Choosing, Agreeing, and Eliminating in Communication Complexity

We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are k fixed Boolean functions f1, ..., fk and Alice and Bob have k inputs x1,..., xk and y1,..., yk, respectively. In the eliminate problem, Alice and Bob should output a vector σ1,..., σk such that fi(xi) ≠ σi for at least one i (i.e., their goal is to eliminate one of the 2k output vectors); in choose, Alice and Bob should return (i, fi(xi,yi)) and in agree they should return fi(xi, yi), for some i. The question, in each of the three cases, is whether one can do better than solving one (say, the first) instance. We study these three problems and prove various positive and negative results.

[1]  D. Sivakumar.  On Membership Comparable Sets , 1999, J. Comput. Syst. Sci..

[2]  Ran Raz,et al.  Direct product results and the GCD problem, in old and new communication models , 1997, STOC '97.

[3]  Ziv Bar-Yossef,et al.  An information statistics approach to data stream and communication complexity , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[4]  Frank Stephan,et al.  Approximable Sets , 1995, Inf. Comput..

[5]  Hartmut Klauck,et al.  Optimal Direct Sum Results for Deterministic and Randomized Decision Tree Complexity , 2010, Inf. Process. Lett..

[6]  Amihood Amir,et al.  Some connections between bounded query classes and non-uniform complexity , 2000, Inf. Comput..

[7]  Ronen Shaltiel.  Towards proving strong direct product theorems , 2003, computational complexity.

[8]  John Gill,et al.  Terse, Superterse, and Verbose Sets , 1993, Inf. Comput..

[9]  Xi Chen,et al.  How to compress interactive communication , 2010, STOC '10.

[10]  Ran Raz,et al.  A direct product theorem , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[11]  Andris Ambainis,et al.  The Communication Complexity of Enumeration, Elimination, and Selection , 2001, J. Comput. Syst. Sci..

[12]  Moni Naor,et al.  Amortized Communication Complexity , 1995, SIAM J. Comput..

[13]  Vikraman Arvind,et al.  Polynomial time truth-table reductions to p-selective sets , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[14]  Jin-Yi Cai,et al.  Enumerative Counting Is Hard , 1989, Inf. Comput..

[15]  C. Jockusch.  Semirecursive sets and positive reducibility , 1968 .

[16]  Michael E. Saks,et al.  Products and Help Bits in Decision Trees , 1999, SIAM J. Comput..

[17]  Eyal Kushilevitz,et al.  Communication Complexity: Index of Notation , 1996 .

[18]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[19]  Ran Raz,et al.  Super-logarithmic depth lower bounds via direct sum in communication complexity , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[20]  Russell Impagliazzo,et al.  Communication complexity towards lower bounds on circuit depth , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[21]  Avi Wigderson,et al.  Monotone Circuits for Connectivity Require Super-Logarithmic Depth , 1990, SIAM J. Discret. Math..

[22]  W. Gasarch,et al.  Bounded Queries in Recursion Theory , 1998 .

[23]  Alan L. Selman,et al.  P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP , 1979, ICALP.

[24]  Avi Wigderson,et al.  A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[25]  Jaikumar Radhakrishnan,et al.  A Direct Sum Theorem in Communication Complexity via Message Compression , 2003, ICALP.

[26]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[27]  Richard Beigel,et al.  One help-bit doesn't help , 1998, STOC '98.

[28]  Martin Kummer.  A Proof of Beigel's Cardinality Conjecture , 1992, J. Symb. Log..

[29]  Lane A. Hemaspaandra,et al.  Theory of Semi-Feasible Algorithms , 2002, Monographs in Theoretical Computer Science An EATCS Series.

[30]  Frank Stephan,et al.  The complexity of oddan , 2000 .

[31]  Frank Stephan,et al.  The complexity of ODDnA , 2000, Journal of Symbolic Logic.

[32]  Eyal Kushilevitz,et al.  Fractional Covers and Communication Complexity , 1995, SIAM J. Discret. Math..

[33]  Bala Kalyanasundaram,et al.  The Probabilistic Communication Complexity of Set Intersection , 1992, SIAM J. Discret. Math..

[34]  Giulia Galbiati.  M. J. Fischer: On the Complexity of 2-Output Boolean Networks , 1981, Theor. Comput. Sci..

[35]  Alexander A. Razborov.  On the Distributional Complexity of Disjontness , 1990, ICALP.

[36]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[37]  Ran Raz,et al.  Super-logarithmic depth lower bounds via the direct sum in communication complexity , 1995, computational complexity.

[38]  Ker-I Ko.  On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[39]  D. Ulig.  On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements , 1974 .

[40]  Andrew Chi-Chih Yao,et al.  Informational complexity and the direct sum problem for simultaneous message complexity , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[41]  Wolfgang J. Paul.  Realizing Boolean Functions on Disjoint sets of Variables , 1976, Theor. Comput. Sci..

[42]  Alan L. Selman,et al.  Analogues of Semicursive Sets and Effective Reducibilities to the Study of NP Complexity , 1982, Inf. Control..

[43]  Alan L. Selman,et al.  Reductions on NP and P-Selective Sets , 1982, Theor. Comput. Sci..