Unexpected upper bounds on the complexity of some communication games
The results that we have presented should serve a as a warning: reducing a problem in complexity theory to a communication game may be only a first small step towards the solution, however intuitively clear the game may look like. Still, if we do not look just for record lower bounds, these reductions may often be, in a sense, very rewarding, as they lead to nice mathematical problems and show connections with other branches of mathematics.
An interesting question is, if it is only an accident that we got stuck at difficult combinatorial problems, or if this is due to the fact that the problems are inherently difficult. In the case of the space-time trade-offs it is almost sure that there exists a different, more accessible, way of getting lower bounds. But notice that the problem of proving nonlinear lower bounds on space-time tradeoffs on branching programs is ridiculously weak if compared to real problems such as “P=NP?”. How difficult problems can we expect to encounter when solving those?
Unable to display preview. Download preview PDF.
- 1.L. Babai, N. Nisan, M. Szegedy, Multiparty protocols and logspace-hard pseudorandom sequences, Journ. of Computer and System Science 45, (1992), 204–232.Google Scholar
- 2.L. Babai, P. Pudlák, V. Rödl, E. Szemeredi, Lower bounds to the complexity of symmetric boolean functions, Theor. Comput. Science 74 (1990), 313–323.Google Scholar
- 3.F.A. Behrend, On sets of integers which contain no three elements in arithmetic progression, Proc. Nat. Acad. Sci. 23, (1946), 331–332.Google Scholar
- 4.A. Chandra, M. Furst, R. Lipton, Multiparty protocols, in Proc. 15-th STOC, 1983, 94–99.Google Scholar
- 5.F.R.K. Chung, R.L. Graham, R.M. Wilson, Quasi-random graphs, Combinatorica 9, (1989), 345–362.Google Scholar
- 6.J. Edmonds and R. Impagliazzo Towards time-space lower bounds on branching programs, manuscript.Google Scholar
- 7.J. Edmonds and R. Impagliazzo About time-space bounds for st-connectivity on branching programs, manuscript.Google Scholar
- 8.P. Erdös, P. Frankl, and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph, Graphs and Combinatorics 2, 113–121 (1986)Google Scholar
- 9.R.L. Graham and V. Rödl, Numbers in Ramsey Theory, In: Surveys in combinatorics (ed. I. Anderson), London Mathematical Society lecture note series 103, pp. 111–153, 1985.Google Scholar
- 10.V. Grolmusz, The BNS lower bound for multi-party protocols is nearly optimal, Information and Computation, to appear.Google Scholar
- 11.J. Håstad, M. Goldmann, On the power of small depth threshold circuits, in Proc. 31-st FOCS, 1990, 610–618.Google Scholar
- 12.M. Karchmer, A. Wigderson, Monotone circuits for connectivity require superlogarithmic depth, in Proc. 20-th STOC, 1988, 539–550.Google Scholar
- 13.N. Nisan, A. Wigderson, Rounds in communication complexity revisited, STOC 1991, 419–429.Google Scholar
- 14.P. Pudlák, V. Rödl, J. Sgall, Boolean circuits, tensor ranks and communication complexity, submitted, (preliminary version Modified ranks of tensors and the size of circuits appeared in Proc. 33-FOCS, 1992).Google Scholar
- 15.P. Pudlák, J. Sgall, An upper bound for a communication game related to spacetime tradeoffs, preprint.Google Scholar
- 16.R.F. Roth, On certain sets of integers, Journ. London Math. Soc. 28, (1953), 104–109.Google Scholar
- 17.E. Szemerédi, Regular partitions of graphs, In: Proc. Coloq. Int. CNRS, pp. 399–401. Paris, CNRS, 1976.Google Scholar
- 18.E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), pp. 199–245.Google Scholar
- 19.L.G. Valiant, Graph-theoretic arguments in low level complexity, in Proc. MFCS 1977, Springer-Verlag LNCS, 162–176.Google Scholar
- 20.A.C.-C. Yao, On ACC and threshold circuits, in Proc. 31-st FOCS, 1990, 619–627.Google Scholar