Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Heuristic initialization of bipartite matching
algorithms
Johannes Langguth
University of Bergen
joint work with Fredrik Manne and Peter Sanders
February 4, 2009
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
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.
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
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|.
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
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|.
Old problem
A classical topic in combinatorial optimization
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Matching in CSC
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Matching in CSC
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Matching in CSC
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Matching in CSC
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Bipartite Matching Algorithms
Exact Algorithms
Search for augmenting pathes
Close relation to Max - Flow
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Bipartite Matching Algorithms
Exact Algorithms
Search for augmenting pathes
Close relation to Max - Flow
Single augmenting pathes: BFS, DFS
Multiple augmenting pathes: Hopcroft-Karp ’73, Alt et al.’91
Push-relabel Algorithms: Goldberg-Tarjan ’88
Running time: O(m √ n)
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Bipartite Matching Algorithms
Exact Algorithms
Search for augmenting pathes
Close relation to Max - Flow
Single augmenting pathes: BFS, DFS
Multiple augmenting pathes: Hopcroft-Karp ’73, Alt et al.’91
Push-relabel Algorithms: Goldberg-Tarjan ’88
Running time: O(m √ n)
Heuristic Initialization
Number of augmenting pathes can be reduced by starting on an
approximate matching
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Matching Heuristics
O(m) running times
Constant approximation ratios: always 1
2 or better
Good heuristic results: often 95
100 or better
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Matching Heuristics
O(m) running times
Constant approximation ratios: always 1
2 or better
Good heuristic results: often 95
100 or better
Simple Greedy
Loop through all unmatched vertices once
Match each vertex with a random (unmatched) vertex
Remove matched vertices from graph
Alternative description:
Repeatedly pick a random edge and match its vertices
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Minimum Degree Heuristic
Mindegree
Repeatedly pick a vertex of minimum degree and match it ...
One sided: with a random neighbour
Two sided: with a neighbour of minimum degree
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Minimum Degree Heuristic
Mindegree
Repeatedly pick a vertex of minimum degree and match it ...
One sided: with a random neighbour
Two sided: with a neighbour of minimum degree
Vertex removal
Remove matched vertices from graph
Static: count degrees once
Dynamic: decrease degrees after removal
Dynamic mindegree requires priority queue
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Example
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Static one-sided mindegree
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Introduction Algorithms New Heuristic Results Outlook
Static one-sided mindegree
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Introduction Algorithms New Heuristic Results Outlook
Static one-sided mindegree
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Introduction Algorithms New Heuristic Results Outlook
Static one-sided mindegree
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Dynamic two-sided mindegree
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Introduction Algorithms New Heuristic Results Outlook
Dynamic two-sided mindegree
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Dynamic two-sided mindegree
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Dynamic two-sided mindegree
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Introduction Algorithms New Heuristic Results Outlook
Karp-Sipser Heuristic
Theorem
Degree 1 Reduction [Karp and Sipser 1981]
Given a graph G = (V,E) and vertices u, w ∈ V with deg(u) = 1
and Γ(u) = {w}, there is always a maximum matching M on G
with {u, w} ∈ M.
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Karp-Sipser Heuristic
Theorem
Degree 1 Reduction [Karp and Sipser 1981]
Given a graph G = (V,E) and vertices u, w ∈ V with deg(u) = 1
and Γ(u) = {w}, there is always a maximum matching M on G
with {u, w} ∈ M.
Karp-Sipser Heuristic
Uses degree 1 reductions only
Equivalent to dynamic mindegree for degree 1 vertices
Equivalent to simple greedy for other vertices
Faster due to shorter priority queue
Offers a good tradeoff (Möhring, Müller-Hannemann ’98)
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Degree 2 Reductions
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Degree 2 Reductions
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Degree 2 Reductions
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Degree 2 Reductions
Theorem
Degree 2 Reduction [Karp and Sipser 1981]
Given a graph G = (V,E) and vertices u, v, w ∈ V with
deg(u) = 2 and Γ(u) = {v, w}, there is always a maximum
matching M on G with {u, v} ∈ M or {u, w} ∈ M.
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Motivation
Observation
Degree 2 reductions yield high solution quality (Magun ’98)
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Motivation
Observation
Degree 2 reductions yield high solution quality (Magun ’98)
Problem
Require flexible, slow data structures
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Motivation
Observation
Degree 2 reductions yield high solution quality (Magun ’98)
Problem
Require flexible, slow data structures
Question
Can we use these reductions with simple and fast data structures ?
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Motivation
Observation
Degree 2 reductions yield high solution quality (Magun ’98)
Problem
Require flexible, slow data structures
Question
Can we use these reductions with simple and fast data structures ?
Solution
Alternating components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Cyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Cyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Cyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Cyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Acyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Acyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Acyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Acyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Acyclic Alternating Components
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Algorithm
Component Based Dynamic Mindegree Algorithm
Pick a vertex v of minimum degree. If it is of...
Degree 1: Perform degree 1 reduction
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Algorithm
Component Based Dynamic Mindegree Algorithm
Pick a vertex v of minimum degree. If it is of...
Degree 1: Perform degree 1 reduction
Degree 2: Find alternating component from v
Match component if it is cyclic
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Algorithm
Component Based Dynamic Mindegree Algorithm
Pick a vertex v of minimum degree. If it is of...
Degree 1: Perform degree 1 reduction
Degree 2: Find alternating component from v
Match component if it is cyclic
Degree 3+: Use dynamic double sided mindegree
Degree 3+: If v belongs to an acyclic component
pick its vertex of maximum external degree and
match it to an external vertex
Repeat until no further vertices can be matched
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Depart from Degree 2 Reduction
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Component Based Dynamic Mindegree Algorithm
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Johannes Langguth Heuristic initialization of bipartite matching algorithms
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Introduction Algorithms New Heuristic Results Outlook
Greedy Enhancement
Observation
Solution quality of Simple Greedy has low correlation with the
solution quality of the Component-based approach
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Greedy Enhancement
Observation
Solution quality of Simple Greedy has low correlation with the
solution quality of the Component-based approach
Idea
If Simple Greedy solution is good then keep it
Otherwise run Component based heuristic
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Greedy Enhancement
Observation
Solution quality of Simple Greedy has low correlation with the
solution quality of the Component-based approach
Idea
If Simple Greedy solution is good then keep it
Otherwise run Component based heuristic
Question
How many vertices are matched in a good solution?
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Greedy Enhancement
Observation
Solution quality of Simple Greedy has low correlation with the
solution quality of the Component-based approach
Idea
If Simple Greedy solution is good then keep it
Otherwise run Component based heuristic
Question
How many vertices are matched in a good solution?
Answer
Threshold of 99.5% matched vertices works well,
up to 99.8% is reasonable
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Heuristics
Simple Greedy
Static mindegree
Karp-Sipser (1s/2s)
Dynamic mindegree (1s/2s)
Component based dynamic mindegree
Greedy enhanced component based dynamic mindegree
(99.5%)
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Heuristics
Simple Greedy
Static mindegree
Karp-Sipser (1s/2s)
Dynamic mindegree (1s/2s)
Component based dynamic mindegree
Greedy enhanced component based dynamic mindegree
(99.5%)
900 matrices
Average size: 67000 Vertices / 365000 Edges
Matrices from the Florida Matrix Collection by Tim Davis
26 generated matrices: HiLo and Rbg Generators
(Setubal ’96, Goldberg ’98)
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Average Runtime (sec.) / unmatched Ratio
0.06
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greedy static
Runtime
Errors
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Average Runtime (sec.) / unmatched Ratio
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greedy static MD 1s MD 2s
Runtime
Errors
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Average Runtime (sec.) / unmatched Ratio
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greedy static MD 1s MD 2s KS 1s KS 2s
Runtime
Errors
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Experiments
Average Runtime (sec.) / unmatched Ratio
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Runtime
Errors
greedy static MD 1s MD 2s KS 1s KS 2s Cbased Gcomp
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Combined Algorithms
Exact Algorithms
Alt, Blum, Paul, Mehlhorn (ABMP) and
Push-Relabel by João C. Setubal
BFS and
Hopcroft-Karp (HK) by Florin Dobrian
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Combined Algorithms
Exact Algorithms
Alt, Blum, Paul, Mehlhorn (ABMP) and
Push-Relabel by João C. Setubal
BFS and
Hopcroft-Karp (HK) by Florin Dobrian
72 matrices
46 matrices from Florida Matrix Collection
26 generated matrices
Average size: 302000 Vertices / 1950000 Edges
Maximum size: 1280000 Vertices / 12738525 Edges
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Results
Average running time in seconds
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greedy static
Push-relabel
ABMP
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Results
Average running time in seconds
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greedy static MD 1s MD 2s
Push-relabel
ABMP
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Results
Average running time in seconds
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2.0
1.5
1.0
0.5
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greedy static MD 1s MD 2s KS 1s KS 2s
Push-relabel
ABMP
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Results
Average running time in seconds
2.5
2.0
1.5
1.0
0.5
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Push-relabel
ABMP
greedy static MD 1s MD 2s KS 1s KS 2s Cbased Gcomp
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Average running in seconds
Other algorithms
2.0
1.5
1.0
0.5
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KS 1s BFS Cbased BFS Gcomp BFS
Heuristic
Exact
Sum
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Average running in seconds
Other algorithms
2.0
1.5
1.0
0.5
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Heuristic
Exact
Sum
KS 1s BFS Cbased BFS Gcomp BFS KS 1s HK Cbased HK Gcomp HK
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Conclusions
Static heuristics do not work well
Dynamic heuristics show simillar quality
Two sided mindegree is superior to one-sided mindegree
One sided Karp-Sipser is good and fast
Component - based heuristic is slower but better
Greedy enhancement pays off
Heuristic solution quality strongly affects exact algorthm time
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Conclusions
Static heuristics do not work well
Dynamic heuristics show simillar quality
Two sided mindegree is superior to one-sided mindegree
One sided Karp-Sipser is good and fast
Component - based heuristic is slower but better
Greedy enhancement pays off
Heuristic solution quality strongly affects exact algorthm time
No free lunch
Reductions have a strong effect on solution quality
Mindegree offers little benefit
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Future Work
Find the best exact algorithm
Parallel impelemtation
Johannes Langguth Heuristic initialization of bipartite matching algorithms

Introduction Algorithms New Heuristic Results Outlook
Future Work
Find the best exact algorithm
Parallel impelemtation
Weighted perfect matching
Weight augmenting pathes
Johannes Langguth Heuristic initialization of bipartite matching algorithms