Related papers: Some Lower Bound Results for Set-Multilinear Arith…
The best known size lower bounds against unrestricted circuits have remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving…
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative…
We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for…
In this paper we improve the lower bound of Chatterjee et al.\ (ECCC 2019) to an $\Omega(n^2)$ lower bound for unlayered Algebraic Branching Programs. We also study the impact layerization has on Algebraic Branching Programs. We exhibit a…
We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two…
Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…
We prove a lower bound of $\Omega(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz,…
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulated problems, in which the columns of the constraint matrix are…
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the…
Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has…
In many applications, we need algorithms which can align partially overlapping point sets and are invariant to the corresponding transformations. In this work, a method possessing such properties is realized by minimizing the objective of…
We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant…
We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful…
We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $\Sigma\Pi\Sigma$…
There has been a lot of interest recently in proving lower bounds on the size of linear programs needed to represent a given polytope P. In a breakthrough paper Fiorini et al. [Proceedings of 44th ACM Symposium on Theory of Computing 2012,…
Valiant's famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential.…
We prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
Recent improvements in adder optimization could be achieved by optimizing the AND-trees occurring within the constructed circuits. The overlap of such trees and its potential for pure size optimization has not been taken into account…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…