1. 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?

3. The Z Graph Do we Pick Edge in 𝐆 (𝟏) ? Fractional Matching? 𝐆 (𝟏)
Graph Appears in Two Batches
𝐆= 𝐆 (𝟏) ∪ 𝐆 (𝟐)
𝐆 (𝟏) Appears
Pick Matching 𝐗 (𝟏) in 𝐆 (𝟏) (Unknown 𝐆 (𝟐) )
Unselected Edges Disappear
𝐆 (𝟐) Appears
Select 𝐗 (𝟐) in 𝐆 (𝟐) s.t. 𝐗 (𝟏) ∪ 𝐗 (𝟐) is a Matching
Goal
Maximize size of 𝐗 (𝟏) ∪ 𝐗 (𝟐)
𝐆 (𝟏)
Do we Pick Edge in 𝐆 (𝟏) ?
Pick w.p. 2 3
Case 1: E[Alg]= & OPT=1
Case 2: E[Alg]= ∗2 & OPT=2
Fractional Matching?
Easier than Integral
𝐆 (𝟐)
𝐆 (𝟐) or
Case 1
Case 2

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 Semi-Streaming Arrival
Single arrival in each step (linear # stages)
Immediate & Irrevocable decisions
Vertex Arrival or Edge Arrival
Semi-Streaming Arrival
O (n) decisions postponed
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?
Which one to pick?
With how much probability?
Graphs Known Where For Every Max Matching 𝐌, Randomly Picking 𝐌 gives < 𝟐 𝟑
𝐆 (𝟏)
𝐆 (𝟐)

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
How to Prove ?
Lemma:
Above algorithm is 𝟐 𝟑 Competitive for Two- Stage Integral Bipartite Matching.

9. Primal-Dual Framework
Offline Bipartite Matching LP
𝑢,𝑣 ∈𝐸 𝑥 𝑢𝑣
𝑣∈𝑛𝑏𝑟(𝑢) 𝑥 𝑢𝑣 ≤1
𝑥 𝑢𝑣 ≥0
𝑢∈𝑉 𝑦 𝑢
𝑦 𝑢 + 𝑦 𝑣 ≥1
𝑦 𝑢 ≥0
max
min
s.t.
s.t.
∀ 𝑢∈𝑉
∀ 𝑢,𝑣 ∈𝐸
∀ 𝑢,𝑣 ∈𝐸
∀ 𝑢∈𝑉
Opt Solution Certificate For 𝒙
Show feasible 𝒚 s.t. ∑ 𝑥 𝑢𝑣 =∑ 𝑦 𝑢
𝛼-Approx Solution Certificate For 𝒙
Show 𝜶-feasible 𝒚 s.t. ∑ 𝑥 𝑢𝑣 =∑ 𝑦 𝑢
i.e., 𝑦 𝑢 + 𝑦 𝑣 ≥𝛼

10. 𝐆 (𝟏) has A Perfect Matching
ALGORITHM
Pick M w.p , & Optimally Augment in Stage 2
Set 𝑋 𝑢𝑣 (1) = 1 when (𝑢,𝑣) is picked in Stage 1
Set 𝑋 𝑢𝑣 (2) = 1 when (𝑢,𝑣) is picked in Stage 2
Set 𝑥 𝑢𝑣 ≜𝐸 𝑋 𝑢𝑣 𝐸 𝑋 𝑢𝑣 2
Lemma: Above algorithm is 𝟐 𝟑 Competitive.
Certificate: 𝒚 s.t. ∑ 𝑥 𝑢𝑣 =∑ 𝑦 𝑢 & 𝑦 𝑢 + 𝑦 𝑣 ≥ 2 3
Set 𝑌 𝑢 (1) = 1 2 when 𝑢 matched in Stage 1
Set 𝑌 𝒖 (𝟐) to be optimal vertex cover for Stage 2, where ∑ 𝑋 𝑢𝑣 (2) = ∑𝑌 𝑢 (2)
Set 𝑦 𝑢 ≜𝐸 𝑌 𝑢 𝐸 𝑌 𝑢 2

11. 𝐆 (𝟏) has A Perfect Matching
Analysis
∑ 𝒙 𝒖𝒗 =∑ 𝒚 𝒖 : Since ∑ 𝑋 𝑢𝑣 1 +∑ 𝑋 𝑢𝑣 (2) =∑ 𝑌 𝑢 (1) + ∑𝑌 𝑢 (2)
𝟐 𝟑 -Feasibility: Case analysis ∀ 𝑢,𝑣 ∈𝐄, 𝑦 𝑢 + 𝑦 𝑣 ≥ 2 3
Both in 𝐆 (𝟏) : 𝐸 𝑌 𝑢 𝐸 𝑌 𝑣 ≥ 2 3 ∗( )
Both not in 𝐆 (𝟏) : 𝐸 𝑌 𝑢 𝐸 𝑌 𝑣 ≥1
Only 𝑢 in 𝐆 (𝟏) : 𝐸 𝑌 𝑢 𝐸 𝑌 𝑢 𝑌 𝑣 2 ≥ 2 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 Ques: Is 𝛼≥ 𝟐 𝟑 ? max 𝛼 𝑓 𝑢 ≤1 𝑦 𝑢 + 𝑦 𝑣 ≥𝛼 𝑥 𝑢𝑣 , 𝑦 𝑢 ≥0
𝑢,𝑣 ∈𝐸 𝑥 𝑢𝑣 = 𝑢∈𝑉 𝑦 𝑢
Instance Optimality:
Gets 𝛼⋅𝑶𝑷𝑻 for every 𝐆 (𝟐)
For every ALG, ∃ 𝐆 (𝟐) where ALG ≤𝛼⋅𝑶𝑷𝑻
s.t.
∀ 𝑢∈𝑉
∀ 𝑢,𝑣 ∈𝐸
𝑓 𝑢 ≜ 𝑣∈𝑛𝑏𝑟(𝑢) 𝑥 𝑢𝑣
Let
≤1−𝑓 𝑢
Ques: Is 𝛼≥ 𝟐 𝟑 ?
𝑦 𝑢 ≥ 𝑓 𝑢 −(1−𝛼)
∀ 𝑢∈𝑉

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 Suppose 𝐆 (𝟏) is 𝜶 Expanding Algorithm Analysis
Here 𝜶≤𝟏
Suppose 𝐆 (𝟏) is 𝜶 Expanding
Every S ′ ⊆𝑆 has 𝑆′ /𝛼 neighbors
Can pick a random matching 𝐌 s.t. ∀u∈𝑆 & ∀𝑣∈𝑇 we have 𝑃𝑟 𝑢∈𝐌 =1 & 𝑃𝑟 𝑣∈𝐌 =𝛼 𝑇 𝑆
Algorithm
Pick M w.p. 1− 𝛼 3 , & Optimally Augment in Stage 2
Analysis
Set 𝑌 𝑢 1 =1−𝜖 & 𝑌 𝑣 1 =𝜖 for 𝜖= 2−𝛼 3−𝛼 when (𝑢,𝑣) picked
For any 𝐆 (𝟐) case-by-case show for every edge (𝑢,𝑣) 𝑦 𝑢 + 𝑦 𝑣 ≥ 2 3

17. Two-Stage Integral Matching
Theorem 1: For Two-Stage Integral Bipartite Matching, There Exists a 𝟐 𝟑 Competitive Tight Algorithm.
Algorithm:
Construct a Matching Skeleton of 𝑮 (𝟏)
Partition into several 𝜶 Expanding Bipartite Subgraphs
Randomly Pick a Max Matching in each Bipartite Subgraph
Optimally Augment in Stage 2
Proof: Show ∃𝒚 s.t.
∑ 𝑥 𝑢𝑣 =∑ 𝑦 𝑢 where 𝑥 𝑢𝑣 =𝐸[ 𝑋 𝑢𝑣 ]
𝑦 𝑢 + 𝑦 𝑣 ≥ 2 3 for every edge (𝑢,𝑣)

18. Bipartite Matching Skeleton
Goel-Kapralov-Khanna
Decompose 𝐆 (𝟏) into ( 𝑆 𝑗 , 𝑇 𝑗 )
𝑆 𝑗 , 𝑇 𝑗 is 𝜶 𝒋 expanding, where 𝛼 𝑗 ≤1
No edge 𝑇 𝑗 to 𝑇 𝑘
No edge 𝑆 𝑗 to 𝑇 𝑘 for 𝛼 𝑗 > 𝛼 𝑘
Algorithm
Select 𝑟 uniformly [0,1]
∀𝑗 pick 𝐌 𝒋 if 𝑟<1− 𝛼 𝑗 3
Analysis
Set 𝑌 𝑢 1 =1− 𝜖 𝑗 & 𝑌 𝑣 1 = 𝜖 𝑗 for 𝜖 𝑗 = 2− 𝛼 𝑗 3− 𝛼 𝑗
For any 𝐆 (𝟐) show for every edge (𝑢,𝑣) 𝑦 𝑢 + 𝑦 𝑣 ≥ 2 3
Algorithm :
Construct a Matching Skeleton of 𝑮 (𝟏)
Randomly pick a Max Matching in each bipartite subgraph
Optimally augment in Stage 2
𝑇 2
𝑆 2
𝑇 1
𝑆 1
𝑇 0
𝑆 0
𝛼 0 =1
𝑆 −1
𝑇 −1
𝑆 −2
𝑇 −2

19. OUTLINE Multi-Stage Matching Examples & Special Cases
Proof Idea: Fractional Bipartite Matching
Proof Idea: Integral Bipartite Matching
Extensions and Open Problems

20. Extensions
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 𝟐 + 𝟏 𝟐 𝑶(𝐬) Competitive Algorithm.

21. General Matching Skeleton
Edmonds-Gallai Decomposition
Proof Idea:
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) : 𝑦 𝑢 + 𝑦 𝑣 ≥ 3 5 for every 𝑢,𝑣 ∈𝐸
𝒏𝒃𝒓′(𝑨) has ≤1 vertex from each odd component

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)

24. summary Questions? Fractional Bipartite Matching
Instance optimal for two stages
Integral Bipartite Matching
2 3 competitive for two-stages
Integral General Matching
𝑂(s) competitive for s-stage s
Open Problems
Beat half for linear # stages?
Other interesting multistage problems?
Questions?

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