Graph Coalition Structure Generation Maria Polukarov University of Southampton Joint work with Tom Voice and Nick Jennings HUJI, 25 th September 2011

Graph Coalition Structure Generation

Maria Polukarov

University of Southampton

Joint work with Tom Voice and Nick Jennings

HUJI, 25th September 2011

Outline

Coalition Structure Generation (CSG) Complete Set Partitioning

CSG over graphs (GCSG) Clustering

Graph Partitioning

Independence of disconnected members (IDM) Results

General graphs

Bounded treewidth graphs

Planar graphs

Future directions

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Outline

Coalition Structure Generation (CSG) Complete Set Partitioning

CSG over graphs (GCSG) Clustering

Graph Partitioning


General graphs


Planar graphs

Future directions

3

Outline

Coalition Structure Generation (CSG) CSG over graphs (GCSG)

Clustering

Graph Partitioning


General graphs


Planar graphs

Future directions4

Outline


Clustering

Graph Partitioning


General graphs


Planar graphs

Future directions5

Outline


Clustering

Graph Partitioning


General graphs


Planar graphs

Future directions6

Outline

Coalition Structure Generation (CSG) CSG over graphs (GCSG) Independence of disconnected members (IDM) Results

General graphs


Planar graphs

Future directions

7

Outline


General graphs


Planar graphs

Future directions

8

Outline


General graphs


Planar graphs

Future directions

9

Outline


General graphs


Planar graphs

Future directions

10

Outline


General graphs


Planar graphs

Future directions

11

Outline

Coalition Structure Generation (CSG) CSG over graphs (GCSG) Independence of disconnected members (IDM) Results Future directions

12

Outline

Coalition Structure Generation (CSG) CSG over graphs (GCSG) Independence of disconnected members (IDM) Results Future directions

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Model

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Coalition Structure Generation

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N = {1,…,n} – set of elements (``agents’’) v: P(N) R – characteristic function

CSG problem: find partition {N1,…,Nm} of N that maximizes Σiv(Ni)

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CSG [notation]

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Graph Coalition Structure Generation

For a graph G = (N,E), a function v: P(N) R is IDM if for all with , and a coalition not containing i and j,

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Independence of disconnected members (IDM)

Each edge (i,j) has a constant weight vij. The edge sum characteristic function I is IDM.

Each edge is labelled by + or – , and let

The correlation characteristic function

is IDM.23

IDM [examples]

Lemma: Given a graph G=(N,E) and an IDM coalition valuation function v(), for any two subsets of nodes A,B, if there are no edges between A\B and B\A then

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IDM [properties]

For a graph G = (N,E), a coalition structure C over N is connected if the induced subgraph of G is connected for all coalitions C in C.

Remark: for an IDM function v and a coalition structure C, there exists a connected structure D such that v(C) = v(D). Moreover, if G is not connected, the problem is solved by finding the optimal structure over each connected component of G and combining the results.

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IDM

Given a connected graph G=(N,E) and an IDM characteristic function v: P(N) R, the Graph Coalition Structure Generation problem over G is to maximize for C a coalition structure over N.

This problem is equivalent to maximizing the

same objective function over all connected coalition structures.

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GCSG

Clustering problems in general do not necessarily fit in our model: some of them have objectives that do not

admit the IDM property (e.g., modularity clustering)

some clustering problems have additional restrictions on feasible graph partitions (e.g., weighted graph partitioning)

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Remark

Results

The GCSG problem is NP-complete on general graphs, even for edge sum characteristic functions

A general instance with |N|=n nodes and |E|=e edges can be solved in time

using O(n2) memory For sparse graphs with e=cn edges, where c is

a constant, this implies the bound of

with a constant

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General graphs

We give general bounds on the computational complexity of the GCSG problem for planar graphs and, more generally, minor free graphs.

We show polynomial time solvability of the

GCSG problem for bounded treewidth graphs.

We prove NP-hardness for planar, and hence, all Kk minor free graphs for k ≥ 5, even for edge sum characteristic functions.

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Minor free graphs

A class of graphs S satisfies an f(n)-separator theorem with constant α < 1 if for all G = (N,E) in S with |N| = n there exist two subgraphs A,B of G such that , the number of nodes in is less than or equal to f(n) and both the number of nodes in A\B and the number of nodes in B\A are ≤ αn.

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Separator theorems

Suppose a class of graphs S is closed under taking subgraphs and there is an increasing function g(n) such that for all G = (N, E) in S with |N| = n, graph G has at most g(n) possible connected coalition structures.

Suppose that S satisfies an f(n)-separator theorem with constant α < 1, and that for any G such a separator can be found in time, where f(n) is an increasing o(n) function and

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Separator theorems

Theorem: For any α < β < 1, an instance of the graph coalition structure generation problem over a graph from S can be solved in I computation steps.

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Separator theorems

Corollary:

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Separator theorems

Theorem: For any graph H with k vertices and , an instance of the graph coalition structure generation problem over an H minor free graph G with n nodes requires computation steps.

Theorem: For any , a general instance of a graph coalition structure generation problem over a planar graph G with n nodes can be

solved in computation steps.

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Bounds for minor free and planar graphs

Theorem: A GCSG problem over the class of graphs of maximum treewidth k can be solved in O(n2) time, for any k.

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Theorem: A GCSG problem over the class of graphs of maximum treewidth k can be solved in O(n) time, for any k.

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Theorem: A general instance of the graph coalition structure generation problem over a graph G with n nodes and a known tree decomposition of width w can be solved in I computation steps.

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Tree decompositions

Lemma: Given a graph G=(N,E) and a tree decomposition (X,T), where X={X1,…,Xm} for m≤n and T is a tree over X. Suppose further that the Xis are numbered in order of shortest distance in T from X1 which can be chosen arbitrarily. Then, for any subset of nodes C,

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Tree decompositions

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Tree decompositions

Lemma:

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Tree decompositions

We prove the bound of by recursively calculating the potential marginal contributions to total coalition structure value for branches of the tree.

Given any constant k, for the class of graphs with maximum treewidth k, a tree decomposition with width at most k can be found in time linear in n, and so the bound of O(n) for the GCSG follows.

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Tree decompositions

Lemma:

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Separator theorems

Theorem: For any graph H with k vertices, an instance of the graph coalition structure generation problem over an H minor free graph G with n nodes requires computation steps for

Theorem: A general instance of a graph coalition structure generation problem over a planar graph G with n nodes can be solved in

computation steps, for

Exponential in , but is almost as good as it can get! 44

Bounds for minor free and planar graphs

Theorem: The class of edge sum graph coalition structure generation problems over planar graphs is NP-complete. Moreover, a 3-SAT problem with m clauses can be represented by a GCSG problem over a planar graph with O(m2) nodes.

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Planar graphs

Proof (short sketch):

• Given a 3-SAT problem with clauses C1, …, Cm, we construct an edge sum GCSG problem over a planar graph of O(m2) nodes which, when solved, reveals a solution to the 3-SAT problem if one exists.

• This graph has components of 5 types.

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Planar graphs

• The contribution of such a component to the value of a coalition structure is at most 3, with equality only if the induced structure over the three outer nodes is either that given by Optimum 1 or that given by Optimum 2.

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Component 1

Edge values

Symbol Optimum 1 Optimum 2

• Similarly, we define two more triangular components

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Components 2 and 3

Edge values

Symbol Optimum 1 Optimum 2

Edge values

Symbol Optimum

Optimum 3

• and a double-line component

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Component 4

Edge values

Symbol Optimum

• We construct a last component out of six copies of Component 1

• For the three points labeled A, B, C, there are two induced coalition structures given in Optimum 1 and Optimum 2, for which the contribution of the edge values in the component is maximal

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Component 5

Construction Symbol Optimum 1 Optimum 2

• We combine the components of the 5 types in certain constructs

• Construct 1 below is such that in any locally optimal coalition structure, nodes X and Y are always in the same coalition and the pair of nodes labeled A lie in the same coalition if and only if the pair of nodes labeled B lie in the same coalition

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Construct 1

Construction Optimum 1 Optimum 2

• In Construct 2, under a locally optimal coalition structure, if the pair of nodes A are together in the same coalition, then the pair of nodes B are in the same coalition, and similarly for the pair of nodes C

• If the pair of nodes A are not in the same coalition, then the pair of nodes B are not in the same coalition, and similarly for the pair of nodes C

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Construct 2


• Construct 3 is similar, except that the state of whether or not the pair of nodes C are in the same coalition as each other is the opposite to the state of whether or not the pair of nodes A are in the same coalition as each other

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Construct 3


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Construct 4


Optimum 5 Optimum 7

Optimum 3

Optimum 4 Optimum 6

• Create a copy of Construct 4 for each clause of the 3-SAT problem, where the three pairs A, B, C are identified with the three literals in the corresponding clause

• A coalition structure over these constructs is identified with a set of logical values for the literals in the clauses (the literal associated with a pair of node is set as true if and only if those nodes are not in the same coalition)

• Use Component 4 to connect the pairs of nodes that represent literals of the same variable or its negation to a series of copies of Constructs 2 and 3

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Graph construction

• This allows us to ensure that any locally optimal coalition structure assigns consistent logical values to variables

• To ensure that the resulting graph is planar, we can replace any pair of Components 4 which cross over with two copies of Construct 1

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Graph construction

• (A ∨ B ∨ B) ∧ (!A ∨ !B ∨ !C) ∧ (!A ∨ B ∨ C)

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Example

• A locally optimal coalition structure exists if and only if the original 3-SAT problem is satisfiable, and given any locally optimal coalition structure, we can identify a solution to the 3-SAT problem

• If a locally optimal coalition structure exists, then a coalition structure is optimal iff it is locally optimal

• The size of this graph is O(m2) ☐

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Finally,

Summary

We gave bounds on computation of the exact optimal coalition structure over general and minor free graphs with an IDM characteristic function.

Proved polynomial time solvability for bounded treewidth graphs and NP-hardness for planar graphs.

Future work: Efficient approximation algorithms are required.

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Thanks!

Thanks!

Documents

Graph Coalition Structure Generation Maria Polukarov University of Southampton Joint work with Tom Voice and Nick Jennings HUJI, 25 th September 2011