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 Gʹ. D is a directed subgraph of Gʹ 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 fixed. In order to do this, we prove a separator theorem about grid graphs, which is interesting on its own right.
@article{788,
title = {Search for the end of a path in the d-dimensional grid and in other graphs},
journal = {ARS MATHEMATICA CONTEMPORANEA},
volume = {14},
year = {2017},
doi = {10.26493/1855-3974.788.89a},
language = {EN},
url = {http://dml.mathdoc.fr/item/788}
}
Gerbner, Dániel; Keszegh, Balázs; Pálvölgyi, Dömötör; Rote, Günter; Wiener, Gábor. Search for the end of a path in the d-dimensional grid and in other graphs. ARS MATHEMATICA CONTEMPORANEA, Tome 14 (2017) . doi : 10.26493/1855-3974.788.89a. http://gdmltest.u-ga.fr/item/788/