Jens Gramm

Graph-Modeled Data Clustering: Exact Algorithms for Clique Generation

We present efficient fixed-parameter algorithms for the NP-complete edge modification problems Cluster Editing and Cluster Deletion. Here, the goal is to make the fewest changes to the edge set of an input graph such that the new graph is a vertex-disjoint union of cliques. Allowing up to k edge additions and deletions (Cluster Editing), we solve this problem in O(2.27k + |V|3) time; allowing only up to k edge deletions (Cluster Deletion), we solve this problem in O(1.77k + |V|3) time. The key ingredients of our algorithms are two easy to implement bounded search tree algorithms and an efficient polynomial-time reduction to a problem kernel of size O(k3). This improves and complements previous work. Finally, we discuss further improvements on search tree sizes using computer-generated case distinctions.

Automated Generation of Search Tree Algorithms for Graph Modification Problems

We present a (seemingly first) framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is two-fold-rapid development and improved upper bounds. Many search tree algorithms for various problems in the literature are based on complicated case distinctions. Our approach may lead to a much simpler process of developing and analyzing these algorithms. Moreover, using the sheer computing power of machines it may also lead to improved upper bounds on search tree sizes (i.e., faster exact solving algorithms) in comparison with previously developed hand-made search trees.

Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

Abstract We present a framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is twofold—rapid development and improved upper bounds. Many search tree algorithms for various problems in the literature are based on complicated case distinctions. Our approach may lead to a much simpler process of developing and analyzing these algorithms. Moreover, using the sheer computing power of machines it may also lead to improved upper bounds on search tree sizes (i.e., faster exact solving algorithms) in comparison with previously developed “hand-made” search trees. Among others, such an example is given with the NP-complete Cluster Editing problem (also known as Correlation Clustering on complete unweighted graphs), which asks for the minimum number of edge additions and deletions to create a graph which is a disjoint union of cliques. The hand-made search tree for Cluster Editing had worst-case size O(2.27k), which now is improved to O(1.92k) due to our new method. (Herein, k denotes the number of edge modifications allowed.)

Faster exact algorithms for hard problems: a parameterized point of view

Abstract Recent times have seen quite some progress in the development of ‘efficient’ exponential-time algorithms for NP-hard problems. These results are also tightly related to the so-called theory of fixed parameter tractability. In this incomplete, personally biased survey, we reflect on some recent developments and prospects in the field of fixed parameter algorithms.

Graph-modeled data clustering: fixed-parameter algorithms for clique generation

We present efficient fixed-parameter algorithms for the NP-complete edge modification problems CLUSTER EDITING and CLUSTER DELETION. Here, the goal is to make the fewest changes to the edge set of an input graph such that the new graph is a vertex-disjoint union of cliques. Allowing up to k edge additions and deletions (CLUSTER EDITING), we solve this problem in O(2.27k + |V|3) time; allowing only up to k edge deletions (CLUSTER DELETION), we solve this problem in O(1.77k + |V|3) time. The key ingredients of our algorithms are two easy to implement bounded search tree algorithms and a reduction to a problem kernel of size O(k3). This improves and complements previous work.

Data reduction and exact algorithms for clique cover

To cover the edges of a graph with a minimum number of cliques is an NP-hard problem with many applications. For this problem we develop efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques.

On Exact and Approximation Algorithms for Distinguishing Substring Selection

The NP-complete Distinguishing Substring Selection problem (DSSS for short) asks, given a set of “good” strings and a set of “bad” strings, for a solution string which is, with respect to Hamming metric, “away” from the good strings and “close” to the bad strings.

Improved fixed-parameter algorithms for two feedback set problems

Settling a ten years open question, we show that the NP-complete Feedback Vertex Set problem is deterministically solvable in O(ck · m) time, where m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. As a second result, we present a fixed-parameter algorithm for the NP-complete Edge Bipartization problem with runtime O(2k · m2).

Pattern matching for arc-annotated sequences

We study pattern matching for arc-annotated sequences. An O(nm) time algorithm is given for the problem to determine whether a length m sequence with nested arc annotation is an arc-preserving subsequence (aps) of a length n sequence with nested arc annotation, called APS(NESTED,NESTED). Arc-annotated sequences and, in particular, those with nested arc annotation are motivated by applications in RNA structure comparison. Our algorithm generalizes results for ordered tree inclusion problems and it is useful for recent fixed-parameter algorithms for LAPCS(NESTED,NESTED), which is the problem of computing a longest arc-preserving common subsequence of two sequences with nested arc annotations. In particular, the presented dynamic programming methodology implies a quadratic-time algorithm for an open problem posed by Vialette.

On the Parameterized Intractability of CLOSEST SUBSTRINGsize and Related Problems

We show that CLOSEST SUBSTRING, one of the most important problems in the field of biological sequence analysis, is W[1]-hard with respect to the number k of input strings (even over a binary alphabet). This problem is therefore unlikely to be solvable in time O(f(k)nc) for any function f and constant c independent of k -- effectively, the problem can be expected to be intractable, in any practical sense, for k ? 3. Our result supports the intuition that CLOSEST SUBSTRING is computationally much harder than the special case of CLOSEST STRING, although both problems are NP-complete and both possess polynomial time approximation schemes. We also prove W[1]-hardness for other parameterizations in the case of unbounded alphabet size. Our main W[1]- hardness result generalizes to CONSENSUS PATTERNS, a problem of similar significance in computational biology.