4,886 research outputs found

### A Variant of the Maximum Weight Independent Set Problem

We study a natural extension of the Maximum Weight Independent Set Problem
(MWIS), one of the most studied optimization problems in Graph algorithms. We
are given a graph $G=(V,E)$, a weight function $w: V \rightarrow \mathbb{R^+}$,
a budget function $b: V \rightarrow \mathbb{Z^+}$, and a positive integer $B$.
The weight (resp. budget) of a subset of vertices is the sum of weights (resp.
budgets) of the vertices in the subset. A $k$-budgeted independent set in $G$
is a subset of vertices, such that no pair of vertices in that subset are
adjacent, and the budget of the subset is at most $k$. The goal is to find a
$B$-budgeted independent set in $G$ such that its weight is maximum among all
the $B$-budgeted independent sets in $G$. We refer to this problem as MWBIS.
Being a generalization of MWIS, MWBIS also has several applications in
Scheduling, Wireless networks and so on. Due to the hardness results implied
from MWIS, we study the MWBIS problem in several special classes of graphs. We
design exact algorithms for trees, forests, cycle graphs, and interval graphs.
In unweighted case we design an approximation algorithm for $d+1$-claw free
graphs whose approximation ratio ($d$) is competitive with the approximation
ratio ($\frac{d}{2}$) of MWIS (unweighted). Furthermore, we extend Baker's
technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page

### Focusing of branes in warped backgrounds

Branes are embedded surfaces in a given background (bulk) spacetime. Assuming
a warped bulk, we investigate, in analogy with the case for geodesics, the
notion of {\em focusing} of families of such embedded, extremal 3--branes in a
five dimensional background . The essential tool behind our analysis, is the
well-known generalised Raychaudhuri equations for surface congruences. In
particular, we find explicit solutions of these equations, which seem to show
that families of 3--branes can focus along lower dimensional submanifolds
depending on where the initial expansions are specified. We conclude with
comments on the results obtained and possibilities about future work along
similar lines.Comment: 11 pages, one figur

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