Michael Dom

Fixed-parameter tractability results for feedback set problems in tournaments

Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and partially improve fixed-parameter tractability results for these problems. We show that Feedback Vertex Set in tournaments is amenable to the novel iterative compression technique. Moreover, we provide data reductions and problem kernels for Feedback Vertex Set and Feedback Arc Set in tournaments, and a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization.

Kernelization Lower Bounds Through Colors and IDs

In parameterized complexity, each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [2009] and Fortnow and Santhanam [2008]. With few exceptions, all known kernelization lower bounds results have been obtained by directly applying this framework. In this article, we show how to combine these results with combinatorial reductions that use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. To follow we give a summary of our main results. All results are under the assumption that the polynomial hierarchy does not collapse to the third level. —We show that the Steiner Tree problem parameterized by the number of terminals and solution size k, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel with at most 2k vertices. —Alon and Gutner [2008] obtain a kpoly(h) kernel for Dominating Set in H-Minor Free Graphs parameterized by h = |H| and solution size k, and ask whether kernels of smaller size exist. We partially resolve this question by showing that Dominating Set in H-Minor Free Graphs does not admit a kernel with size polynomial in k + h. —Harnik and Naor [2007] obtain a “compression algorithm” for the Sparse Subset Sum problem. We show that their algorithm is essentially optimal by showing that the instances cannot be compressed further. —The Hitting Set and Set Cover problems are among the most-studied problems in algorithmics. Both problems admit a kernel of size kO(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the Unique Coverage and Bounded Rank Disjoint Sets problems, admits a polynomial kernel. The existence of polynomial kernels for several of the problems mentioned previously was an open problem explicitly stated in the literature [Alon and Gutner 2008; Betzler 2006; Guo and Niedermeier 2007; Guo et al. 2007; Moser et al. 2007]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [2007], for the problems in question.

Extending the tractability border for closest leaf powers

The nP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most l edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing and “completing” previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we show that Closest 4-Leaf Power is fixed-parameter tractable with respect to parameter l.

Approximation and fixed-parameter algorithms for consecutive ones submatrix problems

We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the NP-hard problem to delete a minimum number of rows or columns from a 0/1-matrix such that the remaining submatrix has the C1P.

Error compensation in leaf root problems

The k-Leaf Root problem is a particular case of graph power problems Here, we study “error correction” versions of k-Leaf Root—that is, for instance, adding or deleting at most l edges to generate a graph that has a k-leaf root We provide several NP-completeness results in this context, and we show that the NP-complete Closest 3-Leaf Root problem (the error correction version of 3-Leaf Root) is fixed-parameter tractable with respect to the number of edge modifications in the given graph Thus, we provide the seemingly first nontrivial positive algorithmic results in the field of error compensation for leaf root problems with k > 2 To this end, as a result of independent interest, we develop a forbidden subgraph characterization of graphs with 3-leaf roots.

The Parameterized Complexity of the Rectangle Stabbing Problem and Its Variants

We study an NP-complete geometric covering problem called d -Dimensional Rectangle Stabbing , where, given a set of axis-parallel d-dimensional hyperrectangles, a set of axis-parallel (d? 1)-dimensional hyperplanes and a positive integer k, the question is whether one can select at most kof the hyperplanes such that every hyperrectangle is intersected by at least one of these hyperplanes. This problem is well-studied from the approximation point of view, while its parameterized complexity remained unexplored so far. Here we show, by giving a nontrivial reduction from a problem called Multicolored Clique , that for d? 3 the problem is W[1]-hard with respect to the parameter k. For the case d= 2, whose parameterized complexity is still open, we consider several natural restrictions and show them to be fixed-parameter tractable.

Approximability and parameterized complexity of consecutive ones submatrix problems

We develop a refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the problem to find a maximum-size submatrix of a 0/1-matrix such that the submatrix has the C1P. Moreover, we achieve a problem kernelization based on simple data reduction rules and provide several search tree algorithms. Finally, we derive inapproximability results.

Closest 4-leaf power is fixed-parameter tractable

The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree-the 4-leaf root-with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r.

Aspects of a Multivariate Complexity Analysis for Rectangle Tiling

Abstract We initiate a parameterized complexity study of the NP-hard problem to tile a positive integer matrix with rectangles, keeping the number of tiles and their maximum weight small. We show that the problem remains NP-hard even for binary matrices only using 1×1 and 2×2-squares as tiles and provide insight into the influence of naturally occurring parameters on the problem’s complexity.

Bounded Degree Closest k-Tree Power Is NP-Complete

An undirected graph G=(V,E) is the k-power of an undirected tree T=(V,E′) if (u,v)∈ E iff u and v are connected by a path of length at most k in T. The tree T is called the tree root of G. Tree powers can be recognized in polynomial time. The thus naturally arising question is whether a graph G can be modified by adding or deleting a specified number of edges such that G becomes a tree power. This problem becomes NP-complete for k≥ 2. Strengthening this result, we answer the main open question of Tsukiji and Chen [COCOON 2004] by showing that the problem remains NP-complete when additionally demanding that the tree roots must have bounded degree.