Approximation Algorithms for 2-Stage
Stochastic-Weighted Matching Problem
INEN 689: Large-Scale
Stochastic Optimization
Dec 6, 2005
Balabhaskar Balasundaram
Svyatoslav Trukhanov
Graphs & Matching
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G = (V,E), |V| = n, |E| = m
Matching is a subset of edges such that no
two of them are incident at the same vertex.
Graphs & Matching
„
„
G = (V,E), |V| = n, |E| = m
Matching is a subset of edges such that no
two of them are incident at the same vertex.
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Maximum Weighted Matching
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G = (V,E) with non-negative edge weights ce on
every edge e in E
Find a matching M such that the sum of the
weights of edges in the matching is a maximum
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0,1-Formulation
Max∑ ce xe
e∈E
subject to :
x
∑
δ
e∈ ( v )
e
≤ 1, ∀v ∈ V
xe ∈ {0,1}, ∀e ∈ E
Applications
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Marriage Problem (Bipartite Matching)
Roommate Problem (General Graph
Matching)
Employee-task assignment
Processor-job assignment
Several other time-tabling and assignment
type problems
2-Stage Stochastic Extension (Kong &
Schaefer, 2004)
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First stage weights are known (ce)
Second stage weights are (des) with
probability ps for each scenario s=1,…,r
First Stage: Find a matching
Second Stage: Augment the matching
Maximize sum of first stage weights and
expected sum of second stage weights
Interpretation
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First stage decision presents you with the first
stage profit on Day 1
Tomorrow is uncertain !
We wish to maximize tomorrow’s expected
profit
Eg. Marriage problem…compatibilities
change after marriage !
2-Stage SIP Formulation (Kong &
Schaefer, 2004)
2 SWMP :
r
Max∑ ce xe + ∑ ps ∑ d es yes
e∈E
s =1
e∈E
subject to :
x
∑
δ
e∈ ( v )
e +
y
∑
δ
e∈ ( v )
s
e
≤ 1, ∀v ∈ V , s = 1, … , r
xe ∈ {0,1}, ∀e ∈ E
yes ∈ {0,1}, ∀e ∈ E , s = 1, … , r
Complexity Results
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Maximum Weighted Matching is polynomial
time solvable. (Edmonds, 1965, see also
Cook et al., 1998)
Complexity Results
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Maximum Weighted Matching is polynomial
time solvable. (Edmonds, 1965, see also
Cook et al., 1998)
Maximum Stochastic-Weighted Matching is
NP-Hard. (Kong & Schaefer, 2004)
Even when G is bipartite.
)
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Complexity Results
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Maximum Weighted Matching is polynomial
time solvable. (Edmonds, 1965, see also
Cook et al., 1998)
Maximum Stochastic-Weighted Matching is
NP-Hard. (Kong & Schaefer, 2004)
Even when G is bipartite.
Surprising ?
)
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Complexity Results
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Maximum Weighted Matching is polynomial
time solvable. (Edmonds, 1965, see also
Cook et al., 1998)
Maximum Stochastic-Weighted Matching is
NP-Hard. (Kong & Schaefer, 2004)
Even when G is bipartite.
Surprising ?
Is it the #scenarios (r) ? No. NP-Hard even
when r=1.
)
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Combinatorial Restatement of 2SWMP
Are there matchings M0, M1,…,Mr in G such that
for s=1,…,r, M0∩ Ms = ∅ , M0∪Ms is a matching in G
r
and
∑ c + ∑ p ∑d
e
e∈M 0
s =1
s
e∈M s
s
e
is a maximum ?
Approximation Algorithms
KS-Algorithm (Kong & Schaefer, 2004)
1. Solve Max Wt Matching on G,ce → Z0,x0
2. For each s, solve Max Wt Matching on G,des
→ Zs, ys
3. If Z0 ≥ ∑rs=1psZs
1.
2.
Then return ZAPX = Z0 and SOLAPX = (x0,0,…,0)
Else return ZAPX = ∑rs=1psZs and SOLAPX =
(0,y1,…yr)
Theorem:
½ ZOPT ≤ ZAPX ≤ ZOPT
Mix & Match Algorithm
Define mixed weight wes = max (ce , des)
M&M-Algorithm
1.
For each s, solve Max Wt Matching on G,wes
→ Mc,s ∪ Md,s
2.
3.
Find Mc = ∩s=1..r Mc,s
Return
and
Z APX =
r
∑ ce +∑ ps
e∈M c
s =1
s
d
∑ e
e∈M d ,s
SOLAPX = ( M c , M d ,1 , … , M d ,r )
Mix & Match Bounds ?
Z APX =
∑
e∈M c
r
ce + ∑ ps
s =1
s
d
∑ e
e∈M d ,s
⎛
⎞
s
ZUB = ∑ ps ⎜ ∑ ce + ∑ d e ⎟
⎜ e∈M
⎟
∈
s =1
e
M
d ,s
⎝ c ,s
⎠
⇒ Z APX ≤ Z OPT ≤ ZUB
r
r
and ZUB − Z APX = ∑ ps
s =1
∑c
e
e∈M c ,s \ M c
= E [weight of c − edges lost w.r.t mixed matching
]
Mix & Match Bounds ?
Z APX =
∑
e∈M c
r
ce + ∑ ps
s =1
s
d
∑ e
e∈M d ,s
⎛
⎞
s
ZUB = ∑ ps ⎜ ∑ ce + ∑ d e ⎟
⎜ e∈M
⎟
∈
s =1
e
M
d ,s
⎝ c ,s
⎠
⇒ Z APX ≤ Z OPT ≤ ZUB
r
r
and ZUB − Z APX = ∑ ps
s =1
∑c
e
e∈M c ,s \ M c
= E [weight of c − edges lost w.r.t mixed matching
So What ?!
]
Current Project Work
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Survey literature on approximation algorithms
for stochastic combinatorial optimization
problems
Implement KS-Algorithm
Implement M&M-Algorithm
Compare performance on randomly
generated instances
Ratio 2 and (NO) PTAS ?
Future Work
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Better Approximation ratios ? Best Possible ?
Faster Algorithms ? Matching takes
O(mn+n2logn) … use linear-time
approximations.
Multistage ?
Random Graphs ?
Future Work
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Better Approximation ratios ? Best Possible ?
Faster Algorithms ? Matching takes
O(mn+n2logn) … use linear-time
approximations.
Multistage ?
Random Graphs ?
Approximating Stochastic Combinatorial
Optimization Problems is a new and hot area !