Comparing the Power of Monadic NP Games

It is well-known that the complexity class NP coincides with the class of problems expressible in existential second-order logic (∑ 1 1 ). Monadic NP is the class of problems expressible in monadic ∑ 1 1 , i.e., ∑ 1 1 with the restriction that the second-order quantifiers range only over sets (as opposed to ranging over, say, binary relations). The author introduced a type of Ehrenfeucht-Fraisse game, called the monadic NP game, to prove that connectivity is not in monadic NP. Later, Ajtai and the author introduced another type of monadic NP game (the “Ajtai-Fagin monadic NP game“) to prove that directed reachability is not in monadic NP. Both games have two players (the spoiler and the duplicator), and involve coloring steps (where the players color nodes of the graphs) and selection steps (where the players select nodes of the graphs, round by round). It is known that the original game and the Ajtai-Fagin game are equivalent, in the sense that both characterize monadic NP. Thus, the duplicator has a winning strategy in the original game for every choice of parameters (number of colors and number of rounds) if and only if the duplicator has a winning strategy in the Ajtai-Fagin game for every choice of parameters. In this paper, we investigate the relationship between these games at a finer level. We show that in one sense, even at a finer level, Ajtai-Fagin monadic NP games are no stronger than the original monadic NP games. Specifically, we show that the families of graphs used in the Ajtai-Fagin game to prove that a problem is not in monadic NP can in principle be used in the original game to prove the same result (where for a given choice of parameters, bigger graphs of the same type are used for the original game than for the Ajtai-Fagin game). This answers an open question of Ajtai and the author. We also show that in another sense, Ajtai-Fagin games are stronger, in that there are situations where the spoiler requires more resources (colors) to win the Ajtai-Fagin game than the original game, when the choices of graphs are fixed. Our analysis gives a nonelementary upper bound, which we conjecture to be optimal, on the number of extra colors that are required for the spoiler to win the Ajtai-Fagin game than the original game.