1 research outputs found
On Complexity of Isoperimetric Problems on Trees
This paper is aimed to investigate some computational aspects of different
isoperimetric problems on weighted trees. In this regard, we consider different
connectivity parameters called {\it minimum normalized cuts}/{\it isoperimteric
numbers} defined through taking minimum of the maximum or the mean of the
normalized outgoing flows from a set of subdomains of vertices, where these
subdomains constitute a {\it partition}/{\it subpartition}. Following the main
result of [A. Daneshgar, {\it et. al.}, {\it On the isoperimetric spectrum of
graphs and its approximations}, JCTB, (2010)], it is known that the
isoperimetric number and the minimum normalized cut both can be described as
-optimization programs, where the latter one does {\it not} admit a
relaxation to the reals. We show that the decision problem for the case of
taking -partitions and the maximum (called the max normalized cut problem
{\rm NCP}) as well as the other two decision problems for the mean version
(referred to as {\rm IPP} and {\rm NCP}) are -complete problems. On
the other hand, we show that the decision problem for the case of taking
-subpartitions and the maximum (called the max isoperimetric problem {\rm
IPP}) can be solved in {\it linear time} for any weighted tree and any . Based on this fact, we provide polynomial time -approximation
algorithms for all different versions of th isoperimetric numbers
considered. Moreover, when the number of partitions/subpartitions, , is a
fixed constant, as an extension of a result of B. Mohar (1989) for the case
(usually referred to as the Cheeger constant), we prove that max and mean
isoperimetric numbers of weighted trees as well as their max normalized cut can
be computed in polynomial time. We also prove some hardness results for the
case of simple unweighted graphs and trees