Lecture Notes in Computer Science 4917

Complementing Missing and Inaccurate Profiling 297
an analogue argument. Finally, we may insert a new directed edge of cost 0 from the
single sink t to the single source s. This will turn any feasible flow to a circulation of
the same cost. Thus we may assume that in fact S=T=∅.
4.1 Constructing the Fixup Graph
= and its measured flow w(v) and w (e)
for the vertices
and edges respectively, as input for the optimal flow fixup, we wish to create a new
Given a graph G ( V , E)
graph ' ( V ',
E')
G = with given minimal and maximal capacity constraints (b, c) for
each edge, herein the fixup graph. This transformation is formally defined below.
Input:
• G t = ( Vt
, Et
)
transformation.
denotes the original graph after applying the vertex
• w (e)
denotes the initial flow estimation for every edge t E e∈ (this is
thoroughly explained in section 5 under "setting the constants").
• k (e)
±
denotes the negative/positive confidence constants on any t E e∈ (see
section 3).
• Let D ( v)
≡ ( w(
el
) ) − ( w(
ek
) ) for every t V v ∈ .
∑
el∈out(
v)
∑
ek∈in(
v)
Output:
• G '= ( V ',
E')
the fixup graph
• b( e'),
c(
e')
minimum/maximum capacities for flow on every edge e'∈ E'
• k (e')
positive confidence constant for any e'∈ E'
(note that infusing negative
flow is not possible so here we do not need a negative confidence constants)
The output graph for the circulation problem is defined as follows:
1. s'�new Vertex, t'�new Vertex
2. b ( t',
s'
) ← 0,
c(
t',
s'
) ← ∞
3. cp ( t',
s'
) ← 0
4. Er ← φ, L ← φ
5. foreach e∈ Et
do:
a. b ( e)
← 0 , (e)
← ∞
c
+
, k'(
e)
← k ( e)
6. foreach e = v,
u ∈ Et
such that u, v ∉ Et
do:
a. ← E ∪ u,
v
Er r
b. k'( ( u,
v ) ) ← k ( ( v,
u ) ), b(
u,
v ) ← 0,
c(
u,
v ) ← w(
e)
−