455 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 , a weight function ,
a budget function , and a positive integer .
The weight (resp. budget) of a subset of vertices is the sum of weights (resp.
budgets) of the vertices in the subset. A -budgeted independent set in
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 . The goal is to find a
-budgeted independent set in such that its weight is maximum among all
the -budgeted independent sets in . 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 -claw free
graphs whose approximation ratio () is competitive with the approximation
ratio () of MWIS (unweighted). Furthermore, we extend Baker's
technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
An Improved Algorithm for Finding Maximum Outerplanar Subgraphs
We study the NP-complete Maximum Outerplanar Subgraph problem. The previous
best known approximation ratio for this problem is 2/3. We propose a new
approximation algorithm which improves the ratio to 7/10
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