Sequential and parallel algorithms for bipartite matching

Outline Introduction Bipartite Matching Parallel Matching Algorithms 2D Partitioning Parallel Algorithm Weighted Bipartit 

Sequential and parallel algorithms for bipartite 

matching 

Johannes Langguth 

April 1, 2011 

Johannes Langguth Sequential and parallel algorithms for bipartite matching


1 Introduction 

2 Bipartite Matching 

3 Parallel Matching Algorithms 

4 2D Partitioning 

5 Parallel Algorithm 

6 Weighted Bipartite Matching 



Bipartite Matching 

Bipartite Cardinality Matching 

Given an undirected, bipartite graph G = (V1,V2,E),E ⊆ V1 × V2 

find a maximum subset M ⊆ E of pairwise nonadjacent edges. 







Perfect Matching 

M is called a perfect matching iff |V1| = |V2| = |M|. 








M is called a perfect matching iff |V1| = |V2| = |M|. 

Old problem 

A classical topic in combinatorial optimization 



Bipartite Matching Algorithms 

Exact Cardinality Algorithms 

Close relation to Max - Flow 

Search for augmenting paths 



Augmenting Paths 

a 

Matching size: 2 

b 



Augmenting Paths 

a 

Matching size: 2 Matching size: 3 

b b 

Johannes Langguth Sequential and parallel algorithms for bipartite matching 

a






Berge’s theorem 

Let G = (V ,E) be a graph and M a matching in G. Then M is of 

maximum cardinality if and only if there is no M-augmenting path 

in G. 







Berge’s theorem 

Let G = (V ,E) be a graph and M a matching in G. Then M is of 

maximum cardinality if and only if there is no M-augmenting path 

in G. 

Obvious Algorithm 

Repeatedly find single augmenting paths using BFS or DFS 

Running time: O(mn) 



Questions 

Questions ? 



Residual Graph 

a 

Orient edges 

b 



Improved Bipartite Matching Algorithms 

Problem 

Residual graph changes after every augmentation. 



Improved Bipartite Matching Algorithms 

Problem 

Residual graph changes after every augmentation. 

Solution 

Find (maximal) sets of disjoint augmenting paths. 



Augmenting Path Based Algorithms 

Hopcroft-Karp Algorithm ’73 

Maximal set of shortest disjoint augmenting paths. 

Uses BFS + DFS 

Running time: O(m √ n) 



Augmenting Path Based Algorithms 

Hopcroft-Karp Algorithm ’73 

Maximal set of shortest disjoint augmenting paths. 

Uses BFS + DFS 


Pothen-Fan Algorithm ’90 

Maximal set of disjoint augmenting paths. 

Uses DFS only 


Fairness makes DFS more stable 

Approx. 3 times faster 



Push-Relabel Based Algorithms 

Goldberg-Tarjan Algorithm ’88 

No explicit augmenting paths 

Speculative path prefix augmentations - “Push” 

Distance Labeling - “Relabel” 


Variable push order 



s,t Network Flow Problem 

1 

1 

1 

1 

1 

0 

0 

0 

0 

0 

Distance Labeling 



Computing Maximum Matching Flow 

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1 

1 

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1 

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Double Push 

1 

1 

1 

3 

1 

2 

2 

2 

0 

0 



Single Push 

1 

1 

1 

3 

1 

2 

2 

2 

2 

0 



Global Relabel 

1 

1 

S 1 

2 

T 

3 

1 

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1 

1 

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3 

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1 2 

0 




3 

1 2 

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Final Pushes 

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Final Matching 

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2 




Global relabelings 

Use BFS from unmatched vertices 


One global relabeling per O(n) 

Actual frequency heavily affects performance 

Push order: 




Global relabelings 

Use BFS from unmatched vertices 


One global relabeling per O(n) 

Actual frequency heavily affects performance 

Push order: 

FIFO (BFS) : Competitive with Pothen-Fan algorithm 

LIFO (DFS) : 20 times slower than FIFO 

Lowest Label First : 20% slower than FIFO 

Highest Label First : more than 20 times slower than FIFO 



Experiments 

Erdős - Rényi Graphs 

Constant average degree 1 ≤ k ≤ 10 

k ≤ 2 trivial 



Experiments 




k = 3 low number of perfect(maximum) matchings 

Hard for augmenting path algorithms 

k > 3 number of perfect(maximum) matchings increases 

Progressively easier 



Random Graphs 



Experiments 




k = 3 low number of perfect(maximum) matchings 

Hard for augmenting path algorithms 

k > 3 number of perfect(maximum) matchings increases 

Progressively easier 

k = 3 is easy for strong heuristics 

Karp-Sipser is often optimal (Aronson, Frieze, Pittel ’98) 



Experiments 

Real world instances 

Sparse matrices 

Varying degree 

“Easy” instances 

Pothen-Fan algorithm is 2 times faster than PR-FIFO here 

Strong initialization does not pay off here 



Experiments 

Real world instances 

Sparse matrices 

Varying degree 

“Easy” instances 

Pothen-Fan algorithm is 2 times faster than PR-FIFO here 

Strong initialization does not pay off here 

Tradeoff 

Push-Relabel algorithm faces a tradeoff between initialization and 

global relabelings 



Parallel Algorithm 

Push - Relabel approach 

Pushes can be performed in any order (Goldberg-Tarjan ’88) 






Basic Assumptions 

Target architechture: traditional supercomputer 

Graph (sparse matrix) is 2D partitioned and distributed 

Graph usually has a perfect matching 

Instances can be huge 






Basic Assumptions 

Target architechture: traditional supercomputer 

Graph (sparse matrix) is 2D partitioned and distributed 

Graph usually has a perfect matching 

Instances can be huge 

Our Objective 

Distributed memory algorithm for large parallel systems 



2D Partitioning 

⎛ 

⎜ 

⎝ 

0 1 0 2 5 

1 1 3 3 0 

0 2 6 6 5 

3 0 1 0 3 

0 1 2 0 5 

⎞ 

⎟ 

⎠ 




⎛ 

⎜ 

⎝ 

0 1 0 2 5 

1 1 3 3 0 

0 2 6 6 5 

3 0 1 0 3 

0 1 2 0 5 

⎞ 

⎟ 

⎠ 

a 

b 

c 

d 

e 

Johannes Langguth Sequential and parallel algorithms for bipartite matching 

u 

v 

w 

x 

y


2D Partitioning is NP-Hard 

CENSORED 



Processor 0 

a 

b 

u 

v 



Processor 1 

a 

b 

c w 

x 

y 



Processor 2 

c 

d 

e 

u 

v 



Processor 3 

d 

e 

w 

x 

y 




a 

b 

c w 

d 

e 

u 

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y 




a 

b 

c w 

d 

e 

u 

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y 



Connectors 

c 

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e 

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Connector Tokens 

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a 

b 

∗ 

d 

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y 



Connector Tokens 

c 

a 

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∗ 

d 

e 

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y 



Token Push 

c 

a 

b 

∗ 

d 

e 

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x 

y 



A New Parallel Algorithm 

Ingredients for a new Algorithm 

1 Maintain locally maximum matching 

2 Find local paths between connectors and free left vertices 

3 Push free left vertices 

4 Use labels to guide pushes 

5 Follow at most m/p edges per round to ensure load balancing 

6 Apply continous global relabelings to update labels 

Challenge 

Distributed memory global relabel is inefficient! 



Path on Processor 2 

4 

2 

0 

1 

3 



Path on Processor 1 and 3 

2 

0 

1 



Solution 

Relaxed Global Relabeling 

1 Perform global relabeling BFS on each processor 

2 Search “jumps” one processor per round 



Global Relabeling Round 1 

2 

0 

1 

1 

1 



Global Relabeling Round 2 

2 

2 

2 1 

2 

0 

1 

1 



Solution 




3 Labels are no longer lower bounds on graph distance 

4 Labels are lower bounds on processor jump distance 

5 Maximum number of jumps J is bounded only by n 

6 Tedious to prove running time 



Solution 




3 Labels are no longer lower bounds on graph distance 

4 Labels are lower bounds on processor jump distance 

5 Maximum number of jumps J is bounded only by n 

6 Tedious to prove running time 

Running time 

O(n(J + r)m) 



Weighted Matching Algorithms 

Exact Weighted Bipartite Algorithms 

Special case of Minimum Cost Flow 

Must find weight augmenting paths 

Can be solved using Shortest Path Algorithms 

Running time: O(n 3 ) 









Approximate Weighted Bipartite Algorithms 

Parallel 1/2 approximation: Bisseling Manne ’07 











2/3 − ǫ approximation: Pettie Sanders ’04 

Running time: O(m) 

Easy on perfect matchings 











2/3 − ǫ approximation: Pettie Sanders ’04 

Running time: O(m) 

Easy on perfect matchings 

Auction algorithms: Bertsekas ’91 



Auction Algorithms 

Unmatched left vertices make bids 

v 

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5 

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4 

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2 

0 



V makes a bid 

Unmatched left vertices make bids 

v 

u 

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6 

7 

4 

4 

2 

0 

Profit = 1 

Profit = 4 



V makes a bid 

v 

u 

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7 

4 

4 

0 

2 ∗ (1 + ǫ) 



U makes a bid 

v 

u 

5 

6 

7 

4 

4 

0 

Profit = 7 − 2 ∗ (1 + ǫ) 

Profit = 4 



U makes a bid 

v 

u 

5 

6 

7 

4 

4 

0 

2 ∗ (1 + ǫ) 2 



End Result 

v 

u 

5 

6 

7 

4 

4 

3 

0 



Auction Algorithm 

Auction algorithm 

“Weighted push-relabel” 

PTAS for maximum weight matching 

Can find maximum cardinality 

Global price updates help: Goldberg Kennedy ’96 









Easy to parallelize: Riedy ’04; Schenk Saate ’10 

Works well on dense instances 

For sparse instances: Global price updates help 









Easy to parallelize: Riedy ’04; Schenk Saate ’10 

Works well on dense instances 

For sparse instances: Global price updates help 

Challenge 

Can we find relaxed parallel global price updates ? 



Questions 

Questions ?

Sequential and parallel algorithms for bipartite matching

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