Search for the end of a path in the d-dimensional grid and in other graphs

We consider the worst-case query complexity of some variants of certain PPAD-complete search problems. Suppose we are given a graph G and a vertex s ∈ V (G). We denote the directed graph obtained from G by directing all edges in both directions by G′. D is a directed subgraph of G′ which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in s. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex v ∈ V (G), and the answer is the set of the edges of D incident to v, together with their directions. We also show lower bounds for the special case when D consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph G is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as ∗Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary. [email protected]. Supported by the Hungarian National Science Fund (OTKA), grant PD 109537 †Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary. [email protected]. Supported by the Hungarian Scientific Research Fund (OTKA), grant PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP003), NK 78439, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and DAAD. ‡Institute of Mathematics, Eötvös University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary. [email protected]. Supported by the Hungarian Scientific Research Fund (OTKA), grant PD 104386 and NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. §Freie Universität Berlin, Institut für Informatik, Takustraße 9, 14195 Berlin, Germany. [email protected]. Supported by the ESF EUROCORES programme EuroGIGAVORONOI, Deutsche Forschungsgemeinschaft (DFG): RO 2338/5-1. ¶Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111, Budapest, Hungary. [email protected]. Supported by the Hungarian Scientific Research Fund (OTKA), grant 108947, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.