Dynamic Matching in Bipartite Graphs

Aaron Bernstein and Cliff Stein

Static Maximum Cardinality Matching

Unweighted graph G Goal: find a maximum matching (MCM) in G Exact algorithms:

O(mn1/2) (Vazirani) O(MM(n)) ~ O(n2.36) (Sankowski) Õ(m10/7) for bipartite graphs (Madry).

Approximate algorithms: (1 + ε)-approximation in time O(m/ε)

Dynamic Exact MCM

Trivial: O(mn1/2) per update O(MM(n)) ~ O(n2.36) per update

Sankowski’s algorithm: O(n1.495) randomized.

Bosek, Leniowski, Sankowski, Zych: O(n1/2) for incremental or decremental

Dynamic Approximate MCMApprox. Update

TimeRandomization

SpecificSetting

Authors

2-or-worse-approximate matching2 O(n) D Trivial

O(1) O(log2(n)) R Onak and Rubinfeld

2 O(log(n)) R Baswana et al.

O(log(n)) O(log2(n)) D Onak and Rubinfeld

(3 + ε) O(m1/3) D Bhattacharya et al.

2 O(log(n)) D Constant Arboricity

Neiman and Solomon

(2 + ε) O(log(n)) D Fractional Matching

Bhattacharya et al.

Better-than-2-approximate matching3/2 O(m1/2) D Neiman and Solomon

(1+ε) O(m1/2) D Gupta and Peng

(1+ε) O(1/ε) D Inc/Decremental,Bipartite

Bosek et al.

What makes for a good approximation?

Best approximation: (1+ε)-approx, O(m1/2) update Does not rely on specific structural properties of

approximate matchings Uses linear-time static (1+ ε)-approx. as a black box.

Algorithm Description: Construct a (1+ ε)-approx. M’ on the current graph M’ remains a (1+ ε)2 approx. for ε μ(G) updates. Amortized update m/μ(G) (can be made worst-case) If μ(G) ≥ m1/2, we have m/μ(G) = O(m1/2) Else, can easily prune edges until m’ ≤ μ(G)2 without

destroying the maximum matching m’/μ(G) = O(μ(G)) = O(m1/2).

What makes for a good approximation?

What about the other better-than-2 approximations? All rely on the following lemma. Lemma: if all augmenting paths w.r.t M have length >

(2k-1) then M is a (1+1/k)-approx. matching. All path lengths > 1 2-approximation All path lengths > 3 3/2-approximation. All path lengths > 1+1/ε (1+O(ε))-approximation.

Augmenting paths seem problematic in dynamic case Little local structure hard to maintain dynamically. Hard to verify their existence/non-existence

Dynamic Matching Lower Bound

Abboud and Williams Conditional lower bound:

Assume 3-sum requires Ω(n2) time Thm: Say that a fully dynamic algorithm A

maintains a matching M such that all augmenting paths w.r.t M have length > 19 Then, A must have amortized update time Ω(m1/3)

Provides some evidence: a better than 1.1-approximation requires Ω(m1/3) time. Potential solution: might be possible to beat this bound

by going beyond an augmenting-paths-based analysis.

Our results

State of the art: (1+ε)-approx., O(m1/2) update, deterministic (3+ε)-approx., O(m1/3) update, deterministic 2-approx., O(log) update, randomized 2-approx., O(log) update, deterministic, constant

arboricity Our results:

Only bipartite graphs (3/2+ε)-approx., O(m1/4) update, deterministic Fastest deterministic algorithm Fastest better-than-2 approximation. Also: (1+ε)-approx., O(log(n)) update, deterministic,

Constant Arboricity

Characterizing Approximate Matchings

Ideally show: A matching M with property P is a (1+ε)-approx. matching

We were unable to do this. Might be able to use LP-duality techniques from

matching in the semi-streaming model.

Matching-Perserving Subgraph

Have a dynamic graph G Want to maintain a large subgraph M with max

degree 1. Idea: Use a matching-perserving subgraph H

First used by Bhattacharya et al (kernel). Update in G Maintain H in G. Update in H Maintain good matching M in H

What we want from H: 1. M is easy to maintain in H 2. Good matching in H is good matching in G 3. Easy to maintain H dynamically.

We develop entirely new subgraph H.

Matching-Perserving Subgraph H

Want: maintain an approx. matching M in H, not G Requirement 1: H should be matching-perserving

µ(H) should be a good approximation to µ(G) Requirement 2: H should be easier to work with

Option 1: H is sparse ~O(n1/2) update Option 2: H has bounded degree B O(B) update Perhaps more complicated properties

Requirement 3: H should be easy to maintain dynamically Approximate matching seems hard to maintain directly

because it lacks local structure. Want definition of H in terms of “local” constraints.

Dynamic matching in bounded degree graphs

Let H be a dynamic graph with max degree Δ Lemma: (Gupta and Peng): can maintain a

(1+ε) matching in H in update time O(Δ) Proof:

Lazy rebuilding update time O(m/µ(G)) Easy to see: m ≤ 2Δ*µ(G) m/µ(G) = O(Δ)

Want the matching-perserving subgraph H to have bounded degree.

Almost-maximal B-matching

The following subgraph H used by Bhattacharya et al. All vertex degrees bounded by B If (u,v) not in H then u or v have degree ≥ B(1-ε)

H easy to work with: bounded degree B Matching-Perserving:

Claim (new): µ(H) ≥ (1/2 – ε) µ(G) They show something less modular Proof surprisingly difficult (and possibly false) for non-

bipartite case. H easy to maintain: maximal completely local. Recourse to maximality cannot go beyond 2-approx. Want: local properties that capture global structure.

Edge Degree Constrained Subgraph (EDCS)

Let λ ~ ε-2 be a small fixed constant. Defn: δ(u) = (weighted) degree of u in H. Defn: for (u,v) in G, Edge Degree(u,v) = δ(u)+δ(v) Defn: a weighted EDCS(B, λ) is a subgraph H of

G with the following properties: Edges on H have “virtual” positive integral weights. The actual subgraph is support(H). 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

Weighted EDCS contains a good approximation.

Defn: a weighted EDCS(B, λ) is a subgraph H of G with the following properties: Edges on H have “virtual” positive integral weights. 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

Thm: let H be a weighted EDCS(B, λ) of a bipartite graph G. Viewing the support of H as an unweighted graph, we have μ(H) ≥ (1-ε) μ(G) Note: H has bounded degree B

Maximal B/2-matching VS. EDCS(B, λ)

Green dotted edges = maximum matching in G

H = maximal B/2-matching in G μ(H) is only a 2-approx to μ(G).

R S

P Q

B/2-matching

Weighted EDCS(B, λ): 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

Edges in G\H violate property 2: must add them to H Edges in H now violate property 1: must remove

them from H.

B/2

0

0

B/2

+

+

+

+

Maintaining a Weighted EDCS

Weighted EDCS(B,λ) properties: 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

Insertion of edge (u,v) in G: If (u,v) doesn’t violate Prop. 2, ignore it. δ(u) and δ(v) increase – incident edges might violate Prop. 1 Rebalance δ(v) by deleting violating edge (v,w) δ(w) decreases – incident edges might violate Prop. 2 Rebalance δ(w) by adding weight to violating edge (w,x) Now need to rebalance δ(x) and so on.

Can show: only need small number of rebalances. Easy: rebalance in time O(degree). Can achieve: rebalance in time O(arboricity)

Relies on dynamic orientation

Weighted EDCS hard to maintain

Weights only used internally to determine H. Final result: unweighted subgraph H that

contains a large matching. When adversary deletes (u,v), if (u,v) had a

large weight must do a lot of fix-up work. Want: Every edge plays a limited role so that

any single edge change can be easily processed.

Unweighted EDCS

Let λ ~ ε-2 be a small fixed constant. Defn: an unweighted EDCS(B, λ) is an unweighted

subgraph H of G with the following properties: 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

Unweighted EDCS

Let λ ~ ε-2 be a small fixed constant. Defn: an unweighted EDCS(B, λ) is an unweighted

subgraph H of G with the following properties: 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G\H, δ(u)+δ(v) ≥ B(1- λ)

Thm: If H is an unweighted EDCS(B, λ) of a bipartite graph G then μ(H) ≥ (2/3 – ε) μ(G)

weighted VS. unweighted: (1+ ε) approximation VS. (3/2+ε) approximation. Unweighted is easier to maintain. Both capture some global structure of the graph.

Unweighted VS. Weighted

Unweighted EDCS: μ(H) = 2n/3. Weighted EDCS: P-S and R-U edges violate Prop. 2.

Must add weight to P-S and R-U edges. Must then remove weight from P-T and Q-U edges (Prop. 1) Must now add weight to Q-T edges (Prop. 2).

(B/2-1)-matching

Blue edges = In H Green edges = In G\H Unweighted EDCS:

1. For edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For edge (u,v) in G\H, δ(u)+δ(v) ≥ B(1- λ)

Weighted EDCS: 1. For edge (u,v) in H, δ(u)+δ(v) ≤ B (same) 2. For edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ)

(B/2-1)-matching

P

Q

R

S

T

U

B/2

B/2 B/2

B/21

1

+ +

+ +

- -

Unbounded Arboricity: Maintaining an Unweighted EDCS

Arboricity = O(m1/2) < O(n) Unweighted: only small # of rebalances per update Time for rebalance: O(m1/2) Gap of λB = Ω(B) between EDCS properties Intuition: only need to rebalance a vertex after λB

degree changes. Requires Unweighted EDCS. “Amortized” rebalance time:

O(m1/2/ λB ) = O(m1/2/B) Analysis actually worst-case.

Small Arboricity Bipartite Graphs

At all times arboricity ≤ α Thm: Weighted EDCS(B,λ) (1+ε)-approx. Thm: Can maintain a weighted EDCS(B,λ) H in

time Õ(α2B) per update in G Algorithm:

Set B ~ ε-2 = O(1) Directly maintain a (1+ ε)-approx. matching M in H. M will be our final matching in G as well. Update in G fix H in time Õ(αB) = Õ(α2) Change to H Fix up M in time O(B) = O(1) M is a (1+ ε)2-approx. matching in G:

|M| is (1+ε)-approximation to μ(H). Extra (1+ε) gap between μ(H) and μ(G).

General Bipartite Graphs

Thm: Unweighted EDCS(B,λ) (3/2+ε)-approx. Thm: Can maintain an unweighted EDCS(B,λ) H

in time O(B + m1/2/B) per update in G. Algorithm:

Set B = O(m1/4) Directly maintain a (1+ ε)-approx. matching M in H. M will be our final matching in G as well. Update in G fix H in time O(B + m1/2/B) = O(m1/4). Changes to H Fix up M in time O(B) = O(m1/4). M is a (3/2+ ε)-approx. matching in G:

|M| is (1+ε)-approximation to μ(H). Extra (3/2+ε) error between μ(H) and μ(G).

EDCS contains a good approximation

Let H be an unweighted EDCS(B,λ). Assume G has a perfect matching: μ(G) = n Want to show: μ(H) ≥ (2/3-ε)n Contradiction Assumption: μ(H) < (2/3-ε)n Let M be some maximum matching in H.

|M| < (2/3-ε)n Can view M as a matching in G, and look at the

residual graph in G w.r.t M.

The residual graph (n vertices per side).

Contradiction Assumption: μ(H) < (2/3-ε)n

S*L

SL

SR

TR

TL T*R

H contains no edges crossing cut. |S*L| = |T*R| ≥ (n/3)(1+ε) Assume: |SL|=|SR|=|TL|=|TR| < n/3 ≥ (n/3)(1+ε) disjoint crossing edges in G\H Assume: crossing edges go from S*L to T*R

Then: average degree of |S*L| U |T*R| ≥ B/2(1-λ) ~ B/2 (Prop. 2)

Can’t fit degree of S*L in smaller SR (Prop. 1) Weighted EDCS: edges from SL to SR take

up even more space in SR (Prop 2.)

1. For edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For edge (u,v) in G\H, δ(u)+δ(v) ≥ B(1- λ

EDCS review

λ = O(ε-2) = O(1) Weighted EDCS(B,λ) H

1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G, δ(u)+δ(v) ≥ B(1- λ) µ(H) ≥ (1- ε) µ(G)

Unweighted EDCS(B, λ) H 1. For each edge (u,v) in H, δ(u)+δ(v) ≤ B 2. For each edge (u,v) in G\H, δ(u)+δ(v) ≥ B(1- λ) µ(H) ≥ (2/3- ε) µ(G)

Bounded Degree B easy to maintain (1+ε)-matching in H

Upcoming Work: Non-Bipartite Graphs

Weighted EDCS only a 3/2-approximation G = complete graph on 6 vertices. µ(G) = 3 H = 2 triangles with edge weights B/4 all vertex degree B/2. But µ(H) = 2

Focus on unweighted EDCS H Can probably show: µ(H) ≥ (3/2-ε) µ(G)

Significantly more involved than in bipartite case Can probably show: can maintain EDCS in

same time as in bipartite case

Open problems: EDCS

Want: faster update times for better-than-2 approximation. We achieve O(m1/4) compared to the previous O(m1/2). Is O(polylog) possible for general bipartite graphs?

Possible approach: faster algorithms for maintaining an EDCS? Randomization Layering

Characterizing Approximate Matchings

Want: if some matching M has property P then it is a (1+ε)-approximation There might already exist LP-duality characterizations,

especially in the semi-streaming literature. Also want: other possible transitional subgraphs

that guarantee a (1+ε)-approximation. The transitional subgraph H needs to have small

bounded degree (or some other property that makes it easy to deal with).

For non-bipartite graphs, we so far only have a characterizations of 3/2-approximate matchings