On the Tractability of (k, i)-Coloring

In an undirected graph, a proper ( k, i )-coloring is an assign- ment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The ( k, i )-coloring problem is to compute the minimum number of colors required for a proper ( k, i )- coloring. This is a generalization of the classic graph colo ring problem. Majumdar et. al. [CALDAM 2017] studied this problem and show ed that the decision version of the ( k, i )-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They aske d if there exists an FPT algorithm with the size of the feedback vertex s et (FVS) as the parameter without using tree-width machinery. We ans wer this in positive by giving a parameterized algorithm with the size o f the FVS as the parameter. We also give a faster and simpler exact algo rithm for ( k, k − 1)-coloring, and make progress on the NP-completeness of sp ecific cases of ( k, i )-colorin

Deterministic Online Call Control in Cellular Networks and Triangle-Free Cellular Networks

Abstract. Wireless Communication Networks based on Frequency Division Multiplexing (FDM in short) plays an important role in the field of communications, in which each request can be satisfied by assigning a frequency. To avoid interference, each assigned frequency must be different to the neighboring assigned frequencies. Since frequency is a scarce resource, the main problem in wireless networks is how to utilize the frequency as fully as possible. In this paper, we consider the call control problem. Given a fixed bandwidth of frequencies and a sequence of communication requests, in handling each request, we must immediately choose an available frequency to accept (or reject) it. The objective of call control problem is to maximize the number of accepted requests. We study the asymptotic performance, i.e., the number of requests in the sequence and the number of available frequencies are very large positive integers. In this paper, we give a 7/3-competitive algorithm for call control problem in cellular network, improving the previous 2.5-competitive result. Moreover, we investigate the triangle-free cellular network, propose a 9/4-competitive algorithm and prove that the lower bound of competitive ratio is at least 5/3.