A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.
Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang gives an improved bound on the maximum size of a $w$-set system that excludes a robust sunflower. In this paper, we use this result to obtain an $\exp(n^{1/2-o(1)})$ lower bound on the monotone circuit size of an explicit $n$-variate monotone function, improving the previous best known $\exp(n^{1/3-o(1)})$ due to And…
Given a graph property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $Φ$. The search for explicit criteria on $Φ$ ensuring that $\#\mathsf{IndSub}(Φ)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some par…
The problem of constructing hazard-free Boolean circuits (those avoiding electronic glitches) dates back to the 1940s and is an important problem in circuit design and even in cybersecurity. We show that a DeMorgan circuit is hazard-free if and only if the circuit produces (purely syntactically) all prime implicants as well as all prime implicates of the Boolean function it computes. This extends to arbitrary DeMorgan circuits a classical result of Eichelberger [IBM J. Res. Develop., 9 (1965)] showing this property for special depth-two circuits. Via an amazingly simple proof, we also strength…
In this paper, we initiate study of the computational power of adaptive and non-adaptive monotone decision trees - decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of non-monotonicity of a given Boolean function. We also study the restriction of the model by restricting (in terms of circuit complexity) the monotone functions that can be queried at each node. This naturally leads to complexity classes of the form DT(mon-C) for any circuit complexity class C, where the hei…
We prove that weighted circuit satisfiability for monotone or antimonotone circuits has no fixed-parameter tractable approximation algorithm with any approximation ratio function $ρ$, unless $FPT\neq W[1]$. In particular, not having such an fpt-approximation algorithm implies that these problems have no polynomial-time approximation algorithms with ratio $ρ(OPT)$ for any nontrivial function $ρ$.
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free …
Circuit lower bounds are important since it is believed that a super-polynomial circuit lower bound for a problem in NP implies that P!=NP. Razborov has proved superpolynomial lower bounds for monotone circuits by using method of approximation. By extending this approach, researchers have proved exponential lower bounds for the monotone network complexity of several different functions. But until now, no one could prove a non-linear lower bound for the non-monotone complexity of any Boolean function in NP. While we show that in this paper by replacement of each Not gates into constant 1 equiva…
Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.
Feasible interpolation is a general technique for proving proof complexity lower bounds. The monotone version of the technique converts, in its basic variant, lower bounds for monotone Boolean circuits separating two NP-sets to proof complexity lower bounds. In a generalized version of the technique, dag-like communication protocols are used instead of monotone Boolean circuits. We study three kinds of protocols and compare their strength. Our results establish the following relationships in the sense of polynomial reducibility: Protocols with equality are at least as strong as protocols with …
Measures of circuit complexity are usually analyzed to ensure the computation of Boolean functions with economy and efficiency. One of these measures is energy complexity, which is related to the number of gates that output true in a circuit for an assignment. The idea behind energy complexity comes from the counting of `firing' neurons in a natural neural network. The initial model is based on threshold circuits, but recent works also have analyzed the energy complexity of traditional Boolean circuits. In this work, we discuss the time complexity needed to compute the best-case energy complex…
We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having $n$ variables and defined over constant-degree expander graphs, have monotone arithmetic complexity $2^{Ω(n)}$. This yields the first strongly exponential lower bound on the monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehuday…
We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $μ\in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monot…
We generalize the feasible interpolation theorem for semantic derivations from K.(1997) by allowing randomized protocols (protocols in the sense of K.(1997). We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for the monotone Karchmer-Wigderson multi-function $KW^m[U,V]$ making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of $U, V$, $U$ closed upwards…
We show that the complexity of minimal monotone circuits implementing a monotone version of the permutation operator on $n$ boolean vectors of length $q$ is $Θ(qn\log n)$. In particular, we obtain an alternative way to prove the known complexity bound $Θ(n\log n)$ for the monotone shift operator on $n$ boolean inputs.
We show that there is a sequence of explicit multilinear polynomials $P_n(x_1,\ldots,x_n)\in \mathbb{R}[x_1,\ldots,x_n]$ with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for $P_n$ must have size $\exp(Ω(n)).$ This builds on (and strengthens) a result of Yehudayoff (2018) who showed a lower bound of $\exp(\tildeΩ(\sqrt{n})).$
We investigate the problem $\#\mathsf{IndSub}(Φ)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $Φ$. This continues the work of Jerrum and Meeks who proved the problem to be $\#\mathrm{W[1]}$-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ is hard for $\#\mathrm{W[…
Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued input variables whose gates compute the sum or the product of their inputs. Positive AC -- that is, AC representing non-negative functions -- subsume many interesting probabilistic models such as probabilistic sentential decision diagram (PSDD) or sum-product network (SPN) on indicator variables. Efficient algorithms for many operations useful in probabilistic reasoning on these models critically depend on imposing structural restrictions to the underlying AC. Generally, adding structural restrictions yields new tracta…
This paper describes about relation between circuit complexity and accept inputs structure in Hamming space by using almost all monotone circuit that emulate deterministic Turing machine (DTM). Circuit family that emulate DTM are almost all monotone circuit family except some NOT-gate which connect input variables (like negation normal form (NNF)). Therefore, we can analyze DTM limitation by using this NNF Circuit family. NNF circuit have symmetry of OR-gate input line, so NNF circuit cannot identify from OR-gate output line which of OR-gate input line is 1. So NNF circuit family cannot comput…
In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P $\neq$ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circu…
Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be…
We consider the class of counting problems,i.e. functions in $\#$P, which are self reducible, and have easy decision version, i.e. for every input it is easy to decide if the value of the function $f(x)$ is zero. For example, $\#$independent-sets of all sizes, is such a problem, and one of the hardest of this class, since it is equivalent to $\#$SAT under multiplicative approximation preserving reductions. Using these two powerful properties, self reducibility and easy decision, we prove that all problems/ functions $f$ in this class can be approximated in probabilistic polynomial time within …
What makes normalized weighted satisfiability tractable
We consider the weighted antimonotone and the weighted monotone satisfiability problems on normalized circuits of depth at most $t \geq 2$ , abbreviated { \sc wsat $^-[t]$ } and { \sc wsat $^+[t]$ }, respectively. These problems model the weighted satisfiability of antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone { \sc cnf-sat}) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of { \sc wsat $^-[t]$ } and { \sc wsat $^+[t]$ } with respect to the…
We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if $S$ is a search problem with critical block sensitivity $b$, then every randomised two-party protocol solving a certain two-party lift of $S$ requires $Ω(b)$ bits of communication. Besides simplicity, our proof has the advantage of generalising to the multi-party setting. We combine these results with new critical block sensitivity lower…
In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not…
We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting plane…
