关于P与NP的问题

A Solution of the P versus NP Problem based on specific property of clique function

Circuit lower bounds are important since it is believed that a super-polynomial circuit lower bound for a problem in NP implies that P!=NP. Razborov has proved superpolynomial lower bounds for monotone circuits by using method of approximation. By extending this approach, researchers have proved exponential lower bounds for the monotone network complexity of several different functions. But until now, no one could prove a non-linear lower bound for the non-monotone complexity of any Boolean function in NP. While we show that in this paper by replacement of each Not gates into constant 1 equiva…

Poising on Ariadne's thread: An algorithm for computing a maximum clique in polynomial time

In this paper, we present a polynomial-time algorithm for the maximum clique problem, which implies P = NP. Our algorithm is based on a continuous game-theoretic representation of this problem and at its heart lies a discrete-time dynamical system. The rule of our dynamical system depends on a parameter such that if this parameter is equal to the maximum-clique size, the iterates of our dynamical system are guaranteed to converge to a maximum clique.

Towards a complexity theory for the congested clique

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique: -- Nondeterminism and beyond: We introduce the…

Does robustness imply tractability? A lower bound for planted clique in the semi-random model

We consider a robust analog of the planted clique problem. In this analog, a set $S$ of vertices is chosen and all edges in $S$ are included; then, edges between $S$ and the rest of the graph are included with probability $\frac{1}{2}$, while edges not touching $S$ are allowed to vary arbitrarily. For this semi-random model, we show that the information-theoretic threshold for recovery is $\tildeΘ(\sqrt{n})$, in sharp contrast to the classical information-theoretic threshold of $Θ(\log(n))$. This matches the conjectured computational threshold for the classical planted clique problem, and thus…

Finding Planted Cliques in Sublinear Time

We study the planted clique problem in which a clique of size k is planted in an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted clique. It is widely believed that it exhibits a statistical-computational gap when computational efficiency is equated with the existence of polynomial time algorithms. We study this problem under a more fine-grained computational lens and consider the following two questions. 1. Do there exist sublinear time algorithms for recovering the planted clique? 2. What is the smallest running time any algorithm can hope to have? We show that bec…

几类有界边宽图的全多项式FPT算法

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

参数化的复杂性理论已经能够对图上NP难优化问题的关键结构属性进行精细分类，从而更好地理解其真正的困难。最近，P中的问题的硬度结果是使用合理的复杂性理论假设实现的，例如：强指数时间假设（SETH），3SUM和所有对最短路径（APSP）。根据这些假设，许多图论问题不承认真正的次二次算法，甚至不承认真正的次三次算法（Williams and Williams，FOCS 2010 and Abboud，Grandoni，Williams，SODA 2015）。解决上述问题的一个核心技术是固定参数多项式时间问题的固定参数算法（P-FPT）。该技术由Abboud、Williams和Wang在SODA 2016中引入，并由Husfeldt（IPEC 2016）和Fomin等人（SODA 2017）继续使用树宽作为参数。将这种技术应用于另一个重要的图形参数candle-width，仍有待完成。在本文中，我们研究了几个图论问题的硬度结果存在，如周期问题（三角形检测，三角形计数，周长，直径），距离问题（直径，偏心率，Gromov双曲，介数中心）和最大匹配。我们提供了硬度的结果和完全多项式FPT算法，使用cnum-width和它的一些上界作为参数（分裂宽度，模宽度和$P\_4$ -稀疏）。我们认为，我们最重要的结果是一个${\cal O}(k^4 \cdot n + m)$时间算法计算的最大匹配，其中$…