相关论文: Ultimate Polynomial Time

The paper proposes a logical model of combinatorial problems, also it gives an example of a problem of the class NP that can not be solved in polynomial time on the dimension of the problem.

We consider the problems of finding the lexicographically minimal (or maximal) satisfying assignment of propositional formulae for different restricted formula classes. It turns out that for each class from our framework, the above problem…

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The…

Schindler recently addressed two versions of the question P $\stackrel{?}{=}$ NP for Turing machines running in transfinite ordinal time. These versions differ in their definition of input length. The corresponding complexity classes are…

The present work proves that P=NP. The proof, presented in this work, is a constructive one: the program of a polynomial time deterministic multi-tape Turing machine M_ExistsAcceptingPath, that determines if there exists an accepting…

Given a sound first-order p-time theory $T$ capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that $T$ must be incomplete. We leave it…

The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby,…

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are…

We prove that P != NP by proving the existence of a class of functions we call Tau, each of whose members satisfies the conditions of one-way functions. Each member of Tau is a function computable in polynomial time, with negligible…

Many combinatorial optimization problems are often considered intractable to solve exactly or by approximation. An example of such problem is maximum clique which -- under standard assumptions in complexity theory -- cannot be solved in…

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: (*) Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In…

I present a single algorithm which solves the clique problems, "What is the largest size clique?", "What are all the maximal cliques?" and the decision problem, "Does a clique of size k exist?" for any given graph in polynomial time. The…

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach…

As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…

We examine possibility to design an efficient solving algorithm for problems of the class \np. It is introduced a classification of \np problems by the property that a partial solution of size $k$ can be extended into a partial solution of…

This is the final article in a series of four articles. Richard Karp has proven that a deterministic polynomial time solution to K-SAT will result in a deterministic polynomial time solution to all NP-Complete problems. However, it is…

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Recent algebraic techniques introduced by Jonsson et al. (SODA 2013) show that the time complexity of the parameterized SAT($\cdot$) problem…

This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity…

The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is sufficient evidence to establish the…