6 research outputs found
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.