The Maximum Exposure Problem

Given a set of points P and axis-aligned rectangles R in the plane, a point p in P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k^2) rectangles, we can expose at least Omega(1/k) of the optimal number of points

Online Maximum k-Coverage

We study an online model for the maximum k-vertex-coverage problem, where given a graph G = (V,E) and an integer k, we ask for a subset A ⊆ V, such that |A | = k and the number of edges covered by A is maximized. In our model, at each step i, a new vertex vi is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k vertices can be kept in memory; if at some point the current solution already contains k vertices, any inclusion of any new vertex in the solution must entail the irremediable deletion of one vertex of the current solution (a vertex not kept when revealed is irremediably deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy 1/2-competitive ratio. We next settle a set-version of the problem, called maximum k-(set)-coverage problem. For this problem we present an algorithm that improves upon former results for the same model for small and moderate values of k

Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

A More Complicated Hardness Proof for Finding Densest Subgraphs in Bounded Degree Graphs

We consider the Densest-Subgraph problem, where a graph and an integer k is given and we search for a subgraph on exactly k vertices that induces the maximum number of edges. We prove that this problem is NP-hard even when the input graph has maximum degree three