Jan 16 Notes
1
Continuing with the Budget Allocation Problem
1.1
Algorithm
P b x
g(x) = ex−1 and yi = j ijBiij .
When node j ∈ R arrives repetitively match dx of j to argmaxi:i∼j {bij (1 − g(yi ))}
P
1.
i xij = 1, i.e., node j is exhausted or
2. mini:i∼j {yi } = 1, i.e., the budgets of nodes connected to j are exhausted.
Let
1.2
until either
Charging Policy
For each
1.3
dx
of
j
which is matched to
i
αi
then increment
by
bij g(yi )dx
and
βj
by
bij (1 − g(yi )) dx.
Analysis of Competitive Ratio
The change in
yi ,
when
dx
arrives along edge
j → i,
is
dyi =
bij dx
Bi , or equivalently
bij dx = Bi dyi .
Consequently, the change in
αi
is
From the analysis of the the water
is thus
ˆ
αi
=
dαi = bij g(yi )dx = Bi g(yi )dyi .
´ yif
level algorithm we have that
g(yi )dyi = g(yif ) − 1e .
0
ˆ
yif
Bi g(yi )dyi = Bi
0
At the end of the algorithm if node
Bi 1 −
0
i
yif
The nal
αi
value
1
g(yi )dyi = Bi g(yif ) −
.
e
yif = 1 then αi = Bi g(1) − 1e =
f
node i and therefore yi < 1. In this
has exhausted its budget, i.e.,
1
is still budget remaining at
e . The alternative is that there f
f
0
case, an adjacent node j will have βj ≥ bij 1 − g(yi ) , as j will always assign to a node i , say, with yi0 ≤ yi .
∗
Consider the optimal set of edge assignment xij . For those edges adjacent to i, the value gathered from
P
∗
these edges is
b
x
,
which
is
at
most
B
,
due
to the budget constraint on i. The value gathered from
i
j ij ij
P
∗
the proposed algorithm is αi from node i and a selected contribution of
j βj xij from the nodes adjacent
to i. From the bounds on αi and βj above,
αi +
X
j
βj x∗ij
X
1
+
bij x∗ij 1 − g(yif )
≥ Bi g(yif ) −
e
j
X
X
1
f
∗
≥
bij xij g(yi ) −
+
bij x∗ij 1 − g(yif ) (from
e
j
j
X
1
f
f
∗
≥
bij xij g(yi ) − + 1 − g(yi )
e
j
X
1
≥
bij x∗ij 1 −
.
e
j
Therefore extending the bounds over all nodes
i,
the constraints
the competitive ratio is at least
j
1−
Exercise: Convert the above analysis of the competitive ratio to a primal-dual proof.
1
X
1
e.
bij x∗ij ≤ Bi )
2
Ranking Algorithm
2.1
Algorithm
Consider a new online matching problem where a permutation
where
n
j
is the number of nodes on the left. When
argmini:i∼j
An equivalent description is for each node
i
π : [n] → [n]
is picked uniformly at random,
on the right arrives match it to
{π(i) : i
is not yet matched} .
on the left to pick a
Yi ∼ [0, 1],
uniformly at random. When
j
on the right arrives match it to
argmini:i∼j
where
Yi 's as ranking the
M : [n] → [n] is the rank.
2.2
Charging Policy
We consider the
For each
2.3
j
if
j
is matched to
i
{Yi : i
is not yet matched} .
nodes on the left from
then set
αi = g(Yi )
and
1
to
n,
i.e.,
∀i, i0
if
Yi ≤ Yi0
then
M (i) ≤ M (i0 ),
βj = 1 − g(Yi ).
Analysis of Competitive Ratio
Yi0 is xed for i0 6= i. Consider the eect of the
matching on i for Yi taken from 0 to 1. As Yi increases from 0 to 1 then its rank against the other nodes will
f
increase. Let yi be the maximum Yi value such that any larger Yi doesn't change i's rank, and therefore its
match. Thus, the expected value of αi over all Yi is
Fix the randomly assigned value of all nodes
i0 6= i,
ˆ
yif
Eyi [αi ] =
0
If
j
is matched to node
Consequently,
a
Yi0 ≤ yif .
βj
i0
Yi
will serve to improve the rank of
i (decrease M (i)).
0
) ≤ M (i)) with
which was previously higher in the ranking (M (i
can only get larger i.e.,
Eyi [αi + βj ] ≥ 1 −
1
g(y)dy = g(yif ) − .
e
then decreasing
may be matched to say
Therefore,
It follows that
3
j
i when Yi = yif
i.e., all
βj ≥ 1 − g(yif ).
1
e and the competitive ratio is at least
1−
1
e.
Vertex Weight Bipartite Matching Problem
Consider an adaptation of the ranking algorithm problem by assigned a matching prot of
node
i
X
max
∀i
3.1
v i ∈ R+ ,
when a
on the left is matched. The objective of the primal problem is consequently
vi .
matched
Charging Policy
For each
j
if
j
is matched to
i
then set
αi = g(Yi )vi
and
βj = (1 − g(Yi )) vi .
Exercise: Extend the algorithm and analysis of the ranking algorithm to the vertex weighted bipartite match-
ing problem.
2