Peter Damaschke

Parameterized enumeration, transversals, and imperfect phylogeny reconstruction

We study parameterized enumeration problems where we are interested in all solutions of limited size rather than just some solution of minimum cardinality. (Actually, we have to enumerate the inclusion-minimal solutions in order to get fixed-parameter tractable (FPT) results.) Two novel concepts are the notion of a full kernel that contains all small solutions and implicit enumeration of solutions in form of compressed descriptions. In particular, we study combinatorial and computational bounds for the transversal hypergraph (vertex covers in graphs is a special case), restricted to hyperedges with at most k elements. As an example, we apply the results and further special-purpose techniques to almost-perfect phylogeny reconstruction, a problem in computational biology.

Threshold group testing

We introduce a natural generalization of the well-studied group testing problem: A test gives a positive (negative) answer if the pool contains at least u (at most l) positive elements, and an arbitrary answer if the number of positive elements is between these fixed thresholds l and u. We show that the p positive elements can be determined up to a constant number of misclassifications, bounded by the gap between the thresholds. This is in a sense the best possible result. Then we study the number of tests needed to achieve this goal if n elements are given. If the gap is zero, the complexity is, similarly to classical group testing, O(plogn) for any fixed u. For the general case we propose a two-phase strategy consisting of a Distill and a Compress phase. We obtain some tradeoffs between classification accuracy and the number of tests.

Even faster parameterized cluster deletion and cluster editing

Cluster Deletion and Cluster Editing ask to transform a graph by at most k edge deletions or edge edits, respectively, into a cluster graph, i.e., disjoint union of cliques. Equivalently, a cluster graph has no conflict triples, i.e., two incident edges without a transitive edge. We solve the two problems in time O^@?(1.415^k) and O^@?(1.76^k), respectively. These results round off our earlier work by considerably improved time bounds. For Cluster Deletion we use a technique that cuts away small connected components that do no longer contribute to the exponential part of the time complexity. As this idea is simple and versatile, it may lead to improvements for several other parameterized graph problems. The improvement for Cluster Editing is achieved by using the full power of an earlier structure theorem for graphs where no edge is in three conflict triples.

Minimum Common String Partition Parameterized

Minimum Common String Partition (MCSP) and related problems are of interest in, e.g., comparative genomics, DNA fingerprint assembly, and ortholog assignment. Given two strings with equal symbol content, the problem is to partition one string into kblocks, kas small as possible, and to permute them so as to obtain the other string. MCSP is NP-hard, and only approximation algorithms are known. Here we show that MCSP is fixed-parameter tractable in suitable parameters, so that practical instances can be efficiently solved to optimality.

Online Search with Time-Varying Price Bounds

Abstract Online search is a basic online problem. The fact that its optimal deterministic/randomized solutions are given by simple formulas (however with difficult analysis) makes the problem attractive as a target to which other practical online problems can be transformed to find optimal solutions. However, since the upper/lower bounds of prices in available models are constant, natural online problems in which these bounds vary with time do not fit in the available models. We present two new models where the bounds of prices are not constant but vary with time in certain ways. The first model, where the upper and lower bounds of (logarithmic) prices have decay speed, arises from a problem in concurrent data structures, namely to maximize the (appropriately defined) freshness of data in concurrent objects. For this model we present an optimal deterministic algorithm with competitive ratio

Parameterized Enumeration, Transversals, and Imperfect Phylogeny Reconstruction

\sqrt{D}

The union of minimal hitting sets: parameterized combinatorial bounds and counting

, where D is the known duration of the game, and a nearly-optimal randomized algorithm with competitive ratio

Nearly optimal strategies for special cases of on-line capital investment

\frac{\ln D}{1+\ln2-\frac{2}{D}}

Point placement on the line by distance data

. We also prove that the lower bound of competitive ratios of randomized algorithms is asymptotically

Competitive group testing and learning hidden vertex covers with minimum adaptivity

\frac{\ln D}{4}