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.

Maritime-Agricultural Communities: Leadership and Independence System of Javanese Village Communities in the Past

This study discusses the leadership and the independence system of Javanese villages in the Past. In maritime village communities - agrarian, there are people who are trusted to lead, with the dynamics of independence. The village naturally regulates its own community based on village bound. The village burden is related to the demands of interests originating from the colonial supra village, forcing the village to be burdened with the interests of supplying goods for the needs of the world market by intensifying feudal housing. This topic is relevant to the village funding program, empowerment of rural communities with substantial funds, but in the implementation, it is still half-hearted because there is no complete belief that the village has actually been proven capable of taking care of itself and meeting its own needs. To discuss this problem, the historical method is used by reviewing important books specifically discussing villages. The results of the study provide evidence that maritime agrarian villages in Java already have leadership independence that practices in managing and meeting among the villagers. Thus, it is undoubted about the ability to channeling the village funds.

Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory and optimization. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update (MWU) method. The direct application of the MWU method does not yield a strict multiplicative approximation algorithm but yield one with an additional additive error term. A key technique to overcome the issue is to introduce a new concept called (η,γ)-reductions for objective functions with parameters η and γ. We show that our scheme outputs a nearly α-approximate solution if there exists an α-approximation algorithm for a subproblem defined by (η,γ)-reductions. This improves approximation ratios in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.