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

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.

Game Self-organization of Hamiltonian Cycle of the Graph

This paper proposes a new application of the stochastic game model to solve the problem of self- organization of the Hamiltonian cycle of a graph. To do this, at the vertices of the undirected graph are placed game agents, whose pure strategies are options for choosing one of the incident edges. A random selection of strategies by all agents forms a set of local paths that begin at each vertex of the graph. Current player payments are defined as loss functions that depend on the strategies of neighboring players that control adjacent vertices of the graph. These functions are formed from a penalty for the choice of opposing strategies by neighboring players and a penalty for strategies that have reduced the length of the local path. Random selection of players’ pure strategies is aimed at minimizing their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution built on the basis of dynamic vectors of mixed strategies. The elements of the vectors of mixed strategies are the probabilities of choosing the appropriate pure strategies that adaptively take into account the values of current losses. The formation of vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which the gradient method of stochastic approximation is used. During the game, the method increases the value of the probabilities of choosing those pure strategies that lead to a decrease in the functions of average losses. For given methods of forming current payments, the result of the stochastic game is the formation of patterns of self-organization in the form of cyclically oriented strategies of game agents. The conditions of convergence of the recurrent method to collectively optimal solutions are ensured by observance of the fundamental conditions of stochastic approximation. The game task is extended to random graphs. To do this, the vertices are assigned the probabilities of recovery failures, which cause a change in the structure of the graph at each step of the game. Realizations of a random graph are adaptively taken into account when searching for Hamiltonian cycles. Increasing the probability of failure slows down the convergence of the stochastic game. Computer simulation of the stochastic game provided patterns of self-organization of agents’ strategies in the form of several local cycles or a global Hamiltonian cycle of the graph, depending on the ways of forming the current losses of players. The reliability of experimental studies is confirmed by the repetition of implementations of self-organization patterns for different sequences of random variables. The results of the study can be used in practice for game-solving NP-complex problems, transport and communication problems, for building authentication protocols in distributed information systems, for collective decision-making in conditions of uncertainty.

The Buck-Passing Game

We consider two classes of games in which players are the vertices of a directed graph. Initially, nature chooses one player according to some fixed distribution and gives the player a buck. This player passes the buck to one of the player’s out-neighbors in the graph. The procedure is repeated indefinitely. In one class of games, each player wants to minimize the asymptotic expected frequency of times that the player receives the buck. In the other class of games, the player wants to maximize it. The PageRank game is a particular case of these maximizing games. We consider deterministic and stochastic versions of the game, depending on how players select the neighbor to which to pass the buck. In both cases, we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. If the graph on which the game is played admits a Hamiltonian cycle, then this is the outcome of prior-free Nash equilibrium in the minimizing game. For the minimizing game, we then use the price of anarchy and stability to measure fairness of these equilibria.

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem . Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O (log n / log log n ) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

Rainbow Pancyclicity in Graph Systems

Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,\ldots,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,\ldots,G_n$ on the same vertex set of size $n$ with $\delta(G_i)\ge (\frac{1}{2}+\varepsilon)n$, there exists a rainbow Hamiltonian cycle. Our main tool is the absorption technique. Additionally, we prove that with $\delta(G_i)\geq \frac{n+1}{2}$ for each $i$, one can find rainbow cycles of length $3,\ldots,n-1$.

Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.