Choosing, Agreeing, and Eliminating in Communication Complexity ? Amos Beimel 1 , Sebastian Ben Daniel 1 , Eyal Kushilevitz 2 , and Enav Weinreb 2 1 Dept. of Computer Science, Ben Gurion University, Beer Sheva, Israel. 2 Dept. of Computer Science, Technion, Haifa, Israel. Abstract. We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are k fixed Boolean functions f 1 ,...,f k and Alice and Bob have k inputs x1,...,x k and y1,...,y k , respectively. In the eliminate problem, Alice and Bob should output a vector σ 1 ,...,σ k such that f i (x i ) 6= σ i for at least one i (i.e., their goal is to eliminate one of the 2 k output vectors); in choose, Alice and Bob should return (i, fi (xi ,yi )) and in agree they should return f i (x i ,y i ), for some i. The question, in each of the three cases, is whether one can do better than solving one (say, the first) in- stance. We study these three problems and prove various positive and negative results. 1 Introduction A basic question in complexity theory is how the complexity of computing k in- dependent instances relates to the complexity of computing one instance. Such problems, called direct sum problems, have been studied for a variety of com- putational models. Broadly, the direct sum question asks (with respect to an arbitrary computational model and any complexity measure): Question 1. Can “solving” k functions f 1 ,...,f k on k independent inputs x 1 ,..., x k (respectively) be done more “efficiently” than just “solving” each f i (x i )? (Of particular interest is the special case where all functions are identical.) Since the inputs are independent, it is tempting to conjecture that in reasonable models the answer is negative. Indeed, it was proved that in several models no significant saving can be obtained; e.g., for decision trees [7, 22, 27, 14]. However, for other models, some savings are possible despite the independence of the inputs, e.g., non-deterministic communication complexity and randomized communication complexity [16, 10], deterministic communication complexity of relations [10], and distributional communication complexity [27]. For other models, the answer is still unknown; e.g., in circuit complexity [11, 23, 29]. Direct sum results are important for understanding the power of a computational model. For example, ? The first author is supported by ISF grant 938/09. The second author is partially supported by the Frankel Center for Computer Science. The third and fourth authors are supported by ISF grant 1310/06.