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Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
Cook and Reckhow 1979 pointed out that NP is not closed under complementation iff there is no propositional proof system that admits polynomial size proofs of all tautologies. Theory of proof complexity generators aims at constructing sets…
A supervised learning algorithm has access to a distribution of labeled examples, and needs to return a function (hypothesis) that correctly labels the examples. The hypothesis of the learner is taken from some fixed class of functions…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
Let $\mathbb C$ be the set of complex numbers, and let $\mathcal P$ be a collection of complex polynomial maps in several variables. Assuming at least one $P\in\mathcal P$ depends on at least two variables, we classify all possibilities for…
We present a new type of monotone submodular functions: \emph{multi-peak submodular functions}. Roughly speaking, given a family of sets $\cF$, we construct a monotone submodular function $f$ with a high value $f(S)$ for every set $S \in…
A new class of functions is presented. The structure of the algorithm, particularly the selection criteria (branching), is used to define the fundamental property of the new class. The most interesting property of the new functions is that…
A computable real function F on [0,1] is constructed such that there exists an exponential time algorithm for the evaluation of the function on [0,1] on Turing machine but there does not exist any polynomial time algorithm for the…
The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes…
We construct a function for almost-complex Riemannian manifolds. Non-vanishing of the function for the almost-complex structure implies the almost-complex structure is not integrable. Therefore the constructed function is an obstruction for…
Optimization is a key task in a number of applications. When the set of feasible solutions under consideration is of combinatorial nature and described in an implicit way as a set of constraints, optimization is typically NP-hard.…
We introduce a method for proving almost sure termination in the context of lambda calculus with continuous random sampling and explicit recursion, based on ranking supermartingales. This result is extended in three ways. Antitone ranking…
We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…
This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP*, that is…
First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the…
We study robustness to test-time adversarial attacks in the regression setting with $\ell_p$ losses and arbitrary perturbation sets. We address the question of which function classes are PAC learnable in this setting. We show that classes…
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…