The Graph Theory, Optimization, Algorithms andComplexity Group!

Gregory Gutin, Anders Yeo,Robert Crowston, Mark Jones and Kokul Karunananthan

Department of Computer ScienceRoyal Holloway

University of London

2011

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The people in the group

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Anders Yeo (Reader). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Anders Yeo (Reader). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Robert Crowston (PhD student). Currently working on FixedParameter Tractability.

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Anders Yeo (Reader). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Robert Crowston (PhD student). Currently working on FixedParameter Tractability.

◮ Mark Jones (PhD student). Currently working on Fixed ParameterTractability.

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Anders Yeo (Reader). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Robert Crowston (PhD student). Currently working on FixedParameter Tractability.

◮ Mark Jones (PhD student). Currently working on Fixed ParameterTractability.

◮ Kokul Karunananthan (MSc by Research student). Currentlyworking on discrete optimization heuristics.

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The people in the group

◮ Gregory Gutin (Professor). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Anders Yeo (Reader). Research interests include Graph Theory,Combinatorics, Combinatorial Optimization, Algorithms, Complexity,Fixed Parameter Tractability and Discrete Mathematics in general.

◮ Robert Crowston (PhD student). Currently working on FixedParameter Tractability.

◮ Mark Jones (PhD student). Currently working on Fixed ParameterTractability.

◮ Kokul Karunananthan (MSc by Research student). Currentlyworking on discrete optimization heuristics.

This group has a few joint papers with Adrian Johnstone and ElizabethScott from the compiler group.

We have also been working on some problems in constraint satisfaction,which is one of Dave Cohens main research areas.

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Fixed Parameter Tractability (FPT)

This is currently one of our main research areas.

We will illustrate the main notations using the Vertex Cover (VC)problem as an example.

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Fixed Parameter Tractability (FPT)

This is currently one of our main research areas.

We will illustrate the main notations using the Vertex Cover (VC)problem as an example.

This is an NP-hard problem.

So if your boss asks you to solve a VC problem on a graph with 500nodes and with a vertex cover of size approximately 30, what do you do?

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Fixed Parameter Tractability (FPT)

This is currently one of our main research areas.

We will illustrate the main notations using the Vertex Cover (VC)problem as an example.

This is an NP-hard problem.

So if your boss asks you to solve a VC problem on a graph with 500nodes and with a vertex cover of size approximately 30, what do you do?

Say you cannot do it as it is an NP-hard problem?

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Fixed Parameter Tractability (FPT)

This is currently one of our main research areas.

We will illustrate the main notations using the Vertex Cover (VC)problem as an example.

This is an NP-hard problem.

So if your boss asks you to solve a VC problem on a graph with 500nodes and with a vertex cover of size approximately 30, what do you do?

Say you cannot do it as it is an NP-hard problem?

NO! Then you will be fired!

So what do you do?

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Fixed Parameter Tractability (FPT)

This is currently one of our main research areas.

We will illustrate the main notations using the Vertex Cover (VC)problem as an example.

This is an NP-hard problem.

So if your boss asks you to solve a VC problem on a graph with 500nodes and with a vertex cover of size approximately 30, what do you do?

Say you cannot do it as it is an NP-hard problem?

NO! Then you will be fired!

So what do you do? You use results from FPT.

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FPT terminology

Formal definition: A parameterised problem is FPT if it can be solved intime O(f (k)nc), where k is the parameter, n is the size of the problem, c

is a constant (not depending on n or k) and f (k) is any computablefunction on k .

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FPT terminology

Formal definition: A parameterised problem is FPT if it can be solved intime O(f (k)nc), where k is the parameter, n is the size of the problem, c

is a constant (not depending on n or k) and f (k) is any computablefunction on k .

VC example: Let G be a graph and let k be any non-negative integer.We want to decide if the size of a minimum vertex cover is at most k .

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FPT terminology

Formal definition: A parameterised problem is FPT if it can be solved intime O(f (k)nc), where k is the parameter, n is the size of the problem, c

is a constant (not depending on n or k) and f (k) is any computablefunction on k .

VC example: Let G be a graph and let k be any non-negative integer.We want to decide if the size of a minimum vertex cover is at most k .

This can be done in O(1.2738k + kn) time (which is bounded byO(1.2738k × n)) so it is FPT.

Note that in this problem the parameter, k , is the size of the solution(this is not always the case).

And 1.273830 + 30 × 500 = 16422.45...

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FPT terminology

Formal definition: A parameterised problem is FPT if it can be solved intime O(f (k)nc), where k is the parameter, n is the size of the problem, c

is a constant (not depending on n or k) and f (k) is any computablefunction on k .

VC example: Let G be a graph and let k be any non-negative integer.We want to decide if the size of a minimum vertex cover is at most k .

This can be done in O(1.2738k + kn) time (which is bounded byO(1.2738k × n)) so it is FPT.

Note that in this problem the parameter, k , is the size of the solution(this is not always the case).

And 1.273830 + 30 × 500 = 16422.45...

Another important concept is kernels.....

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Kernels (or preprocessing)

Formal definition of a kernel: A parameterised problem Π has a kernel, iffor every instance, (I , k), of Π we can in polynomial time find anotherinstance, (I ′, k ′), of Π, such that the problems are equivalent (have thesame answer) and both |I ′| and k ′ are bounded by functions in k .

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Kernels (or preprocessing)

Formal definition of a kernel: A parameterised problem Π has a kernel, iffor every instance, (I , k), of Π we can in polynomial time find anotherinstance, (I ′, k ′), of Π, such that the problems are equivalent (have thesame answer) and both |I ′| and k ′ are bounded by functions in k .

VC example: Let (G , k) be an instance of the VC problem. It is knownthat in time O(m

√n) we can find an instance (G ′, k ′) of the VC problem

such that |V (G ′)| ≤ 2k and k ′ ≤ k and the two instances are equivalent.

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Kernels (or preprocessing)

Formal definition of a kernel: A parameterised problem Π has a kernel, iffor every instance, (I , k), of Π we can in polynomial time find anotherinstance, (I ′, k ′), of Π, such that the problems are equivalent (have thesame answer) and both |I ′| and k ′ are bounded by functions in k .

VC example: Let (G , k) be an instance of the VC problem. It is knownthat in time O(m

√n) we can find an instance (G ′, k ′) of the VC problem

such that |V (G ′)| ≤ 2k and k ′ ≤ k and the two instances are equivalent.

In a paper by A.Yeo and A. Soleimanfallah (a previous PhD student ofour group) we improve the above to 2k − c for any constant c .

This is currently the best known size of the order of a kernel.

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Kernels (or preprocessing)

Formal definition of a kernel: A parameterised problem Π has a kernel, iffor every instance, (I , k), of Π we can in polynomial time find anotherinstance, (I ′, k ′), of Π, such that the problems are equivalent (have thesame answer) and both |I ′| and k ′ are bounded by functions in k .

VC example: Let (G , k) be an instance of the VC problem. It is knownthat in time O(m

√n) we can find an instance (G ′, k ′) of the VC problem

such that |V (G ′)| ≤ 2k and k ′ ≤ k and the two instances are equivalent.

In a paper by A.Yeo and A. Soleimanfallah (a previous PhD student ofour group) we improve the above to 2k − c for any constant c .

This is currently the best known size of the order of a kernel.

Theorem: A problem is FPT if and only if it has a kernel.

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

What happens if we use the number of clauses we want to satisfy as theparameter?

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

What happens if we use the number of clauses we want to satisfy as theparameter?

Answer: The problem becomes trivial. Why?

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

What happens if we use the number of clauses we want to satisfy as theparameter?

Answer: The problem becomes trivial. Why?

Consider a random truth assignment.

The average number of clauses satisfied is 78 |C | (C is the set of clauses).

So if k < 78 |C | the answer is YES, otherwise k ≥ 7

8 |C | and we have akernel.

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

What happens if we use the number of clauses we want to satisfy as theparameter?

Answer: The problem becomes trivial. Why?

Consider a random truth assignment.

The average number of clauses satisfied is 78 |C | (C is the set of clauses).

So if k < 78 |C | the answer is YES, otherwise k ≥ 7

8 |C | and we have akernel.

So, is this problem uninteresting from an FPT perspective?

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Another example: Max-3-SAT

3-SAT is one of the best known problems in computer science (if not thebest known!).

Max-3-SAT is the problem of maximising the number of clauses that canbe satisfied.

What happens if we use the number of clauses we want to satisfy as theparameter?

Answer: The problem becomes trivial. Why?

Consider a random truth assignment.

The average number of clauses satisfied is 78 |C | (C is the set of clauses).

So if k < 78 |C | the answer is YES, otherwise k ≥ 7

8 |C | and we have akernel.

So, is this problem uninteresting from an FPT perspective? NO!

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Parameterise above/below tight bounds

We consider the following problem instead.

Problem: Can we satisfy 78 |C | + k clauses in FPT time?

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Parameterise above/below tight bounds

We consider the following problem instead.

Problem: Can we satisfy 78 |C | + k clauses in FPT time?

Answer: YES! Applying a probabilistic argument combined with somesimple tools from Harmonic analysis we show that the problem has akernel (of polynomial size).

This is done in a paper by N. Alon, G. Gutin, E. Kim, S. Szeider and A.Yeo.

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Parameterise above/below tight bounds

We consider the following problem instead.

Problem: Can we satisfy 78 |C | + k clauses in FPT time?

Answer: YES! Applying a probabilistic argument combined with somesimple tools from Harmonic analysis we show that the problem has akernel (of polynomial size).

This is done in a paper by N. Alon, G. Gutin, E. Kim, S. Szeider and A.Yeo.

This area is probably our main research areas at the moment.

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Other results

Other examples of results by our group include

◮ Maximise the number of equations that can be satisfied whenworking in F2 (Crowston, Gutin, Jones et. al.).

◮ Maximise the number of clauses that can be satisfied inr -satisfiable-SAT for r = 2, 3 (Crowston, Gutin, Jones and Yeo).

◮ Permutation Constraint Satisfaction Problem parameterised aboveaverage has a kernel with a quadratic number of variables (Gutin,Yeo et. al.).

◮ Improve the FPT time complexity of problems such as 3-Hitting setin hypergraphs and minimum feedback vertex set in tournaments(Yeo et. al.).

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Other areas of research

Directed graphs: G. Gutin is a co-author of the book Digraphs: Theory,Algorithms and Applications. In 2010 A. Yeo submittedtwo papers in the area (titled Arc-disjoint spanningsub(di)graphs in Digraphs and Vertex Disjoint Cycles ofDifferent Length in Digraphs).

Transversals in Hypergraphs: A.Yeo collaborates with Prof. MichaelHenning from University of Johannesburg on this topic. In2010/2011 they for example submitted the papers2-Colorings in k-Regular k-Uniform Hypergraphs andTransversals and Matchings in Hypergraphs.

Total Domination in Graphs: A.Yeo collaborates with Prof. MichaelHenning on this topic (it is very closely related totransversals in Hypergraphs).

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Total domination in graphs and transversals in hypergraphsA total dominating set, S , in a graph, G , is a set such that every vertexin the graph has a neighbour in S .

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Total domination in graphs and transversals in hypergraphsA total dominating set, S , in a graph, G , is a set such that every vertexin the graph has a neighbour in S .

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Total domination in graphs and transversals in hypergraphsA total dominating set, S , in a graph, G , is a set such that every vertexin the graph has a neighbour in S .

Let H be the hypergraph with

V (H) = V (G) and

E (H) = {N(x) | x ∈ V (G)}.A vertex set S is a transversal in H if and only if it is a total dominatingset in G .

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Total domination in graphs and transversals in hypergraphsA total dominating set, S , in a graph, G , is a set such that every vertexin the graph has a neighbour in S .

Let H be the hypergraph with

V (H) = V (G) and

E (H) = {N(x) | x ∈ V (G)}.A vertex set S is a transversal in H if and only if it is a total dominatingset in G .

Using this transformation most bounds on total domination are achieved.

E.g. if δ(G) ≥ 4 then a minimum total dominating set contains at most3|V (G)|/7 vertices.

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Conclusion

Our group is very research active in a number of research areas.

Currently our main focus is on fixed parameter tractability (FPT) and A.Yeo is also working on transversals in hypergraphs and total dominationin graphs.

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Conclusion

Our group is very research active in a number of research areas.

Currently our main focus is on fixed parameter tractability (FPT) and A.Yeo is also working on transversals in hypergraphs and total dominationin graphs.

The End

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