Florian Sikora

The ExemplarBreakpointDistance for Non-trivial Genomes Cannot Be Approximated

A promising and active field of comparative genomics consists in comparing two genomes by establishing a one-to-one correspondence (i.e., a matching) between their genes. This correspondence is usually chosen in order to optimize a predefined measure. One such problem is the Exemplar Breakpoint Distance problem (or EBD , for short), which asks, given two genomes modeled by signed sequences of characters, to keep and match exactly one occurrence of each character in the two genomes (a process called exemplarization ), so as to minimize the number of breakpoints of the resulting genomes. Bryant [6] showed that EBD is NP -complete. In this paper, we close the study of the approximation of EBD by showing that no approximation factor can be derived for EBD considering non-trivial genomes --- i.e. genomes that contain duplicated genes.

Complexity insights of the Minimum Duplication problem

The Minimum Duplication problem is a well-known problem in phylogenetics and comparative genomics. Given a set of gene trees, the Minimum Duplication problem asks for a species tree that induces the minimum number of gene duplications in the input gene trees. More recently, a variant of the Minimum Duplication problem, called Minimum Duplication Bipartite, has been introduced in [14], where the goal is to find all pre-duplications, that is duplications that precede, in the evolution, the first speciation with respect to a species tree. In this paper, we investigate the complexity of both Minimum Duplication and Minimum Duplication Bipartite problems. First of all, we prove that the Minimum Duplication problem is APX-hard, even when the input consists of five uniquely leaf-labelled gene trees (progressing on the complexity of the problem). Then, we show that the Minimum Duplication Bipartite problem can be solved efficiently by a randomized algorithm when the input gene trees have bounded depth.

On the parameterized complexity of the repetition free longest common subsequence problem

Longest common subsequence is a widely used measure to compare strings, in particular in computational biology. Recently, several variants of the longest common subsequence have been introduced to tackle the comparison of genomes. In particular, the Repetition Free Longest Common Subsequence (RFLCS) problem is a variant of the LCS problem that asks for a longest common subsequence of two input strings with no repetition of symbols. In this paper, we investigate the parameterized complexity of RFLCS. First, we show that the problem does not admit a polynomial kernel. Then, we present a randomized FPT algorithm for the RFLCS problem, improving the time complexity of the existent FPT algorithm.

Querying Protein-Protein Interaction Networks

Recent techniques increase the amount of our knowledge of interactions between proteins. To filter, interpret and organize this data, many authors have provided tools for querying patterns in the shape of paths or trees in Protein-Protein Interaction networks. In this paper, we propose an exact algorithm for querying graphs pattern based on dynamic programming and color-coding. We provide an implementation which has been validated on real data.

Some Results on more Flexible Versions of Graph Motif

The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Due to the high rate of noise in the biological data, more flexible definitions of the problem have been outlined. We present in this paper two inapproximability results for two different optimization variants of Graph Motif. We also study another definition of the problem, when the connectivity constraint is replaced by modularity. While the problem stays NP-complete, it allows algorithms in FPT for biologically relevant parameterizations.

Parameterized Approximability of Maximizing the Spread of Influence in Networks

In this paper, we consider the problem of maximizing the spread of influence through a social network. Here, we are given a graph G = (V,E), a positive integer k and a threshold value thr(v) attached to each vertex v ∈ V. The objective is then to find a subset of k vertices to “activate” such that the number of activated vertices at the end of a propagation process is maximum. A vertex v gets activated if at least thr(v) of its neighbors are. We show that this problem is strongly inapproximable in fpt-time with respect to (w.r.t.) parameter k even for very restrictive thresholds. For unanimity thresholds, we prove that the problem is inapproximable in polynomial time and the decision version is W[1]-hard w.r.t. parameter k. On the positive side, it becomes r(n)-approximable in fpt-time w.r.t. parameter k for any strictly increasing function r. Moreover, we give an fpt-time algorithm to solve the decision version for bounded degree graphs.

Parameterized exact and approximation algorithms for maximum k-set cover and related satisfiability problems

Given a family of subsets S over a set of elements X and two integers p and k, max k-set cover consists of finding a subfamily T ⊆ S of cardinality at most k, covering at least p elements of X. This problem is W[2]-hard when parameterized by k, and FPT when parameterized by p. We investigate the parameterized approximability of the problem with respect to parameters k and p. Then, we show that max sat-k, a satisfiability problem generalizing max k-set cover, is also FPT with respect to parameter p.

Complexity of Grundy Coloring and Its Variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We show that Grundy Coloring can be solved in time \(O^*(2.443^n)\) on graphs of order n. While the problem is known to be solvable in time \(f(k,w)\cdot n\) for graphs of treewidth w, we prove that under the Exponential Time Hypothesis, it cannot be computed in time \(O^*(c^{w})\), for any constant c. We also study the parameterized complexity of Grundy Coloring parameterized by the number of colors, showing that it is in \(\mathsf {FPT}\) for graphs including chordal graphs, claw-free graphs, and graphs excluding a fixed minor.

Some Results on More Flexible Versions of Graph Motif

The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Due to the high rate of noise in the biological data, more flexible definitions of the problem have been outlined. We present in this paper two inapproximability results for two different optimization variants of Graph Motif: one where the size of the solution is maximized, the other when the number of substitutions of colors to obtain the motif from the solution is minimized. We also study a decision version of Graph Motif where the connectivity constraint is replaced by the well known notion of graph modularity. While the problem remains NP-complete, it allows algorithms in FPT for biologically relevant parameterizations.

Parameterized Inapproximability of Target Set Selection and Generalizations

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value Open image in new window for each vertex v of the graph, find a minimum size vertex-subset to “activate” s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex v is activated during the propagation process if at least Open image in new window of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions f and ρ this problem cannot be approximated within a factor of ρ(k) in f(k) ·n O(1) time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.