scholarly journals Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

2021 ◽  
Author(s):  
Yasaman KalantarMotamedi
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Hamiltonian Cycle ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rigorous Proof ◽  
Complete Problem ◽  
Time Solution ◽  
Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

scholarly journals The Complexity of Constructing Gerechte Designs

10.37236/104 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
E. R. Vaughan
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Latin Squares ◽  
Np Complete

Gerechte designs are a specialisation of latin squares. A gerechte design is an $n\times n$ array containing the symbols $\{1,\dots,n\}$, together with a partition of the cells of the array into $n$ regions of $n$ cells each. The entries in the cells are required to be such that each row, column and region contains each symbol exactly once. We show that the problem of deciding if a gerechte design exists for a given partition of the cells is NP-complete. It follows that there is no polynomial time algorithm for finding gerechte designs with specified partitions unless P=NP.

scholarly journals Complexity of Strong Approximation on the Sphere

2019 ◽  
Author(s):  
Naser T Sardari
Keyword(s):  
Polynomial Time ◽  
Strong Approximation ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Ramanujan Graphs ◽  
Other Hand ◽  
Np Complete

Abstract By assuming some widely believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of four squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture 1.1. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating Lubotzky, Phillips, Sarnak (LPS) Ramanujan graphs, under the same assumptions.

COMPUTATIONAL COMPLEXITY OF THE PERFECT MATCHING PROBLEM IN HYPERGRAPHS WITH SUBCRITICAL DENSITY

2010 ◽  
Vol 21 (06) ◽  
pp. 905-924 ◽  
Author(s):  
MAREK KARPIŃSKI ◽  
ANDRZEJ RUCIŃSKI ◽  
EDYTA SZYMAŃSKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Existence Problem ◽  
Matching Problem ◽  
Uniform Hypergraphs ◽  
The Status ◽  
Np Complete

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for [Formula: see text] and trivial for c ≥ ½, leaving the status of the problem with c in the interval [Formula: see text] widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

scholarly journals On the nonkeys

2016 ◽  
Vol 13 (1) ◽  
pp. 11-15
Author(s):  
Vũ Đức Thi
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Relation Scheme ◽  
Np Complete

In this paper we give some results about nonkeys. We show that for relation scheme the problem decide whether there is a nonkey having cardinality greater than or equal to a give integer m is NP-complete. However, for relation this problem can be solved by a polynomial time algorithm.

scholarly journals P versus NP

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complexity Class ◽  
Decision Problems ◽  
Open Problems ◽  
Precise Statement ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? The precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is P-Sel. P-Sel is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it’s given two instances, a “yes” and a “no” instance, the algorithm can always decide which is the “yes” instance. It is known that if NP is contained in P-Sel, then P = NP. We claim a possible selector for 3SAT and thus, P = NP.

scholarly journals On 2-Colorability Problem for Hypergraphs with P_8-free Incidence Graphs

2019 ◽  
Vol 17 (2) ◽  
pp. 257-263
Author(s):  
Ruzayn Quaddoura
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Incidence Graph ◽  
Coloring Problem ◽  
Vertex Set ◽  
Np Complete

A 2-coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. The hypergraph 2- Coloring Problem is the question whether a given hypergraph is 2-colorable. It is known that deciding the 2-colorability of hypergraphs is NP-complete even for hypergraphs whose hyperedges have size at most 3. In this paper, we present a polynomial time algorithm for deciding if a hypergraph, whose incidence graph is P_8-free and has a dominating set isomorphic to C_8, is 2-colorable or not. This algorithm is semi generalization of the 2-colorability algorithm for hypergraph, whose incidence graph is P_7-free presented by Camby and Schaudt.

Computing the residual Galois representations

2011 ◽  
Author(s):  
Bas Edixhoven
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Galois Representations ◽  
Las Vegas ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complex Numbers ◽  
Rigorous Proof ◽  
Main Result Theorem ◽  
Result Theorem

This chapter proves the main result on the computation of Galois representations. It provides a detailed description of the algorithm and a rigorous proof of the complexity. It first combines the results of chapters 11 and 12 in order to work out the strategy of Chapter 3. This gives the main result, Theorem 14.1.1: a deterministic polynomial time algorithm, based on computations with complex numbers. The crucial transition from approximations to exact values is done, and the proof of Theorem 14.1.1 is finished later in the chapter. The chapter then replaces the complex computations with the computations over finite fields from Chapter 13, and gives a probabilistic (Las Vegas type) polynomial time variant of the algorithm in Theorem 14.1.1.

scholarly journals Interdicting facilities in tree networks

Top ◽  
2021 ◽  
Author(s):  
Nicolas Fröhlich ◽  
Stefan Ruzika
Keyword(s):  
Objective Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Solution Algorithm ◽  
Graph Class ◽  
Np Complete ◽  
Number Of Customers ◽  
General Graphs ◽  
Theoretical Results

AbstractThis article investigates a network interdiction problem on a tree network: given a subset of nodes chosen as facilities, an interdictor may dissect the network by removing a size-constrained set of edges, striving to worsen the established facilities best possible. Here, we consider a reachability objective function, which is closely related to the covering objective function: the interdictor aims to minimize the number of customers that are still connected to any facility after interdiction. For the covering objective on general graphs, this problem is known to be NP-complete (Fröhlich and Ruzika In: On the hardness of covering-interdiction problems. Theor. Comput. Sci., 2021). In contrast to this, we propose a polynomial-time solution algorithm to solve the problem on trees. The algorithm is based on dynamic programming and reveals the relation of this location-interdiction problem to knapsack-type problems. However, the input data for the dynamic program must be elaborately generated and relies on the theoretical results presented in this article. As a result, trees are the first known graph class that admits a polynomial-time algorithm for edge interdiction problems in the context of facility location planning.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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