AMS/MBA 546 (Fall, 2010)
Estie Arkin
Network Flows: Solution sketch to homework set # 3
1). Recall the
weighted max acyclic subgraph problem
: Given a directed graph
D
= (
N,A
), with
weights on the arcs
w
ij
≥
0, find a subset of the arcs
a
′
⊂
A
such that
D
′
= (
N,A
′
) is acyclic, and the weight
of
A
′
,
w
(
A
′
), is as large as possible. Consider the following greedy algorithm: Sort the arcs by weight, from
largest to smallest, consider the arcs in this order, and insert an arc into
A
′
as long as it does not create a
directed cycle with the arcs already in
A
′
. (At the start
A
′
is empty.) Clearly this algorithm yields a feasible
solution, however, the weight of
A
′
can be much less than
opt
, the weight of an optimal solution. Describe
a family of directed graphs for which
w
(
A
′
)
/opt
≤
c/n
, where
c
is some constant (that does not depend on
n
) and
n
is the number of nodes in your graph.
Consider the following directed graph on nodes 1,2,...,
n
with arcs (
i,i
+ 1), for every
i
, of cost 2, and arcs
(
i,j
) for every
i > j
of cost 1. Our greedy algorithm will first pick all arcs of cost 2, and then cannot put
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 Fall '08
 Arkin,E
 Graph Theory, vertex cover, n1, A′

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