Interpolation by a Game

We introduce a notion of a real game (a generalisation of the Karchmer-Wigderson game (cf. [3]) and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monotone protocols (defined in [4, Definition 2.2]) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (by a problem posed in [5, Section 8], in particular). Our main objective is to extend the communication complexity approach of [4, 5] to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree-like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree-like cutting planes proofs for which an exponential lower bound was demonstrated in [2]).     

On the complexity of finding paths in a two‐dimensional domain I: Shortest paths

The computational complexity of finding a shortest path in a two‐dimensional domain is studied in the Turing machine‐based computational model and in the discrete complexity theory. This problem is studied with respect to two formulations of polynomial‐time computable two‐dimensional domains: (A) domains with polynomialtime computable boundaries, and (B) polynomial‐time recognizable domains with polynomial‐time computable distance functions. It is proved that the shortest path problem has the polynomial‐space upper bound for domains of both type (A) and type (B); and it has a polynomial‐space lower bound for the domains of type (B), and has a #P lower bound for the domains of type (A). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Worst case complexity of multivariate Feynman–Kac path integration

We study the multivariate Feynman–Kac path integration problem. This problem was studied in Plaskota et al. (J. Comp. Phys. 164 (2000) 335) for the univariate case. We describe an algorithm based on uniform approximation, instead of the L₂-approximation used in Plaskota et al. (2000). Similarly to Plaskota et al. (2000), our algorithm requires extensive precomputing. We also present bounds on the complexity of our problem. The lower bound is provided by the complexity of a certain integration problem, and the upper bound by the complexity of the uniform approximation problem. The algorithm presented in this paper is almost optimal for the classes of functions for which uniform approximation and integration have roughly the same complexities.     

The quantifier complexity of polynomial‐size iterated definitions in first‐order logic

We refine the constructions of Ferrante‐Rackoff and Solovay on iterated definitions in first‐order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier‐free case and in the case of purely existential or universal quantifiers, we show that Ω(n /log n) quantifiers are necessary and sufficient. The last lower bounds are obtained with the aid of the Yao‐Håstad switching lemma (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the complexity of stochastic integration

We study the complexity of approximating stochastic integrals with error for various classes of functions. For Ito integration, we show that the complexity is of order , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.

     

On the Linear Complexity Profile of Nonlinear Congruential Pseudorandom Number Generators with Dickson Polynomials

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and Monte-Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. Recently, a weak lower bound on the linear complexity profile of a general nonlinear congruential pseudorandom number generator was proven by Gutierrez, Shparlinski and the first author. For most nonlinear generators a much stronger lower bound is expected. Here, we obtain a much stronger lower bound on the linear complexity profile of nonlinear congruential pseudorandom number generators with Dickson polynomials.     

Explicit exponential lower bounds for exact hyperplane covers

We describe an explicit and simple subset of the discrete hypercube which cannot be exactly covered by fewer than exponentially many hyperplanes. The proof exploits a connection to communication complexity, and relies heavily on Razborov's lower bound for disjointness.     

Approximating Boolean functions by OBDDs

In learning theory and genetic programming, OBDDs are used to represent approximations of Boolean functions. This motivates the investigation of the OBDD complexity of approximating Boolean functions with respect to given distributions on the inputs. We present a new type of reduction for one-round communication problems that is suitable for approximations. Using this new type of reduction, we improve a known lower bound on the size of OBDD approximations of the hidden weighted bit function for uniformly distributed inputs to an asymptotically tight bound and prove new results about OBDD approximations of integer multiplication and squaring for uniformly distributed inputs.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

Some Notes on the Linear Complexity of Sidel’nikov-Lempel-Cohn-Eastman Sequences

We continue the study of the linear complexity of binary sequences, independently introduced by Sidel’nikov and Lempel, Cohn, and Eastman. These investigations were originated by Helleseth and Yang and extended by Kyureghyan and Pott. We determine the exact linear complexity of several families of these sequences using well-known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences.     

Gilmore-Lawler bound of quadratic assignment problem

The Gilmore-Lawler bound (GLB) is one of the well-known lower bound of quadratic assignment problem (QAP). Checking whether GLB is tight is an NP-complete problem. In this article, based on Xia and Yuan linearization technique, we provide an upper bound of the complexity of this problem, which makes it pseudo-polynomial solvable. We also pseudopolynomially solve a class of QAP whose GLB is equal to the optimal objective function value, which was shown to remain NP-hard.      

Lower bounds in communication complexity based on factorization norms

We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity. As we show, our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement. Our results also shed some light on the question how much communication can be saved by using entanglement. It is known that entanglement can save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound on the saving in communication is tight almost always. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Limitations of the hyperplane separation technique for bounding the extension complexity of polytopes

The hyperplane separation bound is a lower bound on the extension complexity of a polytope. It is the main tool in Rothvoß's proof of an exponential bound for the matching polytope (Rothvoß, 2017). We show that the technique is sensitive to the choice of slack matrix and does not improve upon the best known lower bounds for spanning tree and completion time polytopes when applied to their canonical slack matrices. Stronger bounds may be obtained by appropriate rescalings and redundancy.     

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m²/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.     

Improved lower bounds on the randomized complexity of graph properties

We prove a lower bound of Ω(n^4/3 log ^1/3n) on the randomized decision tree complexity of any nontrivial monotone n‐vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions of size n. This improves the previous best bound of Ω(n^4/3) due to Hajnal (Combinatorica 11 (1991) 131–143). Our proof works by improving a graph packing lemma used in earlier work, and this improvement in turn stems from a novel probabilistic analysis. Graph packing being a well‐studied subject in its own right, our improved packing lemma and the probabilistic technique used to prove it may be of independent interest. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

On the multiplicative complexity of the Discrete Fourier Transform

Most results in multiplicative complexity assume that the functions to be computed are in the field of constants extended by indeterminates, that is, the variables satisfy no algebraic relation. In this paper we extend some of the known results to the case that some of the variables do satisfy some algebraic relations. We then apply these results to obtaining a lower bound on the multiplicative complexity of the Discrete Fourier Transform. In the special case of computing the Discrete Fourier Transform of a prime number of points, the lower bound is actually attainable.     

On the Complexity of Partial Order Properties

The recognition complexity of ordered set properties is considered in terms of how many questions must be put to an adversary to decide if an unknown partial order has the prescribed property. We prove a lower bound of order n ² for properties that are characterized by forbidden substructures of fixed size. For the properties being connected, and having exactly k comparable pairs, k n ² / 4 we show that the recognition complexity is (n\choose 2). The complexity of interval orders is exactly (n\choose 2) - 1. We further establish bounds for being a lattice, being of height k and having width k.     

Quantum lower bounds by entropy numbers

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional L_p spaces and of Sobolev embeddings.     

Computationally tractable pairwise complexity profile

Quantifying the complexity of systems consisting of many interacting parts has been an important challenge in the field of complex systems in both abstract and applied contexts. One approach, the complexity profile, is a measure of the information to describe a system as a function of the scale at which it is observed. We present a new formulation of the complexity profile, which expands its possible application to high‐dimensional real‐world and mathematically defined systems. The new method is constructed from the pairwise dependencies between components of the system. The pairwise approach may serve as both a formulation in its own right and a computationally feasible approximation to the original complexity profile. We compare it to the original complexity profile by giving cases where they are equivalent, proving properties common to both methods, and demonstrating where they differ. Both formulations satisfy linear superposition for unrelated systems and conservation of total degrees of freedom (sum rule). The new pairwise formulation is also a monotonically nonincreasing function of scale. Furthermore, we show that the new formulation defines a class of related complexity profile functions for a given system, demonstrating the generality of the formalism. © 2013 Wiley Periodicals, Inc. Complexity 18:20–27, 2013