1,750 research outputs found
Exact Algorithms for Maximum Clique: a computational study
We investigate a number of recently reported exact algorithms for the maximum
clique problem (MCQ, MCR, MCS, BBMC). The program code used is presented and
critiqued showing how small changes in implementation can have a drastic effect
on performance. The computational study demonstrates how problem features and
hardware platforms influence algorithm behaviour. The minimum width order
(smallest-last) is investigated, and MCS is broken into its consituent parts
and we discover that one of these parts degrades performance. It is shown that
the standard procedure used for rescaling published results is unsafe.Comment: 40 pages, 14 figures, 10 tables, 12 short java program listings, code
afailable to download at
http://www.dcs.gla.ac.uk/~pat/maxClique/distribution
New Classes of Distributed Time Complexity
A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al.
(FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang &
Pettie (FOCS 2017) -- have advanced our understanding of one of the most
fundamental questions in theory of distributed computing: what are the possible
time complexity classes of LCL problems in the LOCAL model? In essence, we have
a graph problem in which a solution can be verified by checking all
radius- neighbourhoods, and the question is what is the smallest such
that a solution can be computed so that each node chooses its own output based
on its radius- neighbourhood. Here is the distributed time complexity of
.
The time complexity classes for deterministic algorithms in bounded-degree
graphs that are known to exist by prior work are , , , , and . It is also known
that there are two gaps: one between and , and
another between and . It has been conjectured
that many more gaps exist, and that the overall time hierarchy is relatively
simple -- indeed, this is known to be the case in restricted graph families
such as cycles and grids.
We show that the picture is much more diverse than previously expected. We
present a general technique for engineering LCL problems with numerous
different deterministic time complexities, including
for any , for any , and
for any in the high end of the complexity
spectrum, and for any ,
for any , and
for any in the low end; here
is a positive rational number
3-Colourability of Dually Chordal Graphs in Linear Time
A graph G is dually chordal if there is a spanning tree T of G such that any
maximal clique of G induces a subtree in T. This paper investigates the
Colourability problem on dually chordal graphs. It will show that it is
NP-complete in case of four colours and solvable in linear time with a simple
algorithm in case of three colours. In addition, it will be shown that a dually
chordal graph is 3-colourable if and only if it is perfect and has no clique of
size four
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