Monotone Arithmetic Circuits Research Articles

Инверсные входы в арифметических и тропических схемах

It is known that the size of monotone arithmetic (+, ·) circuits can be exponentially decreased by allowing just one division “at the very end,” at the output gate. A natural question is: can the size of (+, ·) circuits be substantially reduced if we allow divisions “at the very beginning,” that is, if besides nonnegative real constants and variables x<sub>1</sub>, …, x<sub>n</sub>, the circuits can also use their reciprocals 1/x<sub>1</sub>, ..., 1/x<sub>n</sub> as inputs. We answer this question in the negative: the gain in circuit size is then always at most quadratic. Over tropical (min, +) and (max, +) semirings, division turns into subtraction; so, reciprocal inputs are then -x<sub>1</sub>, …, -x<sub>n</sub>. We give the same negative answer also for tropical circuits. The question of whether reciprocal inputs can substantially speed up tropical (min, +, max) circuits, remains open.

On $$\epsilon$$-sensitive monotone computations

We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial $$f\in \mathbb {R}[x_1,\dots , x_n]$$ of degree d has an arithmetic circuit of size s then $$(x_1+\dots +x_n+1)^d+\epsilon f$$ has a monotone arithmetic circuit of size $$O(sd^2+n\log n)$$ , for some $$\epsilon >0$$ . Second, if $$f:\{0,1\}^n\rightarrow \{0,1\}$$ is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least $$\varOmega (\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))- 2n)$$ , where J is the all-ones matrix and $$\mathrm{rk}_{+}$$ denotes the nonnegative rank of a matrix. In fact, the quantity $$\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))$$ characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar $$\epsilon $$ -sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.

Tropical Complexity, Sidon Sets, and Dynamic Programming

Many dynamic programming algorithms for discrete 0-1 optimization problems are just special (recursively constructed) tropical (min,+) or (max,+) circuits. A problem is homogeneous if all its feasible solutions have the same number of ones. Jerrum and Snir [J ACM 29 (1982), pp. 874--897] proved that tropical circuit complexity of homogeneous problems coincides with the monotone arithmetic circuit complexity of the corresponding polynomials. So, lower bounds on the monotone arithmetic circuit complexity of these polynomials yield lower bounds on the tropical complexity of the corresponding optimization problems. But the situation with nonhomogeneous problems is entirely different: here the gap between their tropical and arithmetic complexities can be even exponential. In this paper, we improve two classical lower bounds for monotone arithmetic circuits---Schnorr's bound and Hyafil--Valiant's bound---and use these improvements to derive general lower bounds for the tropical circuit complexity of nonhomogeneous optimization problems. In particular, we show that optimization problems, whose sets of feasible solutions are cover free, have large tropical complexity.

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.

A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials

This work suggests a method for deriving lower bounds for the complexity of polynomials with positive real coefficients implemented by circuits of functional elements over the monotone arithmetic basis . Using this method, several new results are obtained. In particular, we construct examples of polynomials of degree in each of the variables with coefficients 0 and 1 having additive monotone complexity and multiplicative monotone complexity as . In this form, the lower bounds derived here are sharp.Bibliography: 72 titles.

An Exponential Lower Bound for the Size of Monotone Real Circuits

We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and OR). Our proof is relatively simple and direct and uses the method of counting bottlenecks. The generalization was proved independently by Pudlák using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus.

A lower bound for monotone arithmetic circuits computing 0–1 permanent

A lower bound for monotone arithmetic circuits computing 0–1 permanent

A direct version of Shamir and Snir's lower bounds on monotone circuit depth

We present direct proofs of the following results of Shamir and Snir [Mathematical System Theory 13 (1980) 301-322] on the depth of monotone arithmetic circuits over rings of characteristic 0: (1) an ω((log p)(log n)) lower bound for computing the product of p n × n matrices; and (2) an ω( n) lower bound for computing the permanent of an n × n matrix.

Size-depth trade-offs for monotone arithmetic circuits

Consider the problem of computing the product a 1 A (1)⋯ A ( t) b , where A (1),…, A ( t) are n × n matrices, a and b are vectors. We show that the size s and depth d of monotone arithmetic circuits for this problem are related as s + n 3d = Ω(tn 3) Thus, a reduction to depth d = o( t) requires an increase from (optimal) size n 2 t to size n 3 t. A similar trade-off is shown for the evaluation of linear recurrences.

Lower bounds on monotone arithmetic circuits with restricted depths

We consider monotone arithmetic circuits with restricted depths to compute monotone multivariate polynomials such as the elementary symmetric functions, convolution of several vectors and raising a matrix to a power. We develop general lower- and upper-bound techniques that seem to generate almost-matching bounds for all the functions considered. These results imply exponential lower bounds for circuits of bounded depths which compute any of these functions. We also obtain several examples for which negation can reduce the size exponentially.