Combinatorics of Monotone Computations

Combinatorics of Monotone Computations

Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length ≤ d, and (ii) arbitrary real-valued non-decreasing functions on ≤d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial ...

Network reliability importance measures : combinatorics and Monte Carlo based computations

In this paper we focus on computational aspects of network reliability importance measure evaluation. It is a well known fact that most network reliability problems are NP-hard and therefore there is a significant gap between theoretical analysis and the ability to compute different reliability parameters for large or even moderate networks. In this paper we present two very efficient combinato...

Finite Limits and Monotone Computations: The Lower Bounds Criterion

Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin non-decreasing real-valued functions as gates. The only requirement is their "local-ity". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; x m) with small enough threshold value minfs; m ? s + 1g, are simplest examples of local gates. T...

The Order Steps of an Analytic Combinatorics

‎Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures‎. ‎This theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines‎, ‎including probability theory‎, ‎statistical physics‎, ‎computational biology and information theory‎. ‎With a caref...

Combinatorics Enumerative Combinatorics 05