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IntroductionMatching
Our Contribution
Space Efficient Approximation Scheme forMaximum Matching in Sparse Graphs
Samir Datta Raghav Kulkarni Anish Mukherjee
Chennai Mathematical Institute
NMI Workshop on Complexity Theory, IIT Gandhinagar
November 04, 2016
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Overview
1 Introduction
2 Matching
3 Our Contribution
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Theorem (Baker ’83, Informal)
A class of problems (many of which are NP-Hard in general) canbe approximated arbitrarily close to the optimal value in linear timefor planar graphs.
Example
Includes problems like
maximum independent set
partition into triangles
minimum vertex-cover
minimum dominating set
... and any MSO definable properties
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Theorem (Baker ’83, Informal)
A class of problems (many of which are NP-Hard in general) canbe approximated arbitrarily close to the optimal value in linear timefor planar graphs.
Example
Includes problems like
maximum independent set
partition into triangles
minimum vertex-cover
minimum dominating set
... and any MSO definable properties
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Theorem (Baker ’83, Informal)
A class of problems (many of which are NP-Hard in general) canbe approximated arbitrarily close to the optimal value in linear timefor planar graphs.
Example
Includes problems like
maximum independent set
partition into triangles
minimum vertex-cover
minimum dominating set
... and any MSO definable properties
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Resulting components have treewidth 3k − 1 [Boadlander]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Resulting components have treewidth 3k − 1 [Boadlander]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Resulting components have treewidth 3k − 1 [Boadlander]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Resulting components have treewidth 3k − 1 [Boadlander]
Solve the problem optimally in each partition in linear time
Union of solutions in all components is within (1− 1/k) OPT
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm
Basic Idea
Partition the vertices into breadth-first search levels
Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k
Resulting components have treewidth 3k − 1 [Boadlander]
Solve the problem optimally in each partition in linear time
Union of solutions in all components is within (1− 1/k) OPT
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ? Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ? Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ? Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ? Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ?
Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Baker’s Algorithm II
But here we are interested in space efficient algorithms,namely algorithms running in Logspace
EJT gives an algorithm for Courcelle’s theorem in Logspace
But for the first part we need to compute distance
Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.
And these classes are not believed to be in Logspace.
Question
Can we get away without using distance ? Not yet
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Overview
1 Introduction
2 Matching
3 Our Contribution
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Matching
A matching M ⊆ E is a set of independent edges
A matching M is called perfect if M covers all vertices of G
M of maximum size is called maximum matching
Augmenting Paths
In an alternating path the edges alternate between M andE \MAn alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Matching
A matching M ⊆ E is a set of independent edges
A matching M is called perfect if M covers all vertices of G
M of maximum size is called maximum matching
Augmenting Paths
In an alternating path the edges alternate between M andE \MAn alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Matching
A matching M ⊆ E is a set of independent edges
A matching M is called perfect if M covers all vertices of G
M of maximum size is called maximum matching
Augmenting Paths
In an alternating path the edges alternate between M andE \M
An alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Matching
A matching M ⊆ E is a set of independent edges
A matching M is called perfect if M covers all vertices of G
M of maximum size is called maximum matching
Augmenting Paths
In an alternating path the edges alternate between M andE \MAn alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm
Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes
The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]
Bipartite Perfect Matching is in quasi-NC [FGT16]
The best hardness known is NL-hardness [CSV84]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm
Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes
The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]
Bipartite Perfect Matching is in quasi-NC [FGT16]
The best hardness known is NL-hardness [CSV84]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm
Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes
The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]
Bipartite Perfect Matching is in quasi-NC [FGT16]
The best hardness known is NL-hardness [CSV84]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm
Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes
The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]
Bipartite Perfect Matching is in quasi-NC [FGT16]
The best hardness known is NL-hardness [CSV84]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching
Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm
Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes
The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]
Bipartite Perfect Matching is in quasi-NC [FGT16]
The best hardness known is NL-hardness [CSV84]
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching in Planar Graphs
Counting perfect matchings in planar graphs is in NC [Vaz88]
Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]
Open Problem
Is the construction version in general planar graphs in NC ?
Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]
Only L-hardness is known for planar graphs [DKLM10].
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching in Planar Graphs
Counting perfect matchings in planar graphs is in NC [Vaz88]
Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]
Open Problem
Is the construction version in general planar graphs in NC ?
Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]
Only L-hardness is known for planar graphs [DKLM10].
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching in Planar Graphs
Counting perfect matchings in planar graphs is in NC [Vaz88]
Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]
Open Problem
Is the construction version in general planar graphs in NC ?
Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]
Only L-hardness is known for planar graphs [DKLM10].
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching in Planar Graphs
Counting perfect matchings in planar graphs is in NC [Vaz88]
Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]
Open Problem
Is the construction version in general planar graphs in NC ?
Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]
Only L-hardness is known for planar graphs [DKLM10].
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Matching in Planar Graphs
Counting perfect matchings in planar graphs is in NC [Vaz88]
Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]
Open Problem
Is the construction version in general planar graphs in NC ?
Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]
Only L-hardness is known for planar graphs [DKLM10].
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Time-Space Tradeoff
Removing non-determinism even for planar reachability leadsto either a quasi-polynomial time blow-up or need large space(O(√n)) [INPVW13, AKNW14]
For general graphs it is even worse, with O(n/2√logn) space
and polynomial time [BBRS]
Previous Results
Approximating maximum matching has been considered bothin time and parallel complexity model
Linear-time [DP14] and NC [HV06] approximation scheme arethe best known complexity bounds here
But work on space efficient approximation seems limited.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Time-Space Tradeoff
Removing non-determinism even for planar reachability leadsto either a quasi-polynomial time blow-up or need large space(O(√n)) [INPVW13, AKNW14]
For general graphs it is even worse, with O(n/2√logn) space
and polynomial time [BBRS]
Previous Results
Approximating maximum matching has been considered bothin time and parallel complexity model
Linear-time [DP14] and NC [HV06] approximation scheme arethe best known complexity bounds here
But work on space efficient approximation seems limited.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Time-Space Tradeoff
Removing non-determinism even for planar reachability leadsto either a quasi-polynomial time blow-up or need large space(O(√n)) [INPVW13, AKNW14]
For general graphs it is even worse, with O(n/2√logn) space
and polynomial time [BBRS]
Previous Results
Approximating maximum matching has been considered bothin time and parallel complexity model
Linear-time [DP14] and NC [HV06] approximation scheme arethe best known complexity bounds here
But work on space efficient approximation seems limited.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Time-Space Tradeoff
Removing non-determinism even for planar reachability leadsto either a quasi-polynomial time blow-up or need large space(O(√n)) [INPVW13, AKNW14]
For general graphs it is even worse, with O(n/2√logn) space
and polynomial time [BBRS]
Previous Results
Approximating maximum matching has been considered bothin time and parallel complexity model
Linear-time [DP14] and NC [HV06] approximation scheme arethe best known complexity bounds here
But work on space efficient approximation seems limited.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Overview
1 Introduction
2 Matching
3 Our Contribution
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.
This result extends to many other sparse graph classes
Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs
But we consider graphs which are not necessarily bipartite
Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity
Solve by reducing it to bounded degree graphs suitably.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.
This result extends to many other sparse graph classes
Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs
But we consider graphs which are not necessarily bipartite
Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity
Solve by reducing it to bounded degree graphs suitably.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.
This result extends to many other sparse graph classes
Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs
But we consider graphs which are not necessarily bipartite
Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity
Solve by reducing it to bounded degree graphs suitably.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.
This result extends to many other sparse graph classes
Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs
But we consider graphs which are not necessarily bipartite
Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity
Solve by reducing it to bounded degree graphs suitably.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.
This result extends to many other sparse graph classes
Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs
But we consider graphs which are not necessarily bipartite
Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity
Solve by reducing it to bounded degree graphs suitably.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Let G be a graph with degrees bounded by a constant d then forany fixed ε > 0, we can find a (1− ε) factor approximation to themaximum matching in Logspace.
The main fact we use here is that any bounded degree graphsalways contains a linear size matching
Many planar graph classes, such as 3-connected planargraphs, are known to be containing a large matching
In fact our algorithm works for any recursively sparse graphcontaining a large matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Let G be a graph with degrees bounded by a constant d then forany fixed ε > 0, we can find a (1− ε) factor approximation to themaximum matching in Logspace.
The main fact we use here is that any bounded degree graphsalways contains a linear size matching
Many planar graph classes, such as 3-connected planargraphs, are known to be containing a large matching
In fact our algorithm works for any recursively sparse graphcontaining a large matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Let G be a graph with degrees bounded by a constant d then forany fixed ε > 0, we can find a (1− ε) factor approximation to themaximum matching in Logspace.
The main fact we use here is that any bounded degree graphsalways contains a linear size matching
Many planar graph classes, such as 3-connected planargraphs, are known to be containing a large matching
In fact our algorithm works for any recursively sparse graphcontaining a large matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Results
Theorem
Let G be a graph with degrees bounded by a constant d then forany fixed ε > 0, we can find a (1− ε) factor approximation to themaximum matching in Logspace.
The main fact we use here is that any bounded degree graphsalways contains a linear size matching
Many planar graph classes, such as 3-connected planargraphs, are known to be containing a large matching
In fact our algorithm works for any recursively sparse graphcontaining a large matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
A Brief Idea
1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths
2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace
3 To convert a planar graph to a bounded degree graph wedelete high degree vertices
4 The number of such vertices is small though possibly stilllinear in the graph size
5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
A Brief Idea
1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths
2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace
3 To convert a planar graph to a bounded degree graph wedelete high degree vertices
4 The number of such vertices is small though possibly stilllinear in the graph size
5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
A Brief Idea
1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths
2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace
3 To convert a planar graph to a bounded degree graph wedelete high degree vertices
4 The number of such vertices is small though possibly stilllinear in the graph size
5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
A Brief Idea
1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths
2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace
3 To convert a planar graph to a bounded degree graph wedelete high degree vertices
4 The number of such vertices is small though possibly stilllinear in the graph size
5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
A Brief Idea
1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths
2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace
3 To convert a planar graph to a bounded degree graph wedelete high degree vertices
4 The number of such vertices is small though possibly stilllinear in the graph size
5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Bounded degree graphs I
We deal with augmenting paths of length at most 2k + 1
Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers
If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths
Lemma
If |M | < (1− 3k )|Mopt| for some k then there are at least
3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.
Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Bounded degree graphs I
We deal with augmenting paths of length at most 2k + 1
Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers
If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths
Lemma
If |M | < (1− 3k )|Mopt| for some k then there are at least
3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.
Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Bounded degree graphs I
We deal with augmenting paths of length at most 2k + 1
Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers
If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths
Lemma
If |M | < (1− 3k )|Mopt| for some k then there are at least
3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.
Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Bounded degree graphs I
We deal with augmenting paths of length at most 2k + 1
Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers
If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths
Lemma
If |M | < (1− 3k )|Mopt| for some k then there are at least
3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.
Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Bounded degree graphs I
We deal with augmenting paths of length at most 2k + 1
Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers
If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths
Lemma
If |M | < (1− 3k )|Mopt| for some k then there are at least
3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.
Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Maximum matching in bounded degree graphs II
Lemma
A β-factor approximation to the maximum independent set can becomputed in Logspace
Colour the paths and the largest colour class works
As the degree is bounded by some D, find at most D disjointforests that partition the edge set
Can be done using Reingold’s algorithm for connectivity
Colour each forest with 2 colours and it gives D bit colours toevery node
This yields a 2D i.e. constant colouring of the graph.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Theorem
In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.
Previous lemma yields large fraction of short paths,augmentable in parallel
A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers
At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Theorem
In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.
Previous lemma yields large fraction of short paths,augmentable in parallel
A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers
At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Theorem
In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.
Previous lemma yields large fraction of short paths,augmentable in parallel
A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers
At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Theorem
In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.
Previous lemma yields large fraction of short paths,augmentable in parallel
A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers
At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)
After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Theorem
In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.
Previous lemma yields large fraction of short paths,augmentable in parallel
A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers
At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Algorithm 1
1 Fix integer k =⌈3ε
⌉.
2 Construct the intersection graph of augmenting paths oflength at most 2k + 1 in G.
3 Let the graph be H with maximum degree≤ D = (2k + 1)2d2k+1
4 Find at most D disjoint forests that partition the edge set.
5 Colour each forest with 2 colours, giving D bit colours toevery node
6 Augment the vertex disjoint augmenting paths in parallel
7 Add the new matching to M
8 Return M
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching
Definition
A graph is tame if all pairs of vertices (a, b) which are endpoints ofa even length isolated path, support at most two of them.
This can be ensured by deleting a set of edges E′ from G
Lemma
The size of the maximum matching in G \ E′ is the same as in G.
Main Lemma
A tame planar graph has a linear sized maximum matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching
Definition
A graph is tame if all pairs of vertices (a, b) which are endpoints ofa even length isolated path, support at most two of them.
This can be ensured by deleting a set of edges E′ from G
Lemma
The size of the maximum matching in G \ E′ is the same as in G.
Main Lemma
A tame planar graph has a linear sized maximum matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching
Definition
A graph is tame if all pairs of vertices (a, b) which are endpoints ofa even length isolated path, support at most two of them.
This can be ensured by deleting a set of edges E′ from G
Lemma
The size of the maximum matching in G \ E′ is the same as in G.
Main Lemma
A tame planar graph has a linear sized maximum matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching: tame graphs
One of the following is true :
Total length of long isolated paths in G′ is large enough
We can transform the graph by case analysis to a minimumdegree 3 planar graph
Lemma
A graph in which the total length of isolated paths is N has amatching of size at least N/4.
Lemma
A min degree 3 planar graph has a matching of size at least n/140.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching: tame graphs
One of the following is true :
Total length of long isolated paths in G′ is large enough
We can transform the graph by case analysis to a minimumdegree 3 planar graph
Lemma
A graph in which the total length of isolated paths is N has amatching of size at least N/4.
Lemma
A min degree 3 planar graph has a matching of size at least n/140.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching: tame graphs
One of the following is true :
Total length of long isolated paths in G′ is large enough
We can transform the graph by case analysis to a minimumdegree 3 planar graph
Lemma
A graph in which the total length of isolated paths is N has amatching of size at least N/4.
Lemma
A min degree 3 planar graph has a matching of size at least n/140.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching: tame graphs
One of the following is true :
Total length of long isolated paths in G′ is large enough
We can transform the graph by case analysis to a minimumdegree 3 planar graph
Lemma
A graph in which the total length of isolated paths is N has amatching of size at least N/4.
Lemma
A min degree 3 planar graph has a matching of size at least n/140.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching: tame graphs
One of the following is true :
Total length of long isolated paths in G′ is large enough
We can transform the graph by case analysis to a minimumdegree 3 planar graph
Lemma
A graph in which the total length of isolated paths is N has amatching of size at least N/4.
Lemma
A min degree 3 planar graph has a matching of size at least n/140.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching III
Theorem
There is a LSAS for maximum matching in planar graphs.
proof
Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′
Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges
So we have a (α− 6/d)n sized matching remaining
Taking d = 122α−ε reduces the problem to find a (1− ε/2)
factor approximation algorithm for bounded degree graphs.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching III
Theorem
There is a LSAS for maximum matching in planar graphs.
proof
Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′
Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges
So we have a (α− 6/d)n sized matching remaining
Taking d = 122α−ε reduces the problem to find a (1− ε/2)
factor approximation algorithm for bounded degree graphs.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching III
Theorem
There is a LSAS for maximum matching in planar graphs.
proof
Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′
Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges
So we have a (α− 6/d)n sized matching remaining
Taking d = 122α−ε reduces the problem to find a (1− ε/2)
factor approximation algorithm for bounded degree graphs.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching III
Theorem
There is a LSAS for maximum matching in planar graphs.
proof
Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′
Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges
So we have a (α− 6/d)n sized matching remaining
Taking d = 122α−ε reduces the problem to find a (1− ε/2)
factor approximation algorithm for bounded degree graphs.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Planar maximum matching III
Theorem
There is a LSAS for maximum matching in planar graphs.
proof
Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′
Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges
So we have a (α− 6/d)n sized matching remaining
Taking d = 122α−ε reduces the problem to find a (1− ε/2)
factor approximation algorithm for bounded degree graphs.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Conclusion
We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs
For planar graphs we require only the following properties:
Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3
So can be extended many other classes of sparse graphs
bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Conclusion
We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs
For planar graphs we require only the following properties:
Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3
So can be extended many other classes of sparse graphs
bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Conclusion
We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs
For planar graphs we require only the following properties:
Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3
So can be extended many other classes of sparse graphs
bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Conclusion
We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs
For planar graphs we require only the following properties:
Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3
So can be extended many other classes of sparse graphs
bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Conclusion
We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs
For planar graphs we require only the following properties:
Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3
So can be extended many other classes of sparse graphs
bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Open Problems
Baker’s Theorem in Logspace ?
Devise an LSAS for maximum matching in general graphs
or at least in arbitrary sparse graphs
Lower bounds in the context of approximation ?
Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Open Problems
Baker’s Theorem in Logspace ?
Devise an LSAS for maximum matching in general graphs
or at least in arbitrary sparse graphs
Lower bounds in the context of approximation ?
Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Open Problems
Baker’s Theorem in Logspace ?
Devise an LSAS for maximum matching in general graphs
or at least in arbitrary sparse graphs
Lower bounds in the context of approximation ?
Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Open Problems
Baker’s Theorem in Logspace ?
Devise an LSAS for maximum matching in general graphs
or at least in arbitrary sparse graphs
Lower bounds in the context of approximation ?
Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Open Problems
Baker’s Theorem in Logspace ?
Devise an LSAS for maximum matching in general graphs
or at least in arbitrary sparse graphs
Lower bounds in the context of approximation ?
Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Thank You
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph H
Maximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
IntroductionMatching
Our Contribution
Packing complex patterns ?
H-Matching
Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.
Approximation and hardness is known for some restricted cases
We give LSAS for graphs with a small balanced separator,
for packing any fixed graph H when degrees are bounded
Otherwise, Packing some special class of patterns
As before, the idea is to delete high degree vertices
and tame the graph by removing some forbidden patterns
Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs
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