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Perfect matchings in O(n log n) time in regular bipartite graphs

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

October 20, 2019

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 1 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 2 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 3 / 28

Introduction

We are given a d-regular bipartite graph on 2n nodes with m = nd edges. We need to find a perfect matching in the graph. This can be done in O(m√n) time using Hopcroft - Karp algorithm. This paper gives a randomized algorithm that finds a perfect matching in O(n log n) time (both in expectation and with high probability).

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 4 / 28

Introduction

We are given a d-regular bipartite graph on 2n nodes with m = nd edges. We need to find a perfect matching in the graph. This can be done in O(m√n) time using Hopcroft - Karp algorithm. This paper gives a randomized algorithm that finds a perfect matching in O(n log n) time (both in expectation and with high probability).

Note

The O(n log n) running time is smaller than the O(nd) time required to read the input and only slightly worse than the O(n) time required to

• utput a perfect matching.
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 4 / 28

Introduction

We are given a d-regular bipartite graph on 2n nodes with m = nd edges. We need to find a perfect matching in the graph. This can be done in O(m√n) time using Hopcroft - Karp algorithm. This paper gives a randomized algorithm that finds a perfect matching in O(n log n) time (both in expectation and with high probability).

Note

The O(n log n) running time is smaller than the O(nd) time required to read the input and only slightly worse than the O(n) time required to

• utput a perfect matching.

Note

It is assumed that the graph is presented in the adjacency array format.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 4 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 5 / 28

Previous Approaches

• H. N. Gabow, G. Karov (1982)

O(m) time algorithm when d is a power of 2.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 6 / 28

Previous Approaches

• H. N. Gabow, G. Karov (1982)

O(m) time algorithm when d is a power of 2.

• R. Cole, K. Ost, S.Schirra (2001)

O(m) time algorithm for general d.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 6 / 28

Previous Approaches

• H. N. Gabow, G. Karov (1982)

O(m) time algorithm when d is a power of 2.

• R. Cole, K. Ost, S.Schirra (2001)

O(m) time algorithm for general d.

• A. Goel, M. Kapralov, and S. Khanna (2009)

˜ O

• min
• m, n2

d

• time for general d.

Also gives a lower bound of ˜ Ω

• min
• m, n2

d

• n the running time of

an algorithm that uses non-adaptive uniform sampling.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 6 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 7 / 28

Construction of H

Let G = (P, Q, E) be the input d-regular bipartite graph and M be a partial matching in G that leaves 2k vertices unmatched.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 8 / 28

Construction of H

Let G = (P, Q, E) be the input d-regular bipartite graph and M be a partial matching in G that leaves 2k vertices unmatched. Construct the directed graph H = (V , E ′) as:

• 1. Orient edges of G from P to Q
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 8 / 28

Construction of H

Let G = (P, Q, E) be the input d-regular bipartite graph and M be a partial matching in G that leaves 2k vertices unmatched. Construct the directed graph H = (V , E ′) as:

• 1. Orient edges of G from P to Q
• 2. Add a source vertex s and add d parallel edges to each unmatched

vertex in P directed out of s.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 8 / 28

Construction of H

Let G = (P, Q, E) be the input d-regular bipartite graph and M be a partial matching in G that leaves 2k vertices unmatched. Construct the directed graph H = (V , E ′) as:

• 1. Orient edges of G from P to Q
• 2. Add a source vertex s and add d parallel edges to each unmatched

vertex in P directed out of s.

• 3. Add a sink vertex t and add d parallel edges from each unmatched

vertex in Q directed into t.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 8 / 28

Construction of H

Let G = (P, Q, E) be the input d-regular bipartite graph and M be a partial matching in G that leaves 2k vertices unmatched. Construct the directed graph H = (V , E ′) as:

• 1. Orient edges of G from P to Q
• 2. Add a source vertex s and add d parallel edges to each unmatched

vertex in P directed out of s.

• 3. Add a sink vertex t and add d parallel edges from each unmatched

vertex in Q directed into t.

• 4. Contract each pair (u, v) ∈ M into a supernode.
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 8 / 28

Construction of H (contd.)

The graph H has n + k + 2 nodes and for each vertex v ∈ V \ {s, t}, the in-degree of v is equal to its out-degree. Also, the out-degree of s equals dk and the in-degree of t equals dk.

Note

Any path from s to t in H gives an augmenting path in G w.r.t. M.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 9 / 28

Construction of H (contd.)

The graph H has n + k + 2 nodes and for each vertex v ∈ V \ {s, t}, the in-degree of v is equal to its out-degree. Also, the out-degree of s equals dk and the in-degree of t equals dk.

Note

Any path from s to t in H gives an augmenting path in G w.r.t. M. Consider the graph H∗, obtained from H by:

• 1. Contracting s and t to s∗
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 9 / 28

Construction of H (contd.)

The graph H has n + k + 2 nodes and for each vertex v ∈ V \ {s, t}, the in-degree of v is equal to its out-degree. Also, the out-degree of s equals dk and the in-degree of t equals dk.

Note

Any path from s to t in H gives an augmenting path in G w.r.t. M. Consider the graph H∗, obtained from H by:

• 1. Contracting s and t to s∗
• 2. Adding a self loop at every vertex
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 9 / 28

Construction of H (contd.)

The graph H has n + k + 2 nodes and for each vertex v ∈ V \ {s, t}, the in-degree of v is equal to its out-degree. Also, the out-degree of s equals dk and the in-degree of t equals dk.

Note

Any path from s to t in H gives an augmenting path in G w.r.t. M. Consider the graph H∗, obtained from H by:

• 1. Contracting s and t to s∗
• 2. Adding a self loop at every vertex

Note

H∗ is a balanced directed graph i.e. the in-degree of every vertex is equal to its out-degree. Out-degree of s∗ equals (dk + 1), for unmatched vertices it equals (d + 1) and for matched vertices it equals d.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 9 / 28

Pseudo-code for algorithm

Input: A d-regular bipartite graph G = (P, Q, E) in adjacency array format. Output: A perfect matching on G Algorithm:

• 1. Set j := 0, M := φ
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 10 / 28

Pseudo-code for algorithm

Input: A d-regular bipartite graph G = (P, Q, E) in adjacency array format. Output: A perfect matching on G Algorithm:

• 1. Set j := 0, M := φ
• 2. Repeatedly run truncated walk(s, bj) for bj := 2
• 4 + 2n

n−j

• n H∗,

until a successfull run is obtained.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 10 / 28

Pseudo-code for algorithm

Input: A d-regular bipartite graph G = (P, Q, E) in adjacency array format. Output: A perfect matching on G Algorithm:

• 1. Set j := 0, M := φ
• 2. Repeatedly run truncated walk(s, bj) for bj := 2
• 4 + 2n

n−j

• n H∗,

until a successfull run is obtained.

• 3. Denote the augmenting path obtained by removing possible loops

from the sequence of steps taken by the walk by p. Set Mj+1 := Mj ⊕ p

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 10 / 28

Pseudo-code for algorithm

Input: A d-regular bipartite graph G = (P, Q, E) in adjacency array format. Output: A perfect matching on G Algorithm:

• 1. Set j := 0, M := φ
• 2. Repeatedly run truncated walk(s, bj) for bj := 2
• 4 + 2n

n−j

• n H∗,

until a successfull run is obtained.

• 3. Denote the augmenting path obtained by removing possible loops

from the sequence of steps taken by the walk by p. Set Mj+1 := Mj ⊕ p

• 4. Set j := j + 1 and go to step 2
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 10 / 28

Pseudo-code for algorithm (contd.)

truncated walk(u, b) is defined as follows:

• 1. If u = t, return SUCCESS
• 2. Set v := sample out edge(u). Set b := b − 1. If b < 0, return FAIL,

else return truncated walk(v, b)

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 11 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 12 / 28

Expected return time to s∗

When 2k vertices are unmatched, the expected return time tk to s∗ is equal to the inverse of the stationary measure of s∗. tk = 1 πs∗ =

• v∈V deg(v)

deg(s∗) = (n − k)d + 2k(d + 1) + (dk + 1) dk + 1 ≤ 1 + (n + k)(d + 1) dk ≤ 4 + 2n k

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 13 / 28

Expected runtime

By the previous analysis, it takes at most

• 4 +

2n (n−j)

• expected time to find

an augmenting path w.r.t Mj. ∴ The expected runtime of the algorithm is:

n−1

• j=0

4 + 2n (n − j) = 4n + 2n

n

• j=1

1 n = O(n log n)

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 14 / 28

High probability analysis

By Markov inequality, a truncated walk of length 2(4 + 2n

n−j ) succeeds

with probability at least 1

2

The probability of failure is less than 1

2

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 15 / 28

High probability analysis

By Markov inequality, a truncated walk of length 2(4 + 2n

n−j ) succeeds

with probability at least 1

2

The probability of failure is less than 1

2

Let Yj be independent exponentially distributed with mean µj :=

bj ln 2.

The probability that augmentation step j takes time more than qbj time is: Pr(Xj ≥ qbj) ≤ 1 2q = exp

• −qbj ln 2

bj

• = Pr(Yj >= qbj)
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 15 / 28

High probability analysis

By Markov inequality, a truncated walk of length 2(4 + 2n

n−j ) succeeds

with probability at least 1

2

The probability of failure is less than 1

2

Let Yj be independent exponentially distributed with mean µj :=

bj ln 2.

The probability that augmentation step j takes time more than qbj time is: Pr(Xj ≥ qbj) ≤ 1 2q = exp

• −qbj ln 2

bj

• = Pr(Yj >= qbj)

Let Y = n−1

j=0 Yj, with mean µ = E(Y ). By Markov’s inequality and

Chernoff Bound, for any t, δ > 0: Pr [Y ≥ (1 + δ)µ] ≤ E[etY ] et(1+δ)µ

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 15 / 28

High probability analysis (contd.)

For any j, and for t < 1

µj ,

E[etYj] = 1 µj ∞ etxe

− x

µj dx =

1 1 − tµj ∴ E[etY ] =

n−1

• j=0

1 1 − tµj ∴ Pr[Y ≥ (1 + δ)µ] ≤ e−t(1+δ)µ n−1

j=0 (1 − tµj)

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 16 / 28

High probability analysis (contd.)

For any j, and for t < 1

µj ,

E[etYj] = 1 µj ∞ etxe

− x

µj dx =

1 1 − tµj ∴ E[etY ] =

n−1

• j=0

1 1 − tµj ∴ Pr[Y ≥ (1 + δ)µ] ≤ e−t(1+δ)µ n−1

j=0 (1 − tµj)

Assume t =

1 2µn−1 . As µn−1 is the largest among all µj’s,

(1 − tµj) ≥ etµj ln 4

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 16 / 28

High probability analysis (contd.)

For any j, and for t < 1

µj ,

E[etYj] = 1 µj ∞ etxe

− x

µj dx =

1 1 − tµj ∴ E[etY ] =

n−1

• j=0

1 1 − tµj ∴ Pr[Y ≥ (1 + δ)µ] ≤ e−t(1+δ)µ n−1

j=0 (1 − tµj)

Assume t =

1 2µn−1 . As µn−1 is the largest among all µj’s,

(1 − tµj) ≥ etµj ln 4 ∴ Pr[Y ≥ (1 + δ)µ] ≤ e

− (1+δ+ln 4)µ

2µn−1

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 16 / 28

High probability analysis (contd.)

µ =

4 ln 2

n−1

j=0 (1 + n n−j ) = 4 ln 2n(1 + Hn) ≥ µn−1H(n), where H(n) is

the nth harmonic number

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 17 / 28

High probability analysis (contd.)

µ =

4 ln 2

n−1

j=0 (1 + n n−j ) = 4 ln 2n(1 + Hn) ≥ µn−1H(n), where H(n) is

the nth harmonic number As H(n) ≥ ln n, we get: Pr[Y ≥ (1 + δ)µ] ≤ n− (1+δ−ln 4)

2

As µ = O(n log n), we get our high probability result.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 17 / 28

Overview

1

Problem Statement

2

Previous Approaches

3

Algorithm

4

Algorithm Analysis

5

Lower bound for deterministic algorithms

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 18 / 28

Lower Bound for deterministic Algorithm

THEOREM: For any 1 ≤ d < n/12, there exists a family of d-regular graphs on which any deterministic algorithm for finding a perfect matching requires Ω(nd) time.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 19 / 28

Family of canonical Graphs

Canonical bipartite graph G(P ∪ {t}, Q ∪ {s}, E) ∈ G(d) P = P1 ∪ P2 Q = Q1 ∪ Q2 |Pi| = |Qi| = 3d for i ∈ {1, 2}

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 20 / 28

Family of canonical Graphs

Canonical bipartite graph G(P ∪ {t}, Q ∪ {s}, E) ∈ G(d) P = P1 ∪ P2 Q = Q1 ∪ Q2 |Pi| = |Qi| = 3d for i ∈ {1, 2}

• 1. Connect s to d arbitrary vertices in P1
• 2. Connect t to d arbitrary vertices in Q2
• 3. Add a matching M′ from Q′

1 ⊂ Q1 to a P′ 2 ⊂ P2 where

|Q′

1| = |P′ 2| = d

• 4. Connect remaining edges in Pi to Qi, i ∈ {1, 2} so that degree of

each vertex is exactly d

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 20 / 28

Adversary Game

Let D be a Deterministic Algorithm for finding perfect matching Let A be an Adaptive Adversary Goal of A is to maximize number of edges that D needs to check for finding perfect matching.

Observation

For D to find a perfect matching, D must discover an edge in M′.

Target

To show that A can force D to examine Ω(d2) edges before revealing an edge in M′.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 21 / 28

Adversary Game (contd.)

Note

It is sufficient to show an Ω(d2) drawn for graphs from G(d) because we can take Θ(n/d) disjoint copies of canonical graphs to create a d-regular graph on n vertices.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 22 / 28

Adversary Game (contd.)

Queries: For any u ∈ P ∪ Q, when D makes query Q(u) to adversary A, A outputs a vertex v that had not been revealed as being adjacent to u. The cost of each Query is 1. Assumption Adversary reveals s, t, and partition of remaining vertices into Pi, Qi, i = 1, 2 along with all edges from s to P1 and from t to Q2 to D at beginning without any cost.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 23 / 28

Sufficient Conditions for subgraph

Lemma :Let Gr(P ∪ {t}, Q ∪ {s}, Er) be simple bipartite with following conditions. (a) vertex s is connected to d distinct vertices in P1 and vertex t is connected to d distinct vertices in Q2. (b) all other edges connect Pi to Qi for some i ∈ {1, 2} (c) degree of each vertex is at most d (d) at least 5d

2 vertices each in both Q1 and P2 have degree strictly less

than d

5

Then for any pair of free vertices u, v such that u ∈ Pi and v ∈ Qi, there exists a canonical graph with same vertices and edge set E such that Er ∪ (u, v) ⊆ E

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 24 / 28

Subgraphs (contd.)

Definition: If a graph Gr satisfies the conditions (a) to (d), then it will be said to be in evasive mode. Otherwise, it will be called in non-evasive mode. Note that initially, the graph will be in evasive mode. Subsequently, the graph can go from evasive mode to non-evasive mode.

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 25 / 28

Adversary strategy

When D makes query for u for some free vertex, if u ∈ P1, A returns edge (u, v) such that v ∈ Q1 if u ∈ P2, A returns edge (u, v) such that v ∈ Q2 if u ∈ Q1, A returns edge (u, v) such that v ∈ P1 if u ∈ Q2, A returns edge (u, v) such that v ∈ P2 Note that for any query, conditions (a), (b) and (c) are not violated. Only condition that can be violated by subsequent queries from D is condiiton (d).

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 26 / 28

Adversary strategy (contd.)

Each query from D can increase the degree of vertices in Q1 or P2 by exactly 1. Looking at condition (d), the number of queries A can respond without going into non-evasive mode is about 5d

2 ∗ d 5

• , which is Ω(d2)
• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 27 / 28

Done...

Shivam Mittal (CS16B038) Utkarsh Patel (CS16B037)

• Ashish Goel, Michael Kapralov and Sanjeev Khanna

Perfect matchings in regular bipartite graphs October 20, 2019 28 / 28