Randomized Selection of Important Separators

Saket Saurabh

The Institute of Mathematical Sciences, India

ASPAK 2014, March 3-8

Some Basic Definitions

I C ⊆ V (G ) is a (p, q)-cluster if |C | ≤ p and d(C ) ≤ q.

I A (p, q)-partition of G is a partition of V (G ) into(p, q)-clusters.

Problem

(p, q)-Partition Parameter: qInput: An undirected graph G and integers p and q.Question: Does G has a (p, q)-partition?

Problem

(p, q)-Partition Parameter: qInput: An undirected graph G and integers p and q.Question: Does G has a (p, q)-partition?

Goal is to show that (p, q)-Partition problem is FPTparameterized by q.

Necessary Condition

Necessary Condition

A necessary condition for the existence of (p, q)-partition is that forevery vertex v ∈ V (G ) there exists a (p, q)-cluster that contains v .

Is this sufficient?

Is this sufficient?

Very surprisingly, it turns out that this trivial necessary condition isactually sufficient for the existence of a (p, q)-partition.

HOW?

Posimodularity

We say that a set function f : 2|V (G)| → R is posimodular if itsatisfies the following inequality for every A,B ⊆ V (G ):

f (A) + f (B) ≥ f (A \ B) + f (B \ A) (1)

Posimodularity

We say that a set function f : 2|V (G)| → R is posimodular if itsatisfies the following inequality for every A,B ⊆ V (G ):

f (A) + f (B) ≥ f (A \ B) + f (B \ A) (1)

The function dG is posimodular.

Let us move forward?

LemmaLet G be an undirected graph and let p, q ≥ 0 be two integers. Ifevery v ∈ V (G ) is contained in some (p, q)-cluster, then G has a(p, q)-partition. Furthermore, given a set of (p, q)-clusters C1, . . . ,Cn whose union is V (G ), a (p, q)-partition can be found inpolynomial time.

Proof.Proof on the board :).

What did we achieve?

We have reduced the problem of (p, q)-Partition to

(p, q)-Cluster Parameter: qInput: An undirected graph G , a veretx v ∈ V (G ) and integersp and q.Question: Does G has (p, q)-cluster containing v?

What did we achieve?

We have reduced the problem of (p, q)-Partition to

(p, q)-Cluster Parameter: qInput: An undirected graph G , a veretx v ∈ V (G ) and integersp and q.Question: Does G has (p, q)-cluster containing v?

Algorithms for (p, q)-Cluster

I For every fixed q, there is an nO(q) time algorithm for(p, q)-Cluster.

I What about parameterized by p + q?

Algorithms for (p, q)-Cluster

I For every fixed q, there is an nO(q) time algorithm for(p, q)-Cluster.

I What about parameterized by p + q?

Algorithms for (p, q)-Cluster

I For every fixed q, there is an nO(q) time algorithm for(p, q)-Cluster.

I What about parameterized by p + q?

There is a 2O(p+q)nO(1) time algorithm for (p, q)-Cluster.

Towards the FPT algorithm

Satellite Problem Parameter: qInput: An undirected graph G , integers p and q, a vertex v ∈V (G ), and a partition (V0,V1, . . . ,Vr ) of V (G ) such that v ∈ V0

and there is no edge between Vi and Vj for any 1 ≤ i < j ≤ r .Question: The task is to find a (p, q)-cluster C satisfying V0 ⊆ Csuch that for every 1 ≤ i ≤ r , either C ∩ Vi = ∅ or Vi ⊆ C .

Towards the FPT algorithm

Satellite Problem Parameter: qInput: An undirected graph G , integers p and q, a vertex v ∈V (G ), and a partition (V0,V1, . . . ,Vr ) of V (G ) such that v ∈ V0

and there is no edge between Vi and Vj for any 1 ≤ i < j ≤ r .Question: The task is to find a (p, q)-cluster C satisfying V0 ⊆ Csuch that for every 1 ≤ i ≤ r , either C ∩ Vi = ∅ or Vi ⊆ C .

For every Vi (1 ≤ i ≤ r), we have to decide whether to include orexclude it from the solution cluster C . If we exclude Vi from C ,then d(C ) increases by, d(Vi ), the number of edges between V0

and Vi . If we include Vi into C , then |C | increases by |C |.

Figure : Instance of Satellite Problem with a solution C . ExcludingV2 and V4 from C decreased the size of C by the gray area, butincreased d(C ) by the red edges.

Satellite Problem

LemmaThe Satellite Problem can be solved in polynomial time.

Proof.We will come back to the proof.

Satellite Problem

LemmaThe Satellite Problem can be solved in polynomial time.

Proof.We will come back to the proof. Hint: Think Knapsack problem.

Objective

We will give a randomized algorithm for (p, q)-Cluster byreducing it to Satellite Problem.

Important Sets

DefinitionWe say that a set X ⊆ V (G ), v 6∈ X is important

1. d(X ) ≤ q,

2. G [X ] is connected,

3. there is no Y ⊃ X , v 6∈ Y such that d(Y ) ≤ d(X ) and G [Y ]is connected.

Important Sets

The following definition connects the notion of important cuts withour problem.

DefinitionWe say that a set X ⊆ V (G ), v 6∈ X is important

1. d(X ) ≤ q,

2. G [X ] is connected,

3. there is no Y ⊃ X , v 6∈ Y such that d(Y ) ≤ d(X ) and G [Y ]is connected.

It is easy to see that X is an important set if and only if ∆(X ) isan important (u, v)-cut of size at most q for every u ∈ X .

Enumerating Important Cuts and Important Sets

TheoremLet X ,Y ⊆ V (G ) be two disjoint sets of vertices in graph G, letk ≥ 0 be an integer, and let Sk be the set of all (X ,Y )-importantcuts of size at most k. Then |Sk | ≤ 4k and Sk can be constructedin time |Sk | · k · (|V (G )|+ |E (G )|).

Minimal (p, q)-cluster

LemmaLet C be an inclusionwise minimal (p, q)-cluster containing v.Then every component of G \ C is an important set.

Proof.On board.

Main Result

LemmaGiven a graph G, vertex v ∈ V (G ), and integers p and q, we canconstruct in time 2O(q) · nO(1) an instance I of the SatelliteProblem such that

I If some (p, q)-cluster contains v , then I is a yes-instance withprobability 2−O(q) (or some 2−O(f (q))),

I If there is no (p, q)-cluster containing v, then I is ano-instance.

Algorithm

I For every u ∈ V (G ), u 6= v , enumerate every important(u, v)-cut of size at most q.

I For every such cut S , put the component K of G \ Scontaining u into the collection X .

I

I

Algorithm

I For every u ∈ V (G ), u 6= v , enumerate every important(u, v)-cut of size at most q.

I For every such cut S , put the component K of G \ Scontaining u into the collection X .

I Let X ′ be a subset of X , where each member K of X ischosen with probability 1

2 independently at random.

I

Algorithm

I For every u ∈ V (G ), u 6= v , enumerate every important(u, v)-cut of size at most q.

I For every such cut S , put the component K of G \ Scontaining u into the collection X .

I Let X ′ be a subset of X , where each member K of X ischosen with probability 1

2 independently at random.

I Let Z be the union of the sets in X ′, let V1, . . . , Vr be theconnected components of G [Z ], and let V0 = V (G ) \ Z . It isclear that V0, V1, . . . , Vr describe an instance I of theSatellite Problem, and a solution for I gives a(p, q)-cluster containing v . Thus we only need to show that ifthere is a (p, q)-cluster C containing v , then I is a yesinstance with desired probability.

Let us ZOOM in :D

I Let Z be the union of the sets in X ′, let V1, . . . , Vr be theconnected components of G [Z ], and let V0 = V (G ) \ Z . It isclear that V0, V1, . . . , Vr describe an instance I of theSatellite Problem, and a solution for I gives a(p, q)-cluster containing v . Thus we only need to show that ifthere is a (p, q)-cluster C containing v , then I is a yesinstance with desired probability.

What do we need?

We show that the reduction works if the union Z of the selectedimportant sets satisfies two constraints: it has to cover every

component of G \ C and it has to be disjoint form the vertices onthe boundary of C . The probability of the event that these

constraints are satisfied is 2−O(f (q)).

I Let C be an inclusionwise minimal (p, q)-cluster containing v .

I Let S be the set of vertices on the boundary of C , i.e., thevertices of C incident to ∆(C ).

I Let K1, . . . , Kt be the components of G \ C .

I Let C be an inclusionwise minimal (p, q)-cluster containing v .

I Let S be the set of vertices on the boundary of C , i.e., thevertices of C incident to ∆(C ).

I Let K1, . . . , Kt be the components of G \ C .

Consider the following two events:(E1) Every component Ki of G \ C is in X ′ (and hence Ki ⊆ Z ).(E2) Z ∩ S = ∅.

Let us do the probability computation onboard :).