On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem

In this paper we investigate the effects of a quantum algorithm which increases the amplitude of the states corresponding to the solutions of the partition problem by a factor of almost two. The study is limited to one iteration.

Minimum Partition of an r − Independence System

Graph partitioning has been studied in the discipline between computer science and applied mathematics. It is a technique to distribute the whole graph data as a disjoint subset to a different device. The minimum graph partition problem with respect to an independence system of a graph has been studied in this paper. The considered independence system consists of one of the independent sets defined by Boutin. We solve the minimum partition problem in path graphs, cycle graphs, and wheel graphs. We supply a relation of twin vertices of a graph with its independence system. We see that a maximal independent set is not always a minimal set in some situations. We also provide realizations about the maximum cardinality of a minimum partition of the independence system. Furthermore, we study the comparison of the metric dimension problem of a graph with the minimum partition problem of that graph.

A new multi-level algorithm for balanced partition problem on large scale directed graphs

Graph partition is a classical combinatorial optimization and graph theory problem, and it has a lot of applications, such as scientific computing, VLSI design and clustering etc. In this paper, we study the partition problem on large scale directed graphs under a new objective function, a new instance of graph partition problem. We firstly propose the modeling of this problem, then design an algorithm based on multi-level strategy and recursive partition method, and finally do a lot of simulation experiments. The experimental results verify the stability of our algorithm and show that our algorithm has the same good performance as METIS. In addition, our algorithm is better than METIS on unbalanced ratio.

The Balanced Connected k-Partition Problem: Polyhedra and Algorithms

The balanced connected k-partition (BCPk) problem consists in partitioning a connected graph into connected subgraphs with similar weights. This problem arises in multiple practical applications, such as police patrolling, image processing, data base and operating systems. In this work, we address the BCPk using mathematical programming. We propose a compact formulation based on flows and a formulation based on separators. We introduce classes of valid inequalities and design polynomial-time separation routines. Moreover, to the best of our knowledge, we present the first polyhedral study for BCPk in the literature. Finally, we report on computational experiments showing that the proposed algorithms significantly outperform the state of the art for BCPk.

A New Multi-level Algorithms for Balanced Partition Problem on Large Scale Directed Graphs

Graph partition is a classical combinatorial optimization and graph theory problem, and it has a lot of applications, such as scientific computing, VLSI design and clustering etc. In this paper, we study the partition problem on large scale directed graphs under a new objective function, a new case of graph partition problem. We firstly propose the modeling of this problem, then design an algorithm based on multi-level strategy and recursive partition method, and finally do a lot of simulation experiments. The experimental results verify the stability of our algorithm and show that our algorithm has the same good performance as METIS. In addition, our algorithm is better than METIS on unbalanced ratio.

Zero-Avoiding Transducers, Length Separable Relations, and the Rational Asymmetric Partition Problem

We consider the problem of partitioning effectively a given irreflexive (and possibly symmetric) rational relation [Formula: see text] into two asymmetric rational relations. This problem is motivated by a recent method of embedding an [Formula: see text]-independent language into one that is maximal [Formula: see text]-independent, where the method requires to use an asymmetric partition of [Formula: see text]. We solve the problem when [Formula: see text] is length-separable, which means that the following two subsets of [Formula: see text] are rational: the subset of word pairs [Formula: see text] where [Formula: see text]; and the subset of word pairs [Formula: see text] where [Formula: see text]. This property is satisfied by all recognizable, all left synchronous, and all right synchronous relations. We leave it as an open problem when [Formula: see text] is not length-separable. We also define zero-avoiding transducers for length-separable relations, which makes our partitioning solution constructive.