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Consider a graph G = (V, E), where V denotes the set of n vertices and E the set of edges. For a (k,v) balanced partition problem, the objective is to partition G into k components of at most size v · (n/k), while minimizing the capacity of the edges between separate components. [1]
The biclique partition problem takes as input an arbitrary undirected graph, and asks for a partition of its edges into a minimum number of complete bipartite graphs. It is NP-hard, but fixed-parameter tractable. The best approximation algorithm known for the problem has an approximation ratio of (/ ). [11] [12]
The input to the algorithm is an undirected graph G = (V, E) with vertex set V, edge set E, and (optionally) numerical weights on the edges in E.The goal of the algorithm is to partition V into two disjoint subsets A and B of equal (or nearly equal) size, in a way that minimizes the sum T of the weights of the subset of edges that cross from A to B.
The dotted line in red represents a cut with three crossing edges. The dashed line in green represents one of the minimum cuts of this graph, crossing only two edges. [1] In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric.
In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut.
For a partition of V into subsets U and W, an edge xy is balanced if either s(xy) = + and x and y are in the same subset, or s(xy) = – and x and y are different subsets. BSP aims at finding a partition with the maximum number b(G) of balanced edges in G. The Edwards-Erdős gives a lower bound on b(G) for every connected signed graph G.
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set,
Let G be a bipartite graph, M a maximum-cardinality matching in G, and V 0 the set of vertices of G unmatched by M (the "free vertices"). Then G can be partitioned into three parts: The E-O-U decomposition. E - the even vertices - the vertices reachable from V 0 by an M-alternating path of even length.
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