Lower Bounds for Tropical Circuits and Dynamic Programs

Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower bounds arguments for tropical circuits, and hence, for dynamic programs.

[1]  Stasys Jukna.  Expanders and time-restricted branching programs , 2005, Theor. Comput. Sci..

[2]  Dima Grigoriev,et al.  Subtraction-free complexity and cluster transformations , 2013, ArXiv.

[3]  Mark Jerrum,et al.  Some Exact Complexity Results for Straight-Line Computations over Semirings , 1982, JACM.

[4]  Igor S. Sergeev,et al.  A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials , 2012 .

[5]  Stasys Jukna.  Combinatorics of Monotone Computations , 1998, Comb..

[6]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[7]  Leslie G. Valiant,et al.  Negation can be exponentially powerful , 1979, Theor. Comput. Sci..

[8]  I. Sergeev.  On the complexity of the gradient of a rational function , 2008 .

[9]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[10]  Kurt Mehlhorn,et al.  Monotone switching circuits and boolean matrix product , 2005, Computing.

[11]  Matthias P. Krieger.  On the Incompressibility of Monotone DNFs , 2005, Theory of Computing Systems.

[12]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[13]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[14]  Martin Tompa,et al.  A Direct Version of Shamir and Snir's Lower Bounds on Monotone Circuit Depth , 1994, Inf. Process. Lett..

[15]  Andrew Chi-Chih Yao,et al.  A lower bound for the monotone depth of connectivity , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[16]  Ran Raz,et al.  Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[17]  Claus-Peter Schnorr,et al.  A Lower Bound on the Number of Additions in Monotone Computations , 1976, Theor. Comput. Sci..

[18]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[19]  Amir Yehudayoff,et al.  Arithmetic Circuits: A survey of recent results and open questions , 2010, Found. Trends Theor. Comput. Sci..

[20]  A. Razborov.  Lower bounds on monotone complexity of the logical permanent , 1985 .

[21]  Laurent Hyafil,et al.  On the parallel evaluation of multivariate polynomials , 1978, SIAM J. Comput..

[22]  Nicholas Pippenger,et al.  On Another Boolean Matrix , 1980, Theor. Comput. Sci..

[23]  L. R. Kerr.  The Effect of Algebraic Structure on the Computational Complexity of Matrix Multiplication , 1970 .

[24]  Игорь Сергеевич Сергеев,et al.  Об одном методе получения нижних оценок сложности монотонных арифметических схем, вычисляющих действительные многочлены@@@A method for obtaining lower bounds for complexity of monotone arithmetic circuits computing real polynomials , 2012 .

[25]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[26]  Leslie G. Valiant,et al.  Fast Parallel Computation of Polynomials Using Few Processors , 1983, SIAM J. Comput..

[27]  Kurt Mehlhorn,et al.  Some remarks on Boolean sums , 1979, Acta Informatica.

[28]  Johan Håstad,et al.  Monotone Circuits for Connectivity Have Depth (log n)2-o(1) , 1998, SIAM J. Comput..

[29]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 2006, Discret. Math..

[30]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[31]  M. Murty.  Ramanujan Graphs , 1965 .

[32]  T. Lindvall.  ON A ROUTING PROBLEM , 2004, Probability in the Engineering and Informational Sciences.

[33]  Neil Hindman,et al.  Ultrafilters and multidimensional Ramsey Theorems , 1989, Comb..

[34]  A. E. Andreev.  A method for obtaining efficient lower bounds for monotone complexity , 1987 .

[35]  W. Marsden.  I and J , 2012 .

[36]  Moshe Morgenstern,et al.  Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q , 1994, J. Comb. Theory, Ser. B.

[37]  Marc Snir,et al.  On the depth complexity of formulas , 1979, Mathematical systems theory.

[38]  Mike Paterson,et al.  Complexity of Monotone Networks for Boolean Matrix Product , 1974, Theor. Comput. Sci..

[39]  Ron Livne,et al.  Ramanujan local systems on graphs , 1997 .