Maximum Matching in the
Online Batch-Arrival Model
26th June, 2017
SAHIL SINGLA
(CARNEGIE MELLON UNIVERSITY)
JOINT WORK WITH EUIWOONG LEE
2
TWO-STAGE MATCHING PROBLEM
• GRAPH EDGES APPEARS
IN TWO
BATCHES/ STAGES
 𝐆 = 𝐆 (𝟏) ∪ 𝐆 (𝟐)
• 𝐆 (𝟏) APPEARS IN STAGE 1
 Pick Matching 𝐗 (𝟏) in 𝐆 (𝟏) (Unknown 𝐆 (𝟐) )
 Unselected Edges Disappear
• 𝐆 (𝟐) APPEARS IN STAGE 2
 Select 𝐗 (𝟐) in 𝐆 (𝟐) s.t. 𝐗 (𝟏) ∪ 𝐗 (𝟐) is a Matching
• GOAL
 Maximize size of 𝐗 (𝟏) ∪ 𝐗 (𝟐)
 Competitive Ratio:
• GREEDY IS HALF COMPETITIVE
𝐄[𝐀𝐋𝐆(𝐆 (𝟏) , 𝐆 (𝟐) )]
𝐎𝐏𝐓(𝐆 (𝟏) , 𝐆 (𝟐) )
CAN WE BEAT HALF?
THE Z GRAPH
3
•
GRAPH APPEARS IN TWO BATCHES

•
𝐆 (𝟏)
•
𝐆 (𝟏) APPEARS

Pick Matching 𝐗 (𝟏) in 𝐆 (𝟏) (Unknown 𝐆 (𝟐) )

Unselected Edges Disappear
𝐆 (𝟐) APPEARS

•
𝐆 (𝟐)
Select 𝐗 (𝟐) in 𝐆 (𝟐) s.t. 𝐗 (𝟏) ∪ 𝐗 (𝟐) is a Matching
GOAL

𝐆 (𝟐)
𝐆 = 𝐆 (𝟏) ∪ 𝐆 (𝟐)
Maximize size of 𝐗 (𝟏) ∪ 𝐗 (𝟐)
• DO WE PICK EDGE IN 𝐆 (𝟏) ?
 Pick w.p.
2
Case 1: E[Alg]= 2
or
3
3
& OPT=1
Case 2: E[Alg]= 2 3 + 1 3 ∗ 2 & OPT=2
• FRACTIONAL MATCHING?
Case 1
Case 2
 Easier than Integral
4
OUR RESULTS
THEOREM 1:
FOR TWO-STAGE INTEGRAL BIPARTITE MATCHING, THERE EXISTS A
𝟐
𝟑 COMPETITIVE TIGHT ALGORITHM.
THEOREM 2:
FOR TWO-STAGE FRACTIONAL BIPARTITE MATCHING, THERE EXISTS
AN INSTANCE OPTIMAL COMPETITIVE ALGORITHM.
INSTANCE OPTIMAL: Given 𝐆 (𝟏) returns 𝛼 s.t.
• Gets 𝛼 ⋅ 𝑶𝑷𝑻 for every 𝐆 (𝟐)
• For every Alg, ∃𝐆 (𝟐) where ALG ≤ 𝛼 ⋅ 𝑶𝑷𝑻
5
PRIOR WORK
• ONLINE ARRIVAL
 Single arrival in each step (linear # stages)
 Immediate & Irrevocable decisions
 Vertex Arrival or Edge Arrival
• SEMI-STREAMING ARRIVAL
 O(n) decisions postponed
 Vertex Arrival or Edge Arrival
• TWO-STAGE STOCHASTIC OPTIMIZATION
 Costs change every stage
 Arrival from a known distribution
6
OUTLINE
• MULTI-STAGE MATCHING
• EXAMPLES & SPECIAL CASES
• PROOF IDEA: FRACTIONAL BIPARTITE MATCHING
• PROOF IDEA: INTEGRAL BIPARTITE MATCHING
• EXTENSIONS
AND
OPEN PROBLEMS
7
RANDOMLY PICK MAX MATCHING?
• FIND A MAX MATCHING
IN
𝐆 (𝟏)
• PICK IT RANDOMLY, AND NOTHING OTHERWISE
• WHAT IF MULTIPLE MAX MATCHINGS?
𝐆 (𝟏)
𝐆 (𝟐)
 Which one to pick?
 With how much probability?
• GRAPHS KNOWN WHERE FOR EVERY MAX MATCHING 𝐌,
RANDOMLY PICKING 𝐌 GIVES < 𝟐 𝟑
8
𝐆 (𝟏) HAS A PERFECT MATCHING
• SUPPOSE 𝐆 (𝟏)
HAS A
PERFECT MATCHING M
 Every vertex with an incident
edge in 𝐆 (𝟏) is matched in M
• PICK M w.p. 𝟐 𝟑, AND NOTHING OTHERWISE
• OPTIMALLY AUGMENT IN STAGE 2
LEMMA:
ABOVE ALGORITHM IS 𝟐 𝟑 COMPETITIVE
STAGE INTEGRAL BIPARTITE MATCHING.
• HOW TO PROVE ?
FOR TWO-
9
PRIMAL-DUAL FRAMEWORK
• OFFLINE BIPARTITE MATCHING LP
𝑥𝑢𝑣
max
min
𝑢,𝑣 ∈𝐸
s.t.
s.t.
𝑥𝑢𝑣 ≤ 1
∀𝑢 ∈𝑉
∀ 𝑢, 𝑣 ∈ 𝐸
𝑦𝑢
𝑢∈𝑉
𝑦𝑢 + 𝑦𝑣 ≥ 1
𝑣∈𝑛𝑏𝑟(𝑢)
∀ 𝑢, 𝑣 ∈ 𝐸
∀𝑢 ∈𝑉
𝑥𝑢𝑣 ≥ 0
• OPT SOLUTION CERTIFICATE FOR 𝒙
 Show feasible 𝒚
s.t.
∑𝑥𝑢𝑣 = ∑𝑦𝑢
• 𝛼-APPROX SOLUTION CERTIFICATE FOR 𝒙
 Show 𝜶-feasible 𝒚 s.t.
i.e., 𝑦𝑢 + 𝑦𝑣 ≥ 𝛼
∑𝑥𝑢𝑣 = ∑𝑦𝑢
𝑦𝑢 ≥ 0
10
𝐆 (𝟏) HAS A PERFECT MATCHING
• ALGORITHM
 Pick M w.p. 2 3, & Optimally Augment in Stage 2
LEMMA: ABOVE ALGORITHM IS 𝟐
𝟑 COMPETITIVE.
(1)
 Set 𝑋𝑢𝑣 = 1 when (𝑢, 𝑣) is picked in Stage 1
(2)
 Set 𝑋𝑢𝑣 = 1 when (𝑢, 𝑣) is picked in Stage 2
1
2
 Set 𝑥𝑢𝑣 ≜ 𝐸 𝑋𝑢𝑣 + 𝐸 𝑋𝑢𝑣
• CERTIFICATE: 𝒚 s.t. ∑𝑥𝑢𝑣 = ∑𝑦𝑢 & 𝑦𝑢 + 𝑦𝑣 ≥ 2
(1)
 Set 𝑌𝑢
(𝟐)
 Set 𝑌𝒖
=1
2
3
when 𝑢 matched in Stage 1
to be optimal vertex cover for Stage 2, where
(2)
(2)
∑𝑋𝑢𝑣 = ∑𝑌𝑢
 Set 𝑦𝑢 ≜ 𝐸 𝑌𝑢
1
+ 𝐸 𝑌𝑢
2
11
𝐆 (𝟏) HAS A PERFECT MATCHING
• ANALYSIS
 ∑𝒙𝒖𝒗 = ∑𝒚𝒖 : Since
1
(2)
(1)
∑𝑋𝑢𝑣 + ∑𝑋𝑢𝑣 = ∑𝑌𝑢

𝟐
(2)
+ ∑𝑌𝑢
𝟑-FEASIBILITY: Case
analysis ∀ 𝑢, 𝑣 ∈ 𝐄, 𝑦𝑢 + 𝑦𝑣 ≥ 2
2
3
1
1
1. Both in 𝐆 (𝟏) :
𝐸 𝑌𝑢
1
+ 𝐸 𝑌𝑣
1
≥ 3 ∗ (2 + 2)
2. Both not in 𝐆 (𝟏) :
𝐸 𝑌𝑢
2
+ 𝐸 𝑌𝑣
2
≥1
3. Only 𝑢 in 𝐆 (𝟏) :
𝐸 𝑌𝑢
1
+ 𝐸 𝑌𝑢
2
+ 𝑌𝑣
2
2
3
1
2
1
3
≥ ∗ + ∗1
Q.E.D.
12
OUTLINE
• MULTI-STAGE MATCHING
• EXAMPLES & SPECIAL CASES
• PROOF IDEA: FRACTIONAL BIPARTITE MATCHING
• PROOF IDEA: INTEGRAL BIPARTITE MATCHING
• EXTENSIONS
AND
OPEN PROBLEMS
13
TWO-STAGE FRACTIONAL MATCHING
THEOREM 2:
FOR TWO-STAGE FRACTIONAL BIPARTITE MATCHING, THERE EXISTS
AN INSTANCE OPTIMAL COMPETITIVE ALGORITHM.
PROOF IDEA:
• CONSTRUCT
AN
LP ON 𝐆 (𝟏)
THAT MAXIMIZES
𝛼
• GETS 𝛼 ⋅ 𝑶𝑷𝑻 FOR EVERY 𝐆 (𝟐)
• FOR EVERY ALG, ∃𝐆 (𝟐)
WHERE
ALG ≤ 𝛼 ⋅ 𝑶𝑷𝑻
Here 𝑶𝑷𝑻 ≜ OPT(𝐆 (𝟏) , 𝐆 (𝟐))
14
A NEW LP
max
𝛼
INSTANCE OPTIMALITY:
s.t.
𝑓𝑢 ≤ 1
∀𝑢 ∈𝑉
∀ 𝑢, 𝑣 ∈ 𝐸
• Gets 𝛼 ⋅ 𝑶𝑷𝑻 for every 𝐆 (𝟐)
• For every ALG, ∃𝐆 (𝟐) where
ALG ≤ 𝛼 ⋅ 𝑶𝑷𝑻
𝑦𝑢 + 𝑦𝑣 ≥ 𝛼
Let 𝑓𝑢 ≜
𝑣∈𝑛𝑏𝑟(𝑢)
𝑥𝑢𝑣 , 𝑦𝑢 ≥ 0
𝑥𝑢𝑣 =
≤ 1 − 𝑓𝑢
𝑢,𝑣 ∈𝐸
∀𝑢 ∈𝑉
𝑥𝑢𝑣
𝑦𝑢
𝑢∈𝑉
𝑦𝑢 ≥ 𝑓𝑢 − (1 − 𝛼)
QUES: Is 𝛼 ≥ 𝟐 𝟑?
15
OUTLINE
• MULTI-STAGE MATCHING
• EXAMPLES & SPECIAL CASES
• PROOF IDEA: FRACTIONAL BIPARTITE MATCHING
• PROOF IDEA: INTEGRAL BIPARTITE MATCHING
• EXTENSIONS
AND
OPEN PROBLEMS
16
𝐆 (𝟏) IS 𝜶 EXPANDING
Here 𝜶 ≤ 𝟏
• SUPPOSE 𝐆 (𝟏)
IS
𝜶 EXPANDING
 Every S ′ ⊆ 𝑆 has 𝑆′ /𝛼 neighbors
𝑆
 Can pick a random matching 𝐌
s.t. ∀u ∈ 𝑆 & ∀𝑣 ∈ 𝑇 we have
𝑃𝑟 𝑢 ∈ 𝐌 = 1 & 𝑃𝑟 𝑣 ∈ 𝐌 = 𝛼
𝑇
• ALGORITHM
 Pick M w.p. 1 − 𝛼 3, & Optimally Augment in Stage 2
• ANALYSIS
 Set 𝑌𝑢
1
= 1 − 𝜖 & 𝑌𝑣
1
2−𝛼
= 𝜖 for 𝜖 = 3−𝛼 when (𝑢, 𝑣) picked
 For any 𝐆 (𝟐) case-by-case show for every edge (𝑢, 𝑣)
𝑦𝑢 + 𝑦𝑣 ≥ 2 3
TWO-STAGE INTEGRAL MATCHING
17
THEOREM 1:
FOR TWO-STAGE INTEGRAL BIPARTITE MATCHING, THERE EXISTS A
𝟐
𝟑 COMPETITIVE TIGHT ALGORITHM.
ALGORITHM:
1.
CONSTRUCT A MATCHING SKELETON OF 𝑮(𝟏)
 Partition into several 𝜶 Expanding Bipartite Subgraphs
2.
3.
RANDOMLY PICK A MAX MATCHING IN EACH BIPARTITE SUBGRAPH
OPTIMALLY AUGMENT IN STAGE 2
PROOF: SHOW ∃𝒚 S.T.
• ∑𝑥𝑢𝑣 = ∑𝑦𝑢 where 𝑥𝑢𝑣 = 𝐸[𝑋𝑢𝑣 ]
• 𝑦𝑢 + 𝑦𝑣 ≥ 2 3 for every edge (𝑢, 𝑣)
BIPARTITE MATCHING SKELETON
18
GOEL-KAPRALOV-KHANNA
•
•
Decompose 𝐆 (𝟏) into (𝑆𝑗 , 𝑇𝑗 )
𝑆𝑗 , 𝑇𝑗 is 𝜶𝒋 expanding, where 𝛼𝑗 ≤ 1
•
No edge 𝑇𝑗 to 𝑇𝑘
•
No edge 𝑆𝑗 to 𝑇𝑘 for 𝛼𝑗 > 𝛼𝑘
ALGORITHM
•
Select 𝑟 uniformly [0,1]
•
∀𝑗 pick 𝐌𝒋 if 𝑟 < 1 −
𝛼𝑗
= 1 − 𝜖𝑗 & 𝑌𝑣
1
𝑇2
𝑆2
𝑇1
𝑆1
𝑇0
𝑆0
𝑆−1
𝑇−1
𝑆−2
𝑇−2
3
ANALYSIS
1
ALGORITHM :
1. Construct a Matching Skeleton of 𝑮(𝟏)
2. Randomly pick a Max Matching in
each bipartite subgraph
3. Optimally augment in Stage 2
= 𝜖𝑗 for 𝜖𝑗 =
2−𝛼𝑗
•
Set 𝑌𝑢
•
For any 𝐆 (𝟐) show for every edge (𝑢, 𝑣)
𝑦𝑢 + 𝑦𝑣 ≥ 2 3
3−𝛼𝑗
𝛼0 = 1
19
OUTLINE
• MULTI-STAGE MATCHING
• EXAMPLES & SPECIAL CASES
• PROOF IDEA: FRACTIONAL BIPARTITE MATCHING
• PROOF IDEA: INTEGRAL BIPARTITE MATCHING
• EXTENSIONS
AND
OPEN PROBLEMS
EXTENSIONS
20
THEOREM 3:
FOR TWO-STAGE FRACTIONAL GENERAL MATCHING, THERE EXISTS
A 𝟑 𝟓 COMPETITIVE ALGORITHM.
THEOREM 4:
FOR S-STAGE INTEGRAL GENERAL MATCHING, THERE EXISTS A
𝟏
𝟐𝑶(𝐬)
COMPETITIVE ALGORITHM.
𝟏
𝟐
+
21
GENERAL MATCHING SKELETON
EDMONDS-GALLAI DECOMPOSITION
PROOF IDEA:
𝒏𝒃𝒓′(𝑨) has ≤ 1
vertex from each
odd component
•
Run Bipartite Algo for 𝑨 ∪ 𝑪 ∪ 𝒏𝒃𝒓′(𝑨)
•
Pick Matching in 𝑫 synchronously with 𝒏𝒃𝒓′ 𝑨
•
Distribute duals to vertices & odd-components
•
Show for any 𝑮(𝟐) : 𝑦𝑢 + 𝑦𝑣 ≥ 3
5
for every 𝑢, 𝑣 ∈ 𝐸
22
OPEN PROBLEMS
PROBLEM 1:
FOR S-STAGE INTEGRAL BIPARTITE MATCHING, DOES THERE EXIST
AN ALGORITHM THAT BEATS HALF BY A CONSTANT?
PROBLEM 2:
FOR TWO-STAGE INTEGRAL GENERAL MATCHING, WHAT IS THE
TIGHT COMPETITIVE RATIO?
We showed it’s > 1
2
and < 2
3
23
OPEN PROBLEMS
PROBLEM 3:
ANY NATURAL ONLINE PROBLEM WITH 𝐨(𝐬) COMPETITIVE
ALGORITHM IN S-STAGE ONLINE-BATCH ARRIVAL MODEL?
• NOT TRUE FOR
 Online Set Cover
 Online Facility Location
 Online Steiner Tree
 Unrelated Load Balancing (makespan minimization)
SUMMARY
24
•
FRACTIONAL BIPARTITE MATCHING
 Instance optimal for two stages
•
INTEGRAL BIPARTITE MATCHING

•
3
competitive for two-stages
INTEGRAL GENERAL MATCHING

•
2
1
1
+
2
2𝑂(s)
competitive for s-stage s
OPEN PROBLEMS
 Beat half for linear # stages?
QUESTIONS?
 Other interesting multistage problems?
25
REFERENCES
•
L. Epstein, A. Levin, D. Segev, and O. Weimann. Improved bounds for online
preemptive matching. STACS’13
•
A. Goel, M. Kapralov, and S. Khanna. `On the communication and streaming
complexity of maximum bipartite matching’. SODA’12
•
D. Golovin, V. Goyal, V. Polishchuk, R. Ravi, and M. Sysikaski. `Improved
approximations for two-stage min-cut and shortest path problems under
uncertainty’. MATH PROG’15
•
R. M. Karp, U. V. Vazirani, and V. V. Vazirani. `An optimal algorithm for on-line
bipartite matching’. STOC’90
•
L. Lovasz and M. D. Plummer. `Matching Theory’. ANN DISC MATH’86
•
A. Mehta. `Online matching and ad allocation’. TCS’12.
•
C. Swamy and D. B. Shmoys. `Approximation algorithms for 2-stage stochastic
optimization problems’. SIGACT’06