Applying Modular Decomposition to Parameterized Cluster Editing Problems

Abstract. For a fixed connected graph H, we consider the NP-complete H-packing problem, where, given an undirected graph G and an integer k ≥ 0, one has to decide whether there exist k vertex-disjoint copies of H in G. We give a problem kernel of O(k |V (H)|−1 ) vertices, that is, we provide a polynomial-time algorithm that reduces a given instance of H-packing to an equivalent instance with at most O(k |V (H)|−1 ) vertices. In particular, this result specialized to H being a triangle improves a problem kernel for Triangle Packing from O(k 3 ) vertices by Fellows et al. [WG 2004] to O(k 2 ) vertices.

Section: Introductionmentioning

confidence: 99%

A Problem Kernelization for Graph Packing

Moser

2009

“…Amit [3] proved the NPhardness and gave a factor-11 approximation based on the relaxation of a linear program. Using a simple branching strategy, the problem can be solved in O(4 k + m) time [18], where m is the number of edges in the graph. Protti et al [18] showed how to construct a problem kernel with 4k 2 + 6k vertices.…”

Section: Bicluster Editingmentioning

confidence: 99%

“…In this section, we present a kernelization algorithm for Bicluster Editing that produces a kernel consisting of at most 4k vertices, improving the kernel consisting of O(k 2 ) vertices given by Protti et al [18]. This kernelization follows the idea of the kernelization algorithm for Cluster Editing in [11] that also produces a kernel consisting of at most 4k vertices.…”

Section: Linear Problem Kernelmentioning

confidence: 99%

See 1 more Smart Citation

Improved Algorithms for Bicluster Editing

Guo

Hüffner

Komusiewicz

et al.

Abstract. The NP-hard Bicluster Editing is to add or remove at most k edges to make a bipartite graph G = (V, E) a vertex-disjoint union of complete bipartite subgraphs. It has applications in the analysis of gene expression data. We show that by polynomial-time preprocessing, one can shrink a problem instance to one with 4k vertices, thus proving that the problem has a linear kernel, improving a quadratic kernel result. We further give a search tree algorithm that improves the running time bound from the trivial O(4 k + |E|) to O(3.24 k + |E|). Finally, we give a randomized 4-approximation, improving a known approximation with factor 11.

“…For a background on fixed-parameter algorithmics we refer to [12,15,29]. Parameterized complexity studies for Cluster Editing were initiated by Gramm et al [21] and have been further pursued in a series of papers [5,6,10,13,20,22,32,33]. A previously shown bound of O(1.92 k + n 3 ) for an n-vertex graph [20] can be improved by combining a linear-time problem kernelization algorithm [13] that yields an instance with O(k 2 ) vertices with the currently best claimed running time of O(1.82 k +n 3 ) [6] to get an algorithm with running time O(1.82 k +n+m), where m is the number of edges in the graph.…”

Section: Introductionmentioning

confidence: 99%

Fixed-Parameter Algorithms for Cluster Vertex Deletion

Hüffner

Komusiewicz

Moser

et al.

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.