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The set coincidence game: Complexity, attainability, and symmetric strategies
JOURNAL
OF COMPUTER
AND
SYSTEM
SCIENCES
39, 376-387 (1989)
The Set Coincidence Complexity,
Attainability, A. G.
Game:
and Symmetric
Strategies*
ROBINSON AND A. J. GOLDMAN
Mathematical Sciences Department The Johns Hopkins University, Baltimore, Maryland 21218 ReceivedJune 9, 1987
We develop two closely related aspects of the complexity theory of the Set Coincidence Game G( V, W), in which two players alternately choose elements not previously chosen from a finite, nonempty set V, and W is a given family of nonempty subsets of V (the “winning sets”). The game is won by that player who first adds an element to the set of “chosen” elements S, so that SE W. We first show that the Set Coincidence Game (in an efticient encoding) is complete in PSPACE. We then develop the theory of attainability which we use to give intuition about the structure of minimal forced wins for either player, and about the “difftculty” of the game. We also show that a well-known type of strategy (“symmetric” or “pairing”) is optimal in the context of our definition of attainability. 0 1989 Academic press, hc.
1.
INTRODUCTION
The Set Coincidence Game G( V, W) is played on a finite nonempty set V of elements. W is a collection of nonempty subsets of V, the winning sets. Players Pl and P2 move alternately, with Pl leading off; at each turn, a player adds a new element to an expanding set S, which was empty at the start of play. If a player’s move causes S to coincide with some w E W, then that player wins (the opponent loses), and play ends. If V is exhausted (i.e., S = V) without a win, then the game is drawn. The fact that both players’ choices contribute to building up a single set S, rather than individual sets S’ and S2, s&ices to differentiate G( V, W) from the morestudied “positional games” of Berge [3], whose more specialized “types 1 and 2” [3], called “amoeba games” (weak and strong) in Beck and Csirmaz [l], in turn include most of the “achievement and avoidance” games of Harary (e.g., [ 11, 121). On the other hand, the diameter and geodesic achievement games of Buckley and Harary [4, 51 are set coincidence games. G( V, W) will be called a forced p-win if one of the players has a strategy assuring a win in no more than p moves, but the opponent has at least one way to prolong * Research supported by a Hughes Doctoral Fellowship (first author), and in part by the National Science Foundation under Grant ECS 81-10035 (second author). First author’s current address: School of Management, University of Massachusetts, Amherst, MA 01003.
376 0022~0000/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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play to a full p moves. (Thus p < n = 1VI, and the winner is Pl or P2 according as p is odd or even.) If G( V, W) is a forced p-win for some p, we call it a forced win. We were led to the games G( V, IV) while investigating the Isolation Game Z(H) proposed by Ringeisen [14]. This game is played beginning with a graph H= (V, E); the players alternately modify the graph, with the objective of being the first to isolate a vertex. The “modifications” allowed by the rules will not be repeated here, but do not explicitly define Z(H) as a Set Coincidence Game; we have shown in [ 151, however, that Z(H) = G( V, IV), where W consists of all vertexneighborhoods in H plus their complements. This observation proved sufficient n [ 151 to permit the analysis of Z(H) for a number of classes of graphs H, but a fully general analysis was thwarted by our inability to find a proof-facilitating recursive structure: the result of a partial play of Z(H) does not seem to correspond to any Z(H’), a consequence of the “symmetry-spoiling” presence of each u E V in the complement of its neighborhood. This motivated imbedding the Isolation Games in a larger class of games which do admit recursive treatment. That the games G( I’, W) indeed form such a class is readily seen. For, consider a partial play of G( I’, W) which has not yielded a win or a draw, and as above, let S denote the set of elements selected so far (by both players). Then the resultant continuation game, denoted G( I’, W, S), is readily seen to coincide with the game G(V-S, W,), where W,= (w-S#@: w E W, S c w}. (This is precisely the notion of “induced hypergraph.“) In particular, G( V, W) is a forced win for Pl iff either (i) it is a forced l-win (i.e., W includes a singleton) or (ii) at least one of the continuation games {G( V, W, {II} ): u E V} is a forced win for its second player. Since checking W for singletons can be regarded as trivial, we see that the problem of determining whether forced winnability by Pl holds can be reduced to the corresponding problem (on a smaller game) for P2. Thus what follows will often concentrate on the latter problem, with assurance that no loss of generality can result. Except where continuation games are involved, the set V of elements affects G( V, W) only via its cardinality n, and the above-mentioned recursive arguments will involve induction on n. We will therefore generally write G(n, W) instead of G( V, W), implicitly assuming V = { 1, 2, .... n}, when no ambiguity is possible. An natural first problem is: given any particular G(n, W), is it a forced win for Pl or a forced win for P2 or a draw? In Section 2 we shall show-for a “tightened” encoding of G(n, W)-that the decision problem for forced-winnability by P2 is PSPACE-complete. This complements the known result [S] that positional games of the first type [3] (and therefore the general class of positional games) are complete in PSPACE. For positional games of the second type, it is known [3] that P2 cannot have a forced win; we do not know the status of the decision problem for forced-winnability by P 1. Another concern here, however, is with the combinatorial optimization problem ZZ(n, p): min{ I WI : G(n, W) a forced p-win} which might confront a game designer required to produce a forced p-win using a
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limited allowance of winning sets. Analysis of n(n, p) is aided by visualizing the Hasse diagram of subsets of V as a digraph D,, with an arc from each node at level 2 of the digraph (these are just the I-sets of V) for 1=0, 1, .... n- 1, to each of the (2 + I)-sets that contains it. Each play of G(n, W) corresponds to a path in D,, beginning at the root-node (4) of D, and rising through nodes SL at successive levels 1 until terminated either by reaching some WE W (a win) or by winlessly reaching the single level-n node V (a draw). Thus S1 denotes the “value” of the expanding set S just after move ;1. We use the term “trajectory” to denote the sequence of subsets of even cardinality encoutered along a path. Note that feasibility of Z7(n, p) is not in question, since choosing W to consist of all level-p nodes certainly yields a forced p-win. In Section 3, we show that in game G(n, 12/), for any fixed strategy of P2, the minimum width of the tree of “attainable” play-trajectories (corresponding to the different strategies for Pl) increases rapidly as the tree rises from level to level, until the mid-level [n/2] is reached. A further result, which seems noteworthy, is that the width-minimizing strategies are precisely those of the “symmetric” or “pairing” type which arise so often in the solution of specific combinatorical games. A parallel is suggested, but not formalized, between the theory developed here and the notion of “entropy” in statistical thermodynamics. Our results suggest the intuition that unless n-p is small (p even), an optimal solution W of n(n, p) must place its meager number of winning sets within D, so as to limit play at the lower levels to a very few trajectories, in the sense that deviations by the winning player P2 are punished by “loss of the win” (permitting the opponent to draw or win), while deviations by Pl are punished by premature loss. As will be shown in a subsequent paper (based on Chapter 5 of [ 16]), formalization of this intuition in the special case of the Isolation Game is sufficient to permit a general analysis of it, with the surprising outcome that (apart from a few identified possible exceptions) these games can be forced-won only either very early (p 6 5) or very late (p = n - 2). Before beginning the body of the paper, we remind the reader of the notation S, defined above, and introduce the notation W, for the family of winning l-sets in G(n, W). The complement of a set B, with respect to some context-specified superset, will be denoted B’.
2. COMPLEXITY In this section, after observing how two other well-known games are related to G(n, W) (one can be transformed into G(n, W), the other can be transformed-to from G(n, W)), we show that a more concise version of the Set Coincidence Game is PSPACE-complete, which places it in the same category of difficulty as (generalized versions of) Hex, Checkers, and the Grundy game. The Grundy Game (Berge [2]) is a two-player game on a simple, directed graph G’ with arc-set A and distinguished “initial” vertex v,,. Pl selects a vertex v, from T(u,) = {v E V: (v,, v) E A }. P2 then selects any vertex v2 from the analogous
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successor-set Z(v,), and so on. If one player selects a vertex vk with f(v,) = @, then that player wins the game. If there are circuits in the graph, then the game need not terminate. A non-negative integer function g(x) defined on V is called a Grundy function if for every DE V, g(v) is the smallest non-negative integer which does not belong to g(T(v)). A graph may admit multiple, unique, or no Grundy functions. If G has a Grundy function, then K= {v E V: g(v) = 0) is a kernel of G (i.e., v E K=> f(v) n K= @ and v 4 Ka T(v) n Kf a). It is easy to see that the player who first chooses a vertex v E K is assured of a win or draw with intelligent play (confining his moves to K). Given an instance of the Set Coincidence Game G(n, IV), if G’ is now taken to be the (immense) graph with vertex set the power set of { 1,2, .... n} and with arc-set A = {(v, MJ)~~v=vu (t>, where t ED’ and 11is not a winning set}, then the Grundy game on G’ with play beginning at @ is equivalent to G(n, IV). The Alternate Hitting Set game (Garey and Johnson [lo]), denoted AH(C), is a two-player game on a collection C of subsets of V= { 1, .... n}. The players alternately choose a new element of I’, until for each c E C, some element of c has been chosen. The player whose choice causes this to happen wins. We may assume that C is a clutter (a family of pairwise incomparable sets), since AH(C) is in an obvious sense unchanged if C is replaced by the family of minimal members of C. If we put W’ = {MIc VI w n c # a, for all c E C}, then AH(C) is clearly equivalent to G(n, W’), which is in turn equivalent to G(n, W), where clutter W consists of all minimal members of W’. Because the “blocking clutter” relationship of W to C is reciprocal (Edmonds and Fulkerson [7]), we actually have a Ill correspondence between “minimal” alternate hitting games and “minimal” set coincidence games on V. Let PSPACE denote the class of all decision problems solvable by a polynomially space bounded Deterministic Turing Machine (DTM). A problem flak PSPACE is called PSPACE-complete if all problems in PSPACE can be polynomial-time transformed to Z7. (The class of PSPACE-complete problems thus consists of the “hardest” problems in the class PSPACE.) Since the class PSPACE can be easily shown to contain NP, such PSPACE-complete problems may be thought of as even “harder” than the problems in NP. Many hard combinatorical games are PSPACE-complete. The fact that so many games (e.g., “generalized’ checkers [lo] and chess [ 181, hex, Grundy game, positional games [3], Boolean Games [ 171) are known to be PSPACE-complete may be due to the fact that one of the problems first shown to be PSPACE-complete (namely Quantified Boolean Satisfiability) lends itself so nicely to transformation into a combinatorical game. For more details the reader is referred to Garey and Johnson [lo], and Even and Tarjan [S]. It is thought that G(n, W) as defined above is not PSPACE-complete, because of the “padding out” which the explicit listing of individual winning sets gives to the input length. We will give a more reasonable “path” encoding of the situation, and then show that the resulting game is PSPACE-complete. It should be observed that 571
39 3-Y
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for the purpose of the complexity analysis, we have, strictly speaking, defined a new game. That new game, however, is easier to treat than its more crudely expressed predecessor, and seems an equally natural account of how one would describe such a game after a little thought. A L-set A in V will be called feasible if it can arise as Sn in some play of G = G(n, W)-i.e., if digraph D, contains a path from a to A none of whose intermediate nodes lie in W. A feasible set A is called safe if A $ W and if when S= A there is no risk of immediate loss, i.e., no choice of a move by the opponent giving the opponent a win on the next level. Given an instance of G(n, W), suppose that V contains nonempty pairwise disjoint subsets X,, X,, .... XK such that (i) the family SF, of safe feasible sets at each level 1~ K of play of G(n, W) consists precisely of the &sets obtained by “unordering” the length-l sequences from X,xX,x
... XX,,
and (ii) the adjunction to any member of SF,_, of any element outside X, risks loss at move K+ 1. The (implicit) product notation asserting this situation is given by
[X,17 CXJ? [X31?...Y[X/cl. Note that this is not an encoding of winning sets, for many different configurations of winning sets can induce a given family of feasible sets. (For example: if n = 8 and W, = /zr then the following winning sets on levels 2 and 3, W, = { (2, 3}, (4, 5j, j6, 71, (7, 8)) W,={(l,4,5}, induce the path encoding [l],
{1,67},
{1>7,8}},
[2, 31.)
Remark. For any Xi, X,, .... X, c V, IUX,l
~~~xx,~,,{u,w}~v-(X,uT)},
where we have abused notation by confounding the sequences in X, x X,X. . . x XL with the corresponding (unordered) A-sets. In addition to the preceding product notation, a path encoding also includes another notion, called subset matching. If there exists a collection of subsets Yi, i = 1, ...) R, such that the game is won at move k iff the set Sk contains one of the sets Y,, then this fact is written down as (kl Y,, i= 1, .... R). If R and the ) Yl’s are “small,” this notation clearly has abbreviative power.
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THE SET COINCIDENCE GAME
(Continuing with the previous example, if the matching subset on level 4 were Y= { 1,6} then W, could be encoded as (41{ 1,6}).) All remaining winning sets are given singly, in the obvious manner. It is easy to see why listing in their entirety the sets abbreviated above might “pad” the game so that G(n, W) might well not be PSPACE-complete. We will refer to our new game as G(n, Wt ). THEOREM
2.1. The decision problem ,for
winnability qf G(n, W,)
hi, P2 is
PSPACE-complete. Proof We will denote this decision problem by D. First we show that D is in PSPACE. Since the game is on n vertices, at most n moves can be made in any play of G(n, W,). In order to analyse any instance of G(n, W,), one could in principle construct a game tree and show, by exhaustive search, that P2 has, or does not have, a forced win. We construct our game tree as a directed, rooted tree with the root representing the initial set of chosen vertices S, = 0, and each node representing a position in the game (ordered to take account of the history of moves to reach that position). The descendants of node x are those nodes representing the result of a play one move farther on from x. A node is called a l-node if it is Pl’s turn to move at that node, and a 2-node if it is P2’s turn to move. The game tree T is of depth at most n and is easily shown to contain at most Cy=, (n!/i!) nodes (many of which correspond to the same set of switched-at vertices but to a different order of switching). For any node x, let W(x) = 1 if P2 has a forced win from node x; W(x) = 0 otherwise. Thus W(x) = 1 if x corresponds to a winning set for P2 (i.e., of even size), W(x) = 0 if x corresponds to a winning set for Pl, while for all nodes that are not winning sets the function W can be defined recursively:
If x is a 2-node, then W(x) = 1 iff there exists at least one descendant d of .Y such that W(d)= 1. If x is a l-node, then W(x) = 1 iff all descendants d of x satisfy W(d) = 1. The decision problem D is to calculate the value of W(root). For details we implicitly refer to Even and Tarjan [8] in showing that D lies in PSPACE. The calculation of W(root) can be done by depth-first search which can halt as soon as a zero W-value is encountered. The stack and workspace for such a search require only O(n2 . log n) bits, since each node is representable by at most n . log n bits and the tree depth is dn. The length it4 of any “reasonable” (cf. [lo, pp. 9-101) input representation of the structure of G(n, W,), is at least n bits. Thus the required search space, at most M + O(n2 .log n), is bounded above by a (low-order) polynomial in the input length A4, so the Set Coincidence Game belongs to PSPACE. We must now give a polynomial-space transformation to D from a problem that is known to be PSPACE-complete. (If D is in P, then we would have P= PSPACE, so D is among the “hardest” of the problems in PSPACE.) We give a transforma-
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tion from QUANTIFIED 3SATISFIABILITY (abbreviated known instance of a PSPACE-complete problem.
Q34AT),
a well-
QUANTIFIED 3SATISFIABILITY. Instance. A well-formed quantified Boolean formula
where E is a conjunction of some set C of 34teral disjunctive clauses ci in the m Boolean variables xi, and each Qi is either a universal or an existential quantifier. The intuitive idea of the transformation is simple. The winning sets for G(n, Wr) at levels before the “end-game” are formed so as to force P2 to play the existential quantifiers and Pl to play the universal quantifiers. Thus the game begins with a “variable-setting” stage during which the Boolean variables xi are successively set to T or F, i.e., xi or its complement Xi is chosen by PI or P2 according as Qi is universal or existential. On any move X, a player is forced to pass if he is not the one allowed to set the next available variable. (This is done by forcing the choice of a “passing” element, pi..) With the variable-setting stage complete, the end-game corresponding to the last two levels is reached: Pl chooses one of the clauses cj, and P2 can complete a winning set iff at least one of the variables in that clause (an xi or its complement Xi) has previously been set to T. (An extra pass is required before Pl can choose cj, if Qm is universal.) For example, if B is
where E has four clauses c, , c2, c3, cd in it, then our path encoding for the levels in the variable-setting stage and the first end-game move is given in product form as
Now for the formal description. Given an m-quantilier instance B of Q3-SAT, the corresponding instance of G(n, IV,-) is defined on a vertex-set V which is the union of
(xi, xi, Xi, Xi: i= 1, .... m},
C,
{p,&PASS},
where J E PASS if one of the players is required to pass on move A, and where xi, X: arise (seee below) only at the last move. (It is clear that max{I: 1~ PASS} < 2m.) The corresponding instance of G(n, Wr) is thus formed on n = 4m + ICI + [PASS1 elements. The determination of PASS, and the product-notation encoding as [X,1, cx21, ...YCXLI, cc1 of the safe feasible sets at all levels but the last, are carried out by the following algorithm:
THE SET COINCIDENCE GAME
383
( 1)
Initialize PASS = a, I = 1, i = 1.
(2)
If i
(3)
If Q, is universal, then set PASS = PASS u {A}, Xi. = {pL}, L = i, and
(4)
Set L = 1,- 1 and stop.
go to (5).
stop. (5) If i is odd and Qi universal, or 1 is even and Qi existential, then set X;.= (xi, Xi}, increment I and i by 1, and return to (2). (6)
Set PASS = PASS u {A}, X, = {pA}, increment I by 1, return to (2).
The remaining winning sets are expressed via subset matching as (where L + 2 denotes the level following the clause-choosing level) (L+21 Y(Lj);I=
1,2, 3;j=
1, ...) IC(),
where three sets Y(Z,j) correspond to each clause ci in the expression E in the instance of Q3-SAT. To each of the three variables z (an xi or X,) in clausej there corresponds one subset Y(I, j) = {z, z’, c, }. It is clear that this transformation may be accomplished in polynomial time, hence in polynomial space. Now suppose that the instance of Q3-SAT is satisfiable. Then in the corresponding instance of G(n, IV’,), P2 can control the choice of settings for variables to be a satisfiable one, and so whichever clause c, Pl chooses at move L, at least one of its variables must be true. Call this variable z. Then z was chosen during the variable-setting stage, and since z is in clause j, P2 can respond to Pl’s choice of cj by choosing element z’, thus forcing the set of chosen vertices at level L + 2 to contain the subset {z, z’, c,}, and hence winning the game. Conversely, if the corresponding instance of G(n, Wr.) is a forced win for P2 then it must be a forced (L + 2)-win, for the specified product-notation structure of W,allows Pl to force the game to last for L + 2 moves. If play lasts to L + 2 moves them Pl must not have been able to choose any settings for the universally quantified variables for which P2 could not set the existentially quantified variables so that all clauses c, could be used by P2 to complete a Y(1,i), i.e., so that at least one of the variables in each clause matches one of the variables previously set to T. Thus B is true, and the theorem is proved. 1 3. ATTAINABILITY We turn now to a formalization and analysis of attainability of subsets of V at the various levels of digraph D,, given a fixed strategy for one of the players. Our interest is in demonstrating the rapid level-by-level growth in the minimum number of attainable sets, as a potent factor in (hence, prior to) the selection of W by a game-designer confronted with problem n(n, p). Therefore, this section works with G(n, 0) rather than any particular G(n, W); paths and trajectories will not be truncated by winning sets.
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We will define a strategy c for either player to consist of (i) a sequence of families A, = A,(a) of attainable sets at the various levels 2, together with (ii) a sequence of substrategies, each defined from the attainable sets at a level of parity opposite the player’s to those on the next level. The definition will involve an intertwined recursion, and also the notion of a map r: A -+ V (where A E 2 “) being nonduplieating if r(a) q!a holds for all a E A. The notation A, = (a{: j= 1,2, .... /AnI } will be used for the family of attainable sets at level A. Our definition is a convenient variant of the “behavioural strategy” concept in extensive-game theory (Kuhn [13]), and is readily adaptable from G(n, @) to G(n, W). P2: A strategy 0 for P2 is a sequence A,,, A,, ci, Al, AX, 03, .... A,, where A,= {OL A 2,+, 02m-1.
.A
2m-1
--+
V
= {a&,-2u
IA,1 = 1, (v}: u$a<,_,,
is any nonduplicating
A2m= {a:,_,
4=0, j= 1, .... IA,,_,l}.
map, and
u {c2m-l(a&,,_-l)}:j=
1, .... IA,,,-,I}.
A strategy and its attainable sets are defined analogously for Pl. (The notation has suppressed the argument c of the level-by-level families An of attainable sets.) We next verify that “attainability” has the properties suggested by its name: LEMMA 3.1. Zf either player uses a fixed strategy ct that is known to the other player, then the other player can force SL to be any a, E Al of his choosing for any i. Furthermore, at any level 1 of the game, Sj.e Ai.
Proof We shall prove the statement in the case where P2 is following the fixed strategy. We shall prove both parts by induction. For 1= 1, Pl can force S, to be any single member of A, = (1, .... n}, so that both statements are true here. Now suppose both statements are true for a given 1= 2m - 1. We shall prove that both statements are true for 1= 2m and 1= 2m + 1. Take any a2,,, E A,,. By the recursive formula, there exists a,,,_ 1E A,,_ 1 such that a 2m=a2m-Iu (02mpl(aZm-1)). BY th e induction hypothesis, Pl can force S2,,,_ , to be a,,,_ 1. Since by the strategy 0, P2 will augment the set a,,_, to a*,,,, we have proved the first statement for 2 = 2m. If J = 2m + 1, let a,,, , be any member of A,, + , . By definition, a*,,, + 1 = azmu {u} for some a,,,, E A,, and some u $ a2m. PI can as above force Slm = azm and can then choose the appropriate element u to complete a2,,,+ 1. and the definition of the attainable sets forces We know S2m-1~A2,_,, Since a similar argument holds for 2m + 1, the lemma is proved. 1 S2m~A2,.
An important concept is that of a symmetric strategy. A player can play a symmetric strategy iff that player’s parity is the same as that of n. If n = 2k, the elements are paired off into k pairs and P2 “completes the pair” at each stage. If n = 2k + 1, Pl switches first on any element, then pairs off the remaining elements and completes the pair for each move.
THE SET COINCIDENCE CiAME
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It is easily seen that for a symmetric strategy rr, IA,,,,1= (,“) if n = 2k, while Th e next theorem shows that the preceding relations IA2m+ll=(k) if n=2k+l. exhibit the symmetric strategies as extremal (minimizing) for all 1A,,1 and IA,, + , /. This optimality provides the intuition behind some of the constructions in [ 151 of forced wins and winning strategies for isolation games I(H), and in other analyses (e.g., [9]) of combinatorical games, as well as for our subsequent introduction (in [16]) of “~-filters.” There appears to be an intriguing conceptual parallel between the cardinality of the family of attainable sets at each level and the concept of “entropy” prevalent in thermodynamics. In the language of this analogy, the following two theorems give lower bounds on the “entropy” of the set of chosen vertices as the game progresses. To attempt to win the game, a player P must use the winning sets in choosing moves that limit the “entropy,” while the opponent P’ tries to increase the entropy beyond what P can subsequently reduce sufliciently to “force” the trajectory to some winning set. THEOREM 3.2. Zf n = 2k, then for every strategy cr jar equality at every even level iff d is symmetric.
P2, IA2,,J >, (:!). wirh
Proof. The proof is by induction on k, with correctness clear for k = 1. For k = 2 we find that (A,1 > 2 unless cs is symmetric, and so the theorem holds. Assume now that the theorem holds for k >, 2. We will show that it holds for k + 1. CLAIM 1. If n = 2(k + 1 ), then for every strategy o,for P2, euch element c uppears in at least (n,k ,) attainable sets on level 2m.
Proof of Claim 1. Suppose Pl chooses element IJon the first move. P2 will reply with a,(v) to give Sz = {u, O,(V)}. We can view the continuation game G( V, (ZI, SZ) as play on n’= 2k elements, level 2(m - 1) in the continuation game corresponding to level 2m in the original game. Hence both v and a,(v) appear in at least (,,,! , ) attainable sets on level 2m, by the induction hypothesis. Since Pl could choose any c’on his first move, it follows that every element is in at least ( mk , ) attainable sets on level 2m. Consequence
of Claim 1.
Since all attainable sets a*,,, E AZ,,, are of size 2~2, we
have, unsing the claim, 2m IA2,,,13W+2)
(m/il),
since each of the 2k + 2 elements must appear in (,? , ) attainable sets on level 2m. Therefore, we have
with equality iff each element appears in exactly (,f. 1) attainable sets on level 2m. If equality holds for all m, then by the induction hypothesis play from each of the sets SZ = {u, a,(v)) must follow some symmetric strategy. 1
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The first of the theorem’s assertions has been proven. For the second, suppose now we have equality in (t) for all m. Then, in particular, the strategy cr gives rise to IA,1 = k + 1 attainable sets on level 2, and every element is in exactly one attainable 2-set. Since n = 2(k + l), we see that the attainable sets a2 pair the elements. Assume, without loss of generality, that this pairing off is given by (1,2), {3,4},..., (n-Ln>. w e now demonstrate that the symmetric strategies induced by e on each of the continuation games {G( V, W, a2): a2 E AZ} all use this same pairing of elements. This follows from: CLAIM 2. On level 4 the attainable sets are precisely those (k l ‘) sets attainable from the symmetric strategy with the pairing { 1,2}, {3,4}, .... (n - 1, n}.
Proof of Claim 2. Suppose {1,2,3,4}$A,. Let a2={1,2}EA2. Then by Claim 1, and the induction hypothesis applied to the continuation of play from a2, a, lies in exactly k attainable sets on level 4. The set a3 = (3,4} u { 1 } E A3 must be contained in at least one member a3 u {o,(a,)} of A,. That member cannot be any of the k already counted, else ( 1,2, 3,4} E Ad. So 1 must appear in members of A, at least k + 1 times. But since equality holds at each m, vertex 1 only appears k times on level 4, so contradiction is established. An identical argument applies to show that { 1,2,2i + 1,2i + 2) E A4 for all i 6 k, and then to show that A, contains the union of any two of the (k+ 1) pairs (1, 2}, {3,4}, .... {n- 1, n}. Since this accounts for the full number (“i ‘) of sets attainable on level 4, the claim is proved. 1 Consequence of Claim. Assuming equality in (7) for all m we have shown that beyond each attainable set on level 2 the action of r~ is symmetric, and with the same pairing as in the attainable sets at level 2. Thus B is symmetric. m The following similar theorems also hold; the logic of their proofs is identical to that for Theorem 3.2. THEOREM 3.3. Zf n = 2k + 1, then every strategy a for PI satisfies IA,,, ,I 2 (i), with equality at every odd level iff the strategy o is a symmetric one on V - {o,(fzI)}. THEOREM 3.4. Ifn = 2k + 1, then every strategy a for P2 satisfies 1A,,1 > (k) for all m; if n = 2k + 2, then every strategy o for Pl satisfies 1A,,,, + , I 2 (k) for all m.
Thus the symmetric strategies are those that permit one player to achieve maximum limitation of the diversity of possible paths treatable by the other player. The first two theorems’ minimum values, which by Theorem 3.4 are also lower bounds in the “opposite parities” cases, corroborate the intuition that the tree of possible trajectories expands so rapidly at levels n/2, then contracts so rapidly-that the designer of a forced p-win with an austere allowance of winning sets will have to place all of them either near the beginning of play (low levels) or near the end of play (high levels). The formalization of this intuition, and the combinatorial analysis to which it leads, will be reported separately (cf. [ 163 ).
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REFERENCES J. BECK AND L. CSIRMAZ,Variations on a game, J. Combin. Theory Ser. A 33 (1982) 297-315. C. BERGE,“Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. C. BERGE,Sur les jeux positionelles, Cahiers Centre Etudes Rech. OpPr. 18 (1976) 91-107. F. BUCKLEYAND F. HARARY,Diameter avoidance games for graphs, Bull. Malaysian Math. SW. 1 (1984) 29-33. 5. F. BUCKLEYAND F. HARARY,Closed geodetic games for graphs, Congr. Numer. 47 (1985) 131-138. 6. J. H. CONWAY,“On Numbers and Games,” Academic Press, New York/London, 1976. 7. J. EDMONUSAND D. R. FULKERSON,Bottleneck extrema, J. Combin. Theory 8 (1970), 299-306. 8. S. EVEN AND R. E. TARJAN, A combinatorial problem which is complete in polynomial space, J. Assoc. Comput. Mach. 23 (1976), 710-719. 9. M. GARDNER,Mathematical games, Sci. Amer. 197 (1957) 145-150. 10. M. R. CAREY AND D. S. JOHNSON,“Computers and Intractability; A Guide to the Theory of NP-Completeness,” Freeman, San Francisco, 1979. 11. F. HARARY,Achievement and avoidance games for graphs, Ann. Discrete Math. 13 (1982), 111--I19. 12. F. HARARY,An achievement game on a toroidal board, in “Lecture Notes in Mathematics, Vol. 1018” (M. Borowiecki, J. W. Kennedy, and M. M. Syslo, Eds.), Springer-Verlag, New York/Berlin, 1983. 13. H. W. KUHN, Extensive games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 570-576. 14. R. D. RINGEISEN,Isolation: A game on a graph, Math. Msg. 47 (1974) 132-138. 15. A. G. ROBINSON AND A. J. GOLDMAN,On Ringeisen’s isolation game, Discrete Math. 76 (1989), in press. 16. A. G. ROBINSONAND A. J. GOLDMAN,“The Isolation Game and Its Generalizations,” Technical Report No. 457, Department of Mathematical Sciences, The Johns Hopkins University, 1986. 17. T. J. SCHAEFER, Complexity of some two-person perfect-information games, J. Comput. Sysfems Sci. 16 (1978), 1855225. 18. J. A. STORER,On the complexity of chess, J. Comput. Systems Sci. 16 ( 1983) 77-100. 1. 2. 3. 4.
OF COMPUTER
AND
SYSTEM
SCIENCES
39, 376-387 (1989)
The Set Coincidence Complexity,
Attainability, A. G.
Game:
and Symmetric
Strategies*
ROBINSON AND A. J. GOLDMAN
Mathematical Sciences Department The Johns Hopkins University, Baltimore, Maryland 21218 ReceivedJune 9, 1987
We develop two closely related aspects of the complexity theory of the Set Coincidence Game G( V, W), in which two players alternately choose elements not previously chosen from a finite, nonempty set V, and W is a given family of nonempty subsets of V (the “winning sets”). The game is won by that player who first adds an element to the set of “chosen” elements S, so that SE W. We first show that the Set Coincidence Game (in an efticient encoding) is complete in PSPACE. We then develop the theory of attainability which we use to give intuition about the structure of minimal forced wins for either player, and about the “difftculty” of the game. We also show that a well-known type of strategy (“symmetric” or “pairing”) is optimal in the context of our definition of attainability. 0 1989 Academic press, hc.
1.
INTRODUCTION
The Set Coincidence Game G( V, W) is played on a finite nonempty set V of elements. W is a collection of nonempty subsets of V, the winning sets. Players Pl and P2 move alternately, with Pl leading off; at each turn, a player adds a new element to an expanding set S, which was empty at the start of play. If a player’s move causes S to coincide with some w E W, then that player wins (the opponent loses), and play ends. If V is exhausted (i.e., S = V) without a win, then the game is drawn. The fact that both players’ choices contribute to building up a single set S, rather than individual sets S’ and S2, s&ices to differentiate G( V, W) from the morestudied “positional games” of Berge [3], whose more specialized “types 1 and 2” [3], called “amoeba games” (weak and strong) in Beck and Csirmaz [l], in turn include most of the “achievement and avoidance” games of Harary (e.g., [ 11, 121). On the other hand, the diameter and geodesic achievement games of Buckley and Harary [4, 51 are set coincidence games. G( V, W) will be called a forced p-win if one of the players has a strategy assuring a win in no more than p moves, but the opponent has at least one way to prolong * Research supported by a Hughes Doctoral Fellowship (first author), and in part by the National Science Foundation under Grant ECS 81-10035 (second author). First author’s current address: School of Management, University of Massachusetts, Amherst, MA 01003.
376 0022~0000/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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play to a full p moves. (Thus p < n = 1VI, and the winner is Pl or P2 according as p is odd or even.) If G( V, W) is a forced p-win for some p, we call it a forced win. We were led to the games G( V, IV) while investigating the Isolation Game Z(H) proposed by Ringeisen [14]. This game is played beginning with a graph H= (V, E); the players alternately modify the graph, with the objective of being the first to isolate a vertex. The “modifications” allowed by the rules will not be repeated here, but do not explicitly define Z(H) as a Set Coincidence Game; we have shown in [ 151, however, that Z(H) = G( V, IV), where W consists of all vertexneighborhoods in H plus their complements. This observation proved sufficient n [ 151 to permit the analysis of Z(H) for a number of classes of graphs H, but a fully general analysis was thwarted by our inability to find a proof-facilitating recursive structure: the result of a partial play of Z(H) does not seem to correspond to any Z(H’), a consequence of the “symmetry-spoiling” presence of each u E V in the complement of its neighborhood. This motivated imbedding the Isolation Games in a larger class of games which do admit recursive treatment. That the games G( I’, W) indeed form such a class is readily seen. For, consider a partial play of G( I’, W) which has not yielded a win or a draw, and as above, let S denote the set of elements selected so far (by both players). Then the resultant continuation game, denoted G( I’, W, S), is readily seen to coincide with the game G(V-S, W,), where W,= (w-S#@: w E W, S c w}. (This is precisely the notion of “induced hypergraph.“) In particular, G( V, W) is a forced win for Pl iff either (i) it is a forced l-win (i.e., W includes a singleton) or (ii) at least one of the continuation games {G( V, W, {II} ): u E V} is a forced win for its second player. Since checking W for singletons can be regarded as trivial, we see that the problem of determining whether forced winnability by Pl holds can be reduced to the corresponding problem (on a smaller game) for P2. Thus what follows will often concentrate on the latter problem, with assurance that no loss of generality can result. Except where continuation games are involved, the set V of elements affects G( V, W) only via its cardinality n, and the above-mentioned recursive arguments will involve induction on n. We will therefore generally write G(n, W) instead of G( V, W), implicitly assuming V = { 1, 2, .... n}, when no ambiguity is possible. An natural first problem is: given any particular G(n, W), is it a forced win for Pl or a forced win for P2 or a draw? In Section 2 we shall show-for a “tightened” encoding of G(n, W)-that the decision problem for forced-winnability by P2 is PSPACE-complete. This complements the known result [S] that positional games of the first type [3] (and therefore the general class of positional games) are complete in PSPACE. For positional games of the second type, it is known [3] that P2 cannot have a forced win; we do not know the status of the decision problem for forced-winnability by P 1. Another concern here, however, is with the combinatorial optimization problem ZZ(n, p): min{ I WI : G(n, W) a forced p-win} which might confront a game designer required to produce a forced p-win using a
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limited allowance of winning sets. Analysis of n(n, p) is aided by visualizing the Hasse diagram of subsets of V as a digraph D,, with an arc from each node at level 2 of the digraph (these are just the I-sets of V) for 1=0, 1, .... n- 1, to each of the (2 + I)-sets that contains it. Each play of G(n, W) corresponds to a path in D,, beginning at the root-node (4) of D, and rising through nodes SL at successive levels 1 until terminated either by reaching some WE W (a win) or by winlessly reaching the single level-n node V (a draw). Thus S1 denotes the “value” of the expanding set S just after move ;1. We use the term “trajectory” to denote the sequence of subsets of even cardinality encoutered along a path. Note that feasibility of Z7(n, p) is not in question, since choosing W to consist of all level-p nodes certainly yields a forced p-win. In Section 3, we show that in game G(n, 12/), for any fixed strategy of P2, the minimum width of the tree of “attainable” play-trajectories (corresponding to the different strategies for Pl) increases rapidly as the tree rises from level to level, until the mid-level [n/2] is reached. A further result, which seems noteworthy, is that the width-minimizing strategies are precisely those of the “symmetric” or “pairing” type which arise so often in the solution of specific combinatorical games. A parallel is suggested, but not formalized, between the theory developed here and the notion of “entropy” in statistical thermodynamics. Our results suggest the intuition that unless n-p is small (p even), an optimal solution W of n(n, p) must place its meager number of winning sets within D, so as to limit play at the lower levels to a very few trajectories, in the sense that deviations by the winning player P2 are punished by “loss of the win” (permitting the opponent to draw or win), while deviations by Pl are punished by premature loss. As will be shown in a subsequent paper (based on Chapter 5 of [ 16]), formalization of this intuition in the special case of the Isolation Game is sufficient to permit a general analysis of it, with the surprising outcome that (apart from a few identified possible exceptions) these games can be forced-won only either very early (p 6 5) or very late (p = n - 2). Before beginning the body of the paper, we remind the reader of the notation S, defined above, and introduce the notation W, for the family of winning l-sets in G(n, W). The complement of a set B, with respect to some context-specified superset, will be denoted B’.
2. COMPLEXITY In this section, after observing how two other well-known games are related to G(n, W) (one can be transformed into G(n, W), the other can be transformed-to from G(n, W)), we show that a more concise version of the Set Coincidence Game is PSPACE-complete, which places it in the same category of difficulty as (generalized versions of) Hex, Checkers, and the Grundy game. The Grundy Game (Berge [2]) is a two-player game on a simple, directed graph G’ with arc-set A and distinguished “initial” vertex v,,. Pl selects a vertex v, from T(u,) = {v E V: (v,, v) E A }. P2 then selects any vertex v2 from the analogous
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successor-set Z(v,), and so on. If one player selects a vertex vk with f(v,) = @, then that player wins the game. If there are circuits in the graph, then the game need not terminate. A non-negative integer function g(x) defined on V is called a Grundy function if for every DE V, g(v) is the smallest non-negative integer which does not belong to g(T(v)). A graph may admit multiple, unique, or no Grundy functions. If G has a Grundy function, then K= {v E V: g(v) = 0) is a kernel of G (i.e., v E K=> f(v) n K= @ and v 4 Ka T(v) n Kf a). It is easy to see that the player who first chooses a vertex v E K is assured of a win or draw with intelligent play (confining his moves to K). Given an instance of the Set Coincidence Game G(n, IV), if G’ is now taken to be the (immense) graph with vertex set the power set of { 1,2, .... n} and with arc-set A = {(v, MJ)~~v=vu (t>, where t ED’ and 11is not a winning set}, then the Grundy game on G’ with play beginning at @ is equivalent to G(n, IV). The Alternate Hitting Set game (Garey and Johnson [lo]), denoted AH(C), is a two-player game on a collection C of subsets of V= { 1, .... n}. The players alternately choose a new element of I’, until for each c E C, some element of c has been chosen. The player whose choice causes this to happen wins. We may assume that C is a clutter (a family of pairwise incomparable sets), since AH(C) is in an obvious sense unchanged if C is replaced by the family of minimal members of C. If we put W’ = {MIc VI w n c # a, for all c E C}, then AH(C) is clearly equivalent to G(n, W’), which is in turn equivalent to G(n, W), where clutter W consists of all minimal members of W’. Because the “blocking clutter” relationship of W to C is reciprocal (Edmonds and Fulkerson [7]), we actually have a Ill correspondence between “minimal” alternate hitting games and “minimal” set coincidence games on V. Let PSPACE denote the class of all decision problems solvable by a polynomially space bounded Deterministic Turing Machine (DTM). A problem flak PSPACE is called PSPACE-complete if all problems in PSPACE can be polynomial-time transformed to Z7. (The class of PSPACE-complete problems thus consists of the “hardest” problems in the class PSPACE.) Since the class PSPACE can be easily shown to contain NP, such PSPACE-complete problems may be thought of as even “harder” than the problems in NP. Many hard combinatorical games are PSPACE-complete. The fact that so many games (e.g., “generalized’ checkers [lo] and chess [ 181, hex, Grundy game, positional games [3], Boolean Games [ 171) are known to be PSPACE-complete may be due to the fact that one of the problems first shown to be PSPACE-complete (namely Quantified Boolean Satisfiability) lends itself so nicely to transformation into a combinatorical game. For more details the reader is referred to Garey and Johnson [lo], and Even and Tarjan [S]. It is thought that G(n, W) as defined above is not PSPACE-complete, because of the “padding out” which the explicit listing of individual winning sets gives to the input length. We will give a more reasonable “path” encoding of the situation, and then show that the resulting game is PSPACE-complete. It should be observed that 571
39 3-Y
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for the purpose of the complexity analysis, we have, strictly speaking, defined a new game. That new game, however, is easier to treat than its more crudely expressed predecessor, and seems an equally natural account of how one would describe such a game after a little thought. A L-set A in V will be called feasible if it can arise as Sn in some play of G = G(n, W)-i.e., if digraph D, contains a path from a to A none of whose intermediate nodes lie in W. A feasible set A is called safe if A $ W and if when S= A there is no risk of immediate loss, i.e., no choice of a move by the opponent giving the opponent a win on the next level. Given an instance of G(n, W), suppose that V contains nonempty pairwise disjoint subsets X,, X,, .... XK such that (i) the family SF, of safe feasible sets at each level 1~ K of play of G(n, W) consists precisely of the &sets obtained by “unordering” the length-l sequences from X,xX,x
... XX,,
and (ii) the adjunction to any member of SF,_, of any element outside X, risks loss at move K+ 1. The (implicit) product notation asserting this situation is given by
[X,17 CXJ? [X31?...Y[X/cl. Note that this is not an encoding of winning sets, for many different configurations of winning sets can induce a given family of feasible sets. (For example: if n = 8 and W, = /zr then the following winning sets on levels 2 and 3, W, = { (2, 3}, (4, 5j, j6, 71, (7, 8)) W,={(l,4,5}, induce the path encoding [l],
{1,67},
{1>7,8}},
[2, 31.)
Remark. For any Xi, X,, .... X, c V, IUX,l
~~~xx,~,,{u,w}~v-(X,uT)},
where we have abused notation by confounding the sequences in X, x X,X. . . x XL with the corresponding (unordered) A-sets. In addition to the preceding product notation, a path encoding also includes another notion, called subset matching. If there exists a collection of subsets Yi, i = 1, ...) R, such that the game is won at move k iff the set Sk contains one of the sets Y,, then this fact is written down as (kl Y,, i= 1, .... R). If R and the ) Yl’s are “small,” this notation clearly has abbreviative power.
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(Continuing with the previous example, if the matching subset on level 4 were Y= { 1,6} then W, could be encoded as (41{ 1,6}).) All remaining winning sets are given singly, in the obvious manner. It is easy to see why listing in their entirety the sets abbreviated above might “pad” the game so that G(n, W) might well not be PSPACE-complete. We will refer to our new game as G(n, Wt ). THEOREM
2.1. The decision problem ,for
winnability qf G(n, W,)
hi, P2 is
PSPACE-complete. Proof We will denote this decision problem by D. First we show that D is in PSPACE. Since the game is on n vertices, at most n moves can be made in any play of G(n, W,). In order to analyse any instance of G(n, W,), one could in principle construct a game tree and show, by exhaustive search, that P2 has, or does not have, a forced win. We construct our game tree as a directed, rooted tree with the root representing the initial set of chosen vertices S, = 0, and each node representing a position in the game (ordered to take account of the history of moves to reach that position). The descendants of node x are those nodes representing the result of a play one move farther on from x. A node is called a l-node if it is Pl’s turn to move at that node, and a 2-node if it is P2’s turn to move. The game tree T is of depth at most n and is easily shown to contain at most Cy=, (n!/i!) nodes (many of which correspond to the same set of switched-at vertices but to a different order of switching). For any node x, let W(x) = 1 if P2 has a forced win from node x; W(x) = 0 otherwise. Thus W(x) = 1 if x corresponds to a winning set for P2 (i.e., of even size), W(x) = 0 if x corresponds to a winning set for Pl, while for all nodes that are not winning sets the function W can be defined recursively:
If x is a 2-node, then W(x) = 1 iff there exists at least one descendant d of .Y such that W(d)= 1. If x is a l-node, then W(x) = 1 iff all descendants d of x satisfy W(d) = 1. The decision problem D is to calculate the value of W(root). For details we implicitly refer to Even and Tarjan [8] in showing that D lies in PSPACE. The calculation of W(root) can be done by depth-first search which can halt as soon as a zero W-value is encountered. The stack and workspace for such a search require only O(n2 . log n) bits, since each node is representable by at most n . log n bits and the tree depth is dn. The length it4 of any “reasonable” (cf. [lo, pp. 9-101) input representation of the structure of G(n, W,), is at least n bits. Thus the required search space, at most M + O(n2 .log n), is bounded above by a (low-order) polynomial in the input length A4, so the Set Coincidence Game belongs to PSPACE. We must now give a polynomial-space transformation to D from a problem that is known to be PSPACE-complete. (If D is in P, then we would have P= PSPACE, so D is among the “hardest” of the problems in PSPACE.) We give a transforma-
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tion from QUANTIFIED 3SATISFIABILITY (abbreviated known instance of a PSPACE-complete problem.
Q34AT),
a well-
QUANTIFIED 3SATISFIABILITY. Instance. A well-formed quantified Boolean formula
where E is a conjunction of some set C of 34teral disjunctive clauses ci in the m Boolean variables xi, and each Qi is either a universal or an existential quantifier. The intuitive idea of the transformation is simple. The winning sets for G(n, Wr) at levels before the “end-game” are formed so as to force P2 to play the existential quantifiers and Pl to play the universal quantifiers. Thus the game begins with a “variable-setting” stage during which the Boolean variables xi are successively set to T or F, i.e., xi or its complement Xi is chosen by PI or P2 according as Qi is universal or existential. On any move X, a player is forced to pass if he is not the one allowed to set the next available variable. (This is done by forcing the choice of a “passing” element, pi..) With the variable-setting stage complete, the end-game corresponding to the last two levels is reached: Pl chooses one of the clauses cj, and P2 can complete a winning set iff at least one of the variables in that clause (an xi or its complement Xi) has previously been set to T. (An extra pass is required before Pl can choose cj, if Qm is universal.) For example, if B is
where E has four clauses c, , c2, c3, cd in it, then our path encoding for the levels in the variable-setting stage and the first end-game move is given in product form as
Now for the formal description. Given an m-quantilier instance B of Q3-SAT, the corresponding instance of G(n, IV,-) is defined on a vertex-set V which is the union of
(xi, xi, Xi, Xi: i= 1, .... m},
C,
{p,&PASS},
where J E PASS if one of the players is required to pass on move A, and where xi, X: arise (seee below) only at the last move. (It is clear that max{I: 1~ PASS} < 2m.) The corresponding instance of G(n, Wr) is thus formed on n = 4m + ICI + [PASS1 elements. The determination of PASS, and the product-notation encoding as [X,1, cx21, ...YCXLI, cc1 of the safe feasible sets at all levels but the last, are carried out by the following algorithm:
THE SET COINCIDENCE GAME
383
( 1)
Initialize PASS = a, I = 1, i = 1.
(2)
If i
(3)
If Q, is universal, then set PASS = PASS u {A}, Xi. = {pL}, L = i, and
(4)
Set L = 1,- 1 and stop.
go to (5).
stop. (5) If i is odd and Qi universal, or 1 is even and Qi existential, then set X;.= (xi, Xi}, increment I and i by 1, and return to (2). (6)
Set PASS = PASS u {A}, X, = {pA}, increment I by 1, return to (2).
The remaining winning sets are expressed via subset matching as (where L + 2 denotes the level following the clause-choosing level) (L+21 Y(Lj);I=
1,2, 3;j=
1, ...) IC(),
where three sets Y(Z,j) correspond to each clause ci in the expression E in the instance of Q3-SAT. To each of the three variables z (an xi or X,) in clausej there corresponds one subset Y(I, j) = {z, z’, c, }. It is clear that this transformation may be accomplished in polynomial time, hence in polynomial space. Now suppose that the instance of Q3-SAT is satisfiable. Then in the corresponding instance of G(n, IV’,), P2 can control the choice of settings for variables to be a satisfiable one, and so whichever clause c, Pl chooses at move L, at least one of its variables must be true. Call this variable z. Then z was chosen during the variable-setting stage, and since z is in clause j, P2 can respond to Pl’s choice of cj by choosing element z’, thus forcing the set of chosen vertices at level L + 2 to contain the subset {z, z’, c,}, and hence winning the game. Conversely, if the corresponding instance of G(n, Wr.) is a forced win for P2 then it must be a forced (L + 2)-win, for the specified product-notation structure of W,allows Pl to force the game to last for L + 2 moves. If play lasts to L + 2 moves them Pl must not have been able to choose any settings for the universally quantified variables for which P2 could not set the existentially quantified variables so that all clauses c, could be used by P2 to complete a Y(1,i), i.e., so that at least one of the variables in each clause matches one of the variables previously set to T. Thus B is true, and the theorem is proved. 1 3. ATTAINABILITY We turn now to a formalization and analysis of attainability of subsets of V at the various levels of digraph D,, given a fixed strategy for one of the players. Our interest is in demonstrating the rapid level-by-level growth in the minimum number of attainable sets, as a potent factor in (hence, prior to) the selection of W by a game-designer confronted with problem n(n, p). Therefore, this section works with G(n, 0) rather than any particular G(n, W); paths and trajectories will not be truncated by winning sets.
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We will define a strategy c for either player to consist of (i) a sequence of families A, = A,(a) of attainable sets at the various levels 2, together with (ii) a sequence of substrategies, each defined from the attainable sets at a level of parity opposite the player’s to those on the next level. The definition will involve an intertwined recursion, and also the notion of a map r: A -+ V (where A E 2 “) being nonduplieating if r(a) q!a holds for all a E A. The notation A, = (a{: j= 1,2, .... /AnI } will be used for the family of attainable sets at level A. Our definition is a convenient variant of the “behavioural strategy” concept in extensive-game theory (Kuhn [13]), and is readily adaptable from G(n, @) to G(n, W). P2: A strategy 0 for P2 is a sequence A,,, A,, ci, Al, AX, 03, .... A,, where A,= {OL A 2,+, 02m-1.
.A
2m-1
--+
V
= {a&,-2u
IA,1 = 1, (v}: u$a<,_,,
is any nonduplicating
A2m= {a:,_,
4=0, j= 1, .... IA,,_,l}.
map, and
u {c2m-l(a&,,_-l)}:j=
1, .... IA,,,-,I}.
A strategy and its attainable sets are defined analogously for Pl. (The notation has suppressed the argument c of the level-by-level families An of attainable sets.) We next verify that “attainability” has the properties suggested by its name: LEMMA 3.1. Zf either player uses a fixed strategy ct that is known to the other player, then the other player can force SL to be any a, E Al of his choosing for any i. Furthermore, at any level 1 of the game, Sj.e Ai.
Proof We shall prove the statement in the case where P2 is following the fixed strategy. We shall prove both parts by induction. For 1= 1, Pl can force S, to be any single member of A, = (1, .... n}, so that both statements are true here. Now suppose both statements are true for a given 1= 2m - 1. We shall prove that both statements are true for 1= 2m and 1= 2m + 1. Take any a2,,, E A,,. By the recursive formula, there exists a,,,_ 1E A,,_ 1 such that a 2m=a2m-Iu (02mpl(aZm-1)). BY th e induction hypothesis, Pl can force S2,,,_ , to be a,,,_ 1. Since by the strategy 0, P2 will augment the set a,,_, to a*,,,, we have proved the first statement for 2 = 2m. If J = 2m + 1, let a,,, , be any member of A,, + , . By definition, a*,,, + 1 = azmu {u} for some a,,,, E A,, and some u $ a2m. PI can as above force Slm = azm and can then choose the appropriate element u to complete a2,,,+ 1. and the definition of the attainable sets forces We know S2m-1~A2,_,, Since a similar argument holds for 2m + 1, the lemma is proved. 1 S2m~A2,.
An important concept is that of a symmetric strategy. A player can play a symmetric strategy iff that player’s parity is the same as that of n. If n = 2k, the elements are paired off into k pairs and P2 “completes the pair” at each stage. If n = 2k + 1, Pl switches first on any element, then pairs off the remaining elements and completes the pair for each move.
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It is easily seen that for a symmetric strategy rr, IA,,,,1= (,“) if n = 2k, while Th e next theorem shows that the preceding relations IA2m+ll=(k) if n=2k+l. exhibit the symmetric strategies as extremal (minimizing) for all 1A,,1 and IA,, + , /. This optimality provides the intuition behind some of the constructions in [ 151 of forced wins and winning strategies for isolation games I(H), and in other analyses (e.g., [9]) of combinatorical games, as well as for our subsequent introduction (in [16]) of “~-filters.” There appears to be an intriguing conceptual parallel between the cardinality of the family of attainable sets at each level and the concept of “entropy” prevalent in thermodynamics. In the language of this analogy, the following two theorems give lower bounds on the “entropy” of the set of chosen vertices as the game progresses. To attempt to win the game, a player P must use the winning sets in choosing moves that limit the “entropy,” while the opponent P’ tries to increase the entropy beyond what P can subsequently reduce sufliciently to “force” the trajectory to some winning set. THEOREM 3.2. Zf n = 2k, then for every strategy cr jar equality at every even level iff d is symmetric.
P2, IA2,,J >, (:!). wirh
Proof. The proof is by induction on k, with correctness clear for k = 1. For k = 2 we find that (A,1 > 2 unless cs is symmetric, and so the theorem holds. Assume now that the theorem holds for k >, 2. We will show that it holds for k + 1. CLAIM 1. If n = 2(k + 1 ), then for every strategy o,for P2, euch element c uppears in at least (n,k ,) attainable sets on level 2m.
Proof of Claim 1. Suppose Pl chooses element IJon the first move. P2 will reply with a,(v) to give Sz = {u, O,(V)}. We can view the continuation game G( V, (ZI, SZ) as play on n’= 2k elements, level 2(m - 1) in the continuation game corresponding to level 2m in the original game. Hence both v and a,(v) appear in at least (,,,! , ) attainable sets on level 2m, by the induction hypothesis. Since Pl could choose any c’on his first move, it follows that every element is in at least ( mk , ) attainable sets on level 2m. Consequence
of Claim 1.
Since all attainable sets a*,,, E AZ,,, are of size 2~2, we
have, unsing the claim, 2m IA2,,,13W+2)
(m/il),
since each of the 2k + 2 elements must appear in (,? , ) attainable sets on level 2m. Therefore, we have
with equality iff each element appears in exactly (,f. 1) attainable sets on level 2m. If equality holds for all m, then by the induction hypothesis play from each of the sets SZ = {u, a,(v)) must follow some symmetric strategy. 1
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The first of the theorem’s assertions has been proven. For the second, suppose now we have equality in (t) for all m. Then, in particular, the strategy cr gives rise to IA,1 = k + 1 attainable sets on level 2, and every element is in exactly one attainable 2-set. Since n = 2(k + l), we see that the attainable sets a2 pair the elements. Assume, without loss of generality, that this pairing off is given by (1,2), {3,4},..., (n-Ln>. w e now demonstrate that the symmetric strategies induced by e on each of the continuation games {G( V, W, a2): a2 E AZ} all use this same pairing of elements. This follows from: CLAIM 2. On level 4 the attainable sets are precisely those (k l ‘) sets attainable from the symmetric strategy with the pairing { 1,2}, {3,4}, .... (n - 1, n}.
Proof of Claim 2. Suppose {1,2,3,4}$A,. Let a2={1,2}EA2. Then by Claim 1, and the induction hypothesis applied to the continuation of play from a2, a, lies in exactly k attainable sets on level 4. The set a3 = (3,4} u { 1 } E A3 must be contained in at least one member a3 u {o,(a,)} of A,. That member cannot be any of the k already counted, else ( 1,2, 3,4} E Ad. So 1 must appear in members of A, at least k + 1 times. But since equality holds at each m, vertex 1 only appears k times on level 4, so contradiction is established. An identical argument applies to show that { 1,2,2i + 1,2i + 2) E A4 for all i 6 k, and then to show that A, contains the union of any two of the (k+ 1) pairs (1, 2}, {3,4}, .... {n- 1, n}. Since this accounts for the full number (“i ‘) of sets attainable on level 4, the claim is proved. 1 Consequence of Claim. Assuming equality in (7) for all m we have shown that beyond each attainable set on level 2 the action of r~ is symmetric, and with the same pairing as in the attainable sets at level 2. Thus B is symmetric. m The following similar theorems also hold; the logic of their proofs is identical to that for Theorem 3.2. THEOREM 3.3. Zf n = 2k + 1, then every strategy a for PI satisfies IA,,, ,I 2 (i), with equality at every odd level iff the strategy o is a symmetric one on V - {o,(fzI)}. THEOREM 3.4. Ifn = 2k + 1, then every strategy a for P2 satisfies 1A,,1 > (k) for all m; if n = 2k + 2, then every strategy o for Pl satisfies 1A,,,, + , I 2 (k) for all m.
Thus the symmetric strategies are those that permit one player to achieve maximum limitation of the diversity of possible paths treatable by the other player. The first two theorems’ minimum values, which by Theorem 3.4 are also lower bounds in the “opposite parities” cases, corroborate the intuition that the tree of possible trajectories expands so rapidly at levels
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