Graph Coalition Structure Generation

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

Outline • Coalition Structure Generation (CSG) • CSG over graphs (GCSG) • Clustering • Graph Partitioning • Independence of disconnected members (IDM) • Results • General graphs • Bounded treewidth graphs • Planar graphs • Future directions

Outline • Coalition Structure Generation (CSG) • CSG over graphs (GCSG) • Independence of disconnected members (IDM) • Results • General graphs • Bounded treewidth graphs • Planar graphs • Future directions

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

Model

Coalition Structure Generation

CSG [notation] • 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)

Graph Coalition Structure Generation

Independence of disconnected members (IDM) • For a graph G = (N,E), a function v: P(N)  Ris IDM if for all with , and a coalition not containing i and j,

IDM [examples] • 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.

IDM [properties] • 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

IDM • 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.

GCSG • 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.

Remark • 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)

Results

General graphs • 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

Minor free 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.

Separator theorems • 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.

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

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.

Separator theorems • Corollary:

Bounds for minor free and planar graphs • 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.

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

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

Tree decompositions • 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.

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,

Tree decompositions

Tree decompositions • Lemma:

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.

Separator theorems • Lemma:

Bounds for minor free and planar graphs • 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!

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.

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.

Component 1 Edge values Symbol Optimum 1 Optimum 2 • 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.

Components 2 and 3 • Similarly, we define two more triangular components Edge values Symbol Optimum 1 Optimum 2 Optimum 3 Edge values Symbol Optimum

Component 4 • and a double-line component Edge values Symbol Optimum

Component 5 Construction Symbol Optimum 1 Optimum 2 • 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

Construct 1 • 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 Optimum 2 Construction Optimum 1

Graph Coalition Structure Generation