On the complexity of branching programs and decision trees for clique functions

Exponential lower bounds on the complexity of computing the clique functions in the Boolean decision-tree model are proved. For one-time-only branching programs, large polynomial lower bounds are proved for <italic>k</italic>-clique functions if <italic>k</italic> is fixed, and exponential lower bounds if <italic>k</italic> increases with <italic>n</italic>. Finally, the hierarchy of the classes BP<subscrpt><italic>d</italic></subscrpt>(<italic>P</italic>) of all sequences of Boolean functions that may be computed by <italic>d</italic>-times only branching programs of polynomial size is introduced. It is shown constructively that BP<subscrpt>1</subscrpt>(<italic>P</italic>) is a proper subset of BP<subscrpt>2</subscrpt>(<italic>P</italic>).

[1]  Ingo Wegener,et al.  Optimal Decisions Trees and One-Time-Only Branching Programs for Symmetric Boolean Functions , 1984, CAAP.

[2]  Paul E. Dunne,et al.  Lower bounds on the complexity of 1-time only branching programs , 1985, FCT.

[3]  Ingo Wegener.  On the Complexity of Slice Functions , 1985, Theor. Comput. Sci..

[4]  William Joseph Masek,et al.  A fast algorithm for the string editing problem and decision graph complexity , 1976 .

[5]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[6]  Ingo Wegener,et al.  More on the Complexity of Slice Functions , 1986, Theor. Comput. Sci..

[7]  Pavel Pudlák.  A Lower Bound on Complexity of Branching Programs (Extended Abstract) , 1984, MFCS.

[8]  Allan Borodin,et al.  Bounds for width two branching programs , 1983, SIAM J. Comput..

[9]  Vojtech Rödl,et al.  Two lower bounds for branching programs , 1986, STOC '86.

[10]  Ingo Wegener,et al.  Time-Space Trade-offs for Branching Programs , 1986, J. Comput. Syst. Sci..

[11]  John E. Savage,et al.  The Complexity of Computing , 1976 .

[12]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[13]  Journal of the Association for Computing Machinery , 1961, Nature.

[14]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[15]  Leslie G. Valiant,et al.  Exponential lower bounds for restricted monotone circuits , 1983, STOC.

[16]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[17]  Ingo Wegener.  On the Complexity of Branching Programs and Decision Trees for Clique Functions , 1987, TAPSOFT, Vol.1.

[18]  Stephan Waack,et al.  Lower Bounds on The Complexity of Real-Time Branching Programs , 1988, RAIRO Theor. Informatics Appl..

[19]  Andrew C. Yao,et al.  Lower bounds by probabilistic arguments , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[20]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[21]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[22]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.