Inhalt des Dokuments
The Parameterized Complexity of Positional Games
Abdallah Saffidine (University of New South Wales)
We study the parameterized complexity of several positional games.
Our main result is that Short Generalized Hex is W-complete
parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999.
Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic
where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures
for a formula in this fragment is W-complete when parameterized by formula size.
We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices.
In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first
player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first
player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise.
Short Maker-Maker, Short Maker-Breaker, and Short Enforcer-Avoider are respectively AW[*]-, W-, and co-W-complete parameterized by the
number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first
level of the W-hierarchy when the winning condition only depends on which vertices one player has been able to pick, but AW[*]-complete
when it depends on which vertices both players have picked. However, some positional games with highly structured board and winning
configurations are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when
parameterized by the number of moves.
This is joint work with Édouard Bonnet, Serge Gaspers, Antonin Lambilliotte, and Stefan Rümmele (ICALP 2017).
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