Maximum Matching in the Online Batch-Arrival Model 26th June, 2017 Sahil Singla (Carnegie Mellon University) Joint work with Euiwoong Lee

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 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]= 2 3 & OPT=1 Case 2: E[Alg]= 2 3 + 1 3 ∗2 & OPT=2 Fractional Matching? Easier than Integral 𝐆 (𝟐) 𝐆 (𝟐) or Case 1 Case 2

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

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

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

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)

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.

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

𝐆 (𝟏) has A Perfect Matching ALGORITHM Pick M w.p. 2 3 , & 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 1 +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 1 +𝐸 𝑌 𝑢 2

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 1 +𝐸 𝑌 𝑣 1 ≥ 2 3 ∗( 1 2 + 1 2 ) Both not in 𝐆 (𝟏) : 𝐸 𝑌 𝑢 2 +𝐸 𝑌 𝑣 2 ≥1 Only 𝑢 in 𝐆 (𝟏) : 𝐸 𝑌 𝑢 1 +𝐸 𝑌 𝑢 2 + 𝑌 𝑣 2 ≥ 2 3 ∗ 1 2 + 1 3 ∗1 Q.E.D.

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

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) )

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

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

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

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)

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

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

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.

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

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

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 Questions? Fractional Bipartite Matching Instance optimal for two stages Integral Bipartite Matching 2 3 competitive for two-stages Integral General Matching 1 2 + 1 2 𝑂(s) competitive for s-stage s Open Problems Beat half for linear # stages? Other interesting multistage problems? Questions?

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