Abstract. In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a discrete time process on Z defined by parameters (p 1 , . . . p M ) ∈ [0, 1] M for some positive integer M , where the walker upon the i th visit to z ∈ Z moves to z + 1 with probability p i (mod M ) , and moves to z − 1 with probability 1 − p i (mod M ) . We give an explicit formula in terms of the parameters (p 1 , . . . , p M ) which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case thatall behaviors are possible, and may depend on the order of the p i . Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.

We define the following parameter of connected graphs. For a given graph G = (V, E) we place one agent in each vertex v ∈ V . Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of G (not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge become acquainted, and the process continues. We define the acquaintance time of a graph G, denoted by AC(G), to be the minimal number of rounds required until every two agents are acquainted. We first study the acquaintance time for some natural families of graphs including the path, expanders, the binary tree, and the complete bipartite graph. We also show that for all n ∈ N and for all positive integers k ≤ n 1.5 there exists an n-vertex graph G such that k/c ≤ AC(G) ≤ c · k for some universal constant c ≥ 1. We also prove that for all n-vertex connected graphs G we have AC(G) = O( n 2 log(n)/ log log(n) ), thus improving the trivial upper bound of O(n 2 ) achieved by sequentially letting each agent perform depth-first search along some spanning tree of G. Studying the computational complexity of this problem, we prove that for any constant t ≥ 1 the problem of deciding that a given graph G has AC(G) ≤ t or AC(G) ≥ 2t is N P-complete. That is, AC(G) is N P-hard to approximate within multiplicative factor of 2, as well as within any additive constant factor. On the algorithmic side, we give a deterministic algorithm that given an n-vertex graph G with AC(G) = 1 finds a strategy for acquaintance that consists of n/c matchings in time n c+O(1) . We also design a randomized polynomial time algorithm that given an n-vertex graph G with AC(G) = 1 finds with high probability an O(log(n))-rounds strategy for acquaintance.

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Loosely speaking, a proximity-oblivious (property) tester is a randomized algorithm that makes a constant number of queries to a tested object and distinguishes objects that have a predetermined property from those that lack it. Specifically, for some threshold probability c, objects having the property are accepted with probability at least c, whereas objects that are ǫ-far from having the property are accepted with probability at most c − F (ǫ), where F : (0, 1] → (0, 1] is some fixed monotone function. (We stress that, in contrast to standard testers, a proximity-oblivious tester is not given the proximity parameter.)The foregoing notion, introduced by Goldreich and Ron (STOC 2009), was originally defined with respect to c = 1, which corresponds to one-sided error (proximity-oblivious) testing. Here we study the two-sided error version of proximity-oblivious testers; that is, the (general) case of arbitrary c ∈ (0, 1]. We show that, in many natural cases, two-sided error proximity-oblivious testers are more powerful than one-sided error proximity-oblivious testers; that is, many natural properties that have no one-sided error proximity-oblivious testers do have a two-sided error proximity-oblivious tester.

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