892 research outputs found
Coalition structure generation over graphs
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph
Scaling Analysis of Affinity Propagation
We analyze and exploit some scaling properties of the Affinity Propagation
(AP) clustering algorithm proposed by Frey and Dueck (2007). First we observe
that a divide and conquer strategy, used on a large data set hierarchically
reduces the complexity to , for a
data-set of size and a depth of the hierarchical strategy. For a
data-set embedded in a -dimensional space, we show that this is obtained
without notably damaging the precision except in dimension . In fact, for
larger than 2 the relative loss in precision scales like
. Finally, under some conditions we observe that there is a
value of the penalty coefficient, a free parameter used to fix the number
of clusters, which separates a fragmentation phase (for ) from a
coalescent one (for ) of the underlying hidden cluster structure. At
this precise point holds a self-similarity property which can be exploited by
the hierarchical strategy to actually locate its position. From this
observation, a strategy based on \AP can be defined to find out how many
clusters are present in a given dataset.Comment: 28 pages, 14 figures, Inria research repor
Learning Multi-Tree Classification Models with Ant Colony Optimization
Ant Colony Optimization (ACO) is a meta-heuristic for solving combinatorial optimization problems, inspired by the behaviour of biological ant colonies. One of the successful applications of ACO is learning classification models (classifiers). A classifier encodes the relationships between the input attribute values and the values of a class attribute in a given set of labelled cases and it can be used to predict the class value of new unlabelled cases. Decision trees have been widely used as a type of classification model that represent comprehensible knowledge to the user. In this paper, we propose the use of ACO-based algorithms for learning an extended multi-tree classification model, which consists of multiple decision trees, one for each class value. Each class-based decision trees is responsible for discriminating between its class value and all other values available in the class domain. Our proposed algorithms are empirically evaluated against well-known decision trees induction algorithms, as well as the ACO-based Ant-Tree-Miner algorithm. The results show an overall improvement in predictive accuracy over 32 benchmark datasets. We also discuss how the new multi-tree models can provide the user with more understanding and knowledge-interpretability in a given domain
DASH: Dynamic Approach for Switching Heuristics
Complete tree search is a highly effective method for tackling MIP problems, and over the years, a plethora of branching heuristics have been introduced. Recently, portfolio algorithms have taken the process a step further, trying to predict the best heuristic for each instance at hand. This thesis identifies a method which decides the best time to switch the branching heuristic and it is shown how such\na system can be trained efficientl
Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering
We prove the following theorem. Given a planar graph and an integer ,
it is possible in polynomial time to randomly sample a subset of vertices
of with the following properties: (i) induces a subgraph of of
treewidth , and (ii) for every connected subgraph
of on at most vertices, the probability that covers the whole
vertex set of is at least , where is the number of vertices of .
Together with standard dynamic programming techniques for graphs of bounded
treewidth, this result gives a versatile technique for obtaining (randomized)
subexponential parameterized algorithms for problems on planar graphs, usually
with running time bound . The technique can be applied to problems expressible as
searching for a small, connected pattern with a prescribed property in a large
host graph, examples of such problems include Directed -Path, Weighted
-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up
to this point, it was open whether these problems can be solved in
subexponential parameterized time on planar graphs, because they are not
amenable to the classic technique of bidimensionality. Furthermore, all our
results hold in fact on any class of graphs that exclude a fixed apex graph as
a minor, in particular on graphs embeddable in any fixed surface
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