Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

We consider the problem of discovering overlapping communities in networks that we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems, we seek a collection <i>S^′</i> ⊆ <i>S</i> consisting of at least <i>k</i> sets subject to certain disjointness restrictions. In the <i>r</i>-Set Packing with <i>t</i>-Membership, each element of <i>U</i> belongs to at most <i>t</i> sets of <i>S^′</i>, while in <i>r</i>-Set Packing with <i>t</i>-Overlap, each pair of sets in <i>S^′</i> overlaps in at most <i>t</i> elements. For both problems, each set of <i>S</i> has at most <i>r</i> elements. Similarly, both of our Graph Packing problems seek a collection <i>K</i> of at least <i>k</i> subgraphs in a graph <i>G</i>, each isomorphic to a graph <i>H</i> ∈ <i>H</i>. In <i>H</i>-Packing with <i>t</i>-Membership, each vertex of <i>G</i> belongs to at most <i>t</i> subgraphs of <i>K</i>, while in <i>H</i>-Packing with <i>t</i>-Overlap, each pair of subgraphs in <i>K</i> overlaps in at most <i>t</i> vertices. For both problems, each member of <i>H</i> has at most <i>r</i> vertices and <i>m</i> edges, where <i>t</i>, <i>r</i>, and <i>m</i> are constants. Here, we show NP-completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the <i>H</i>-Packing with <i>t</i>-Membership problem analogous to the Kirkpatrick and Hell dichotomy [Kirkpatrick and Hell 1978]. Using polynomial parameter transformations, we reduce the <i>r</i>-Set Packing with <i>t</i>-Membership to a problem kernel with <i>O</i>((<i>r</i> + 1)^<i>r</i><i>k^r</i>) elements and the <i>H</i>-Packing with <i>t</i>-Membership and its edge version to problem kernels with <i>O</i>((<i>r</i> + 1)^<i>r</i><i>k^r</i>) and <i>O</i>((<i>m</i> + 1)^<i>m</i><i>k^m</i>) vertices, respectively. On the other hand, by generalizing [Fellows et al. 2008; Moser 2009], we achieve a kernel with <i>O</i>(<i>r^rk</i>^{<i>r</i> − <i>t</i> − 1}) elements for the <i>r</i>-Set Packing with <i>t</i>-Overlap and kernels with <i>O</i>(<i>r^rk</i>^{<i>r</i> − <i>t</i> − 1}) and <i>O</i>(<i>m^mk</i>^{<i>m</i> − <i>t</i> − 1}) vertices for the <i>H</i>-Packing with <i>t</i>-Overlap and its edge version, respectively. In all cases, <i>k</i> is the input parameter, while <i>t</i>, <i>r</i>, and <i>m</i> are constants.