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 MCM
Approx.
Update
Time
Random Specific
ization
Setting
Authors
2-or-worse-approximate matching
2
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 matching
3/2
O(m1/2)
D
Neiman and Solomon
(1+ε)
O(m1/2)
D
Gupta and Peng
(1+ε)
O(1/ε)
D
Inc/Decremental,
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:
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
 Construct
 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:



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.



Assume 3-sum requires Ω(n2) time
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 nonbipartite 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:

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)
 Edges
Weighted EDCS contains a good approximation.

Defn: a weighted EDCS(B, λ) is a subgraph H of
G with the following properties:
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)
 Edges

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)
P
Q
R
S
B/2
+
0
+

Green dotted edges =
0
maximum matching in G
+
 H = maximal B/2-matching in G
 μ(H) is only a 2-approx to μ(G).
B/2
+

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.

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
B/2
P
S

1
+
+


Q
T
B/2
B/2
-
-

1
+


R
U
B/2
+
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)
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).
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:

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:
 Set
is (1+ε)-approximation to μ(H).
 Extra (1+ε) gap between μ(H) and μ(G).
 |M|
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:

is (1+ε)-approximation to μ(H).
 Extra (3/2+ε) error between μ(H) and μ(G).
 |M|
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

TL
T*R



SL
TR
S*L
SR


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.
achieve O(m1/4) compared to the previous O(m1/2).
 Is O(polylog) possible for general bipartite graphs?
 We

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
