. Yamasaki, Shannon switching games without terminals III, submitted. Y. Yamasaki, A difficulty in particular Shannon-like games, Discrete Appl. Math., to appear.
[4] [S] [6] [7] [S] [9]
and Hypergraphs (North Holland, Amsterdam, 1973). J.H. Conway and R.K. Guy, Winning ways (Academic
Press, New York, 1982).
481
111 (1993) 481-483
Shannon-like
games are difficult
Y6hei Y amasaki College of Biomedical Received
Technology,
Osaka University.
Toyonaka,
Osaka 560, Japan
22 July 1991
Abstract Yamasaki,
Y., Shannon-like
games are difficult, Discrete
Mathematics
111 (1993) 481483.
We deal with an analogy of the Shannon switching game, called n-pair games, played on a graph with n pairs of terminals (+lr -1), (+2, -2), . . . . (+., -.). H ere these 2n terminals are not necessarily distinct and a coincidence of terminals is indicated by =, Each player at his turn labels an edge unlabeled as yet. Short intends to make a path consisting of the edges labeled by him between one of the given n pairs of terminals and Cut intends to obstruct it. The classical strategy theory can be generalized for the so-called double-matroid games, including 2-pair games [cf. Yamasaki (1978, 1989)]. However, certain particular 3-pair games are ‘difficult’ (cf. Yamasaki, to appear).
The simplest strategy one may imagine is the matching strategy: each move depends only on the preceding move of the opponent. It cannot be applied even for l-pair games, i.e. classical Shannon games. Fig. 1 gives a situation in a l-pair game where no matching strategy exists for Short although he is winning, that is, he can win even playing second. The next strategy is more global. A winning player-Short, in Fig. l-takes a pair of disjoint subsets of edges of the same size. He chooses an element of the subset different from the one in which his opponent has chosen the preceding move. This strategy is applied generally to the double-matroid games, but not to the n-pair games. Short in Fig. 2 and Cut in Fig. 3 are winning, but neither of them has such a strategy. How can we obtain an improved strategy? While our strategy requires a sort of correspondence, we would expect in any position where a player rc is winning that the strategy for 7c is as follows: There is an even size subset S of edges. If the opponent the corresponding move can be found in it.
of 7cplays in S, then
Even this sort of strategy exists for neither of the winning player in Figs. 2 and 3, because he is no more winning when an edge is labeled by his opponent, although there are odd number of edges in the given graph. Correspondence Machikaneyama-cho, 0012-365X/93/$06.00
to: Yohei Yamasaki, College Toyonaka, 560, Japan. 0
1993-Elsevier
of Biomedical
Science Publishers
Technology,
Osaka
B.V. All rights reserved
University,
l-l,
482
Y. Yamasaki
Fig. 1. l-pair
+z
+I
=
=
game (n= 1).
-4
+3
-3
=
-5
0
A
+s =
-2
+4
=
--I
Fig. 2. 5-pair game (n=5).
+,
-1
=
+z
-2
=
+3
-3
Fig. 3. 3-pair game (n = 3).
Table 1 Game
2-Pair 3-Pair Hex
game game
Correspondence strategy
Complexity
Exists Does not exist Does not exist
Polynomial ????? PSPACE-complete
(cf. C71)
Table 1 lists the complexity and the existence of the correspondence strategy of several classes of games. Here hex is a class of Shannon switching games on vertices whose complexity is PSPACE-complete [4] and has no correspondence strategy. The 2-pair game is polynomial because of its correspondence strategy. How is the ?????-part? To prove that it is polynomial, one needs a quite different idea from the correspondence strategy. On the other hand, if one states some hardness of it, then his proof must distinguish the 2-pair game from the 3-pair game, provided PSPACE-complete is not polynomial.
Shannon-like games are dijficult
483
References [l] [2]
C. Berge, Graphs E.R. Berlekamph,
[3]
S. Even and R.E. Tarjan, A combinatorial problem which is complete in polynomial space, Proc. 7th ACM Symp. on the Theory of Computing, May 1975. S. Reisch, Hex ist PSPACE-vollstlndig, Acta Inform. 15 (1981). Y. Yamasaki, Theory of division games, Publ. Res. Inst. Math. Sci. 17 (1978) 337-358. Y. Yamasaki, Shannon switching games without terminals II, Graphs and Combin. 5 (1989) 275-282. Y. Yamasaki, Combinatorial Games: Back and Front (in Japanese) (Springer, Tokyo, 1989). Y. Yamasaki, Shannon switching games without terminals III, submitted. Y. Yamasaki, A difficulty in particular Shannon-like games, Discrete Appl. Math., to appear.
[4] [S] [6] [7] [S] [9]
and Hypergraphs (North Holland, Amsterdam, 1973). J.H. Conway and R.K. Guy, Winning ways (Academic
Press, New York, 1982).