Cooley, Madison; Greene, Casey; Issac, Davis; Pividori, Milton; Sullivan, Blair Parameterized Algorithms for Identifying Gene Co-Expression Modules via Weighted Clique DecompositionApplied and Computational Discrete Algorithms (ACDA) 2021: 111–122
We present a new combinatorial model for identifying regulatory modules in gene co-expression data using a decomposition into weighted cliques. To capture complex interaction effects, we generalize the previously-studied weighted edge clique partition problem. As a first step, we restrict ourselves to the noise-free setting, and show that the problem is fixed parameter tractable when parameterized by the number of modules (cliques). We present two new algorithms for finding these decompositions, using linear programming and integer partitioning to determine the clique weights. Further, we implement these algorithms in Python and test them on a biologically-inspired synthetic corpus generated using real-world data from transcription factors and a latent variable analysis of co-expression in varying cell types.
Borndörfer, Ralf; Casel, Katrin; Issac, Davis; Niklanovits, Aikaterini; Schwartz, Stephan; Zeif, Ziena Connected k-Partition of k-Connected Graphs and c-Claw-Free GraphsApproximation Algorithms for Combinatorial Optimization Problems (APPROX) 2021: 27:1–27:14
A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. In these settings, it is often desirable to partition into a fixed number of parts of roughly of the same size or weight. The resulting computational problem is called Balanced Connected Partition (BCP). The two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on \(c\)-claw-free graphs, the class of graphs that do not have \(K_{1,c}\) as an induced subgraph, and present efficient \((c-1)\)-approximation algorithms for both objectives. In particular, for \(3\)-claw-free graphs, also simply known as claw-free graphs, we obtain a \(2\)-approximation. Due to the claw-freeness of line graphs, this also implies a \(2\)-approximation for the edge-partition version of BCP in general graphs. A harder connected partition problem arises from demanding a connected partition into k parts that have (possibly) heterogeneous target weights \(w_1, \dots, w_k\). In the 1970s Győri and Lovász showed that if \(G\) is \(k\)-connected and the target weights sum to the total size of \(G\), such a partition exists. However, to this day no polynomial algorithm to compute such partitions exists for \(k > 4\). Towards finding such a partition \(T_1, \dots, T_k\) in \(k\)-connected graphs for general \(k\), we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each \(w_i\) is greater than the weight of the heaviest vertex. In particular, we give a \(3\)-approximation for both the lower and the upper bounded version i.e. we guarantee that each \(T_i\) has weight at least \(\frac{w_i}{3}\) or that each \(T_i\) has weight most \(3 w_i\), respectively. Also, we present a both-side bounded version that produces a connected partition where each \(T_i\) has size at least \(\frac{w_i}{3}\) and at most \(\max(\{r, 3\}) w_i \), where \(r \geq 1\) is the ratio between the largest and smallest value in \(w_1, \dots, w_k\). In particular for the balanced version, i.e. \(w_1 = w_2 =, \dots , = w k\), this gives a partition with \(\frac{w_i3 leq w(T_i) \leq 3 w_i\).
Casel, Katrin; Friedrich, Tobias; Issac, Davis; Niklanovits, Aikaterini; Zeif, Ziena Balanced Crown Decomposition for Connectivity ConstraintsEuropean Symposium on Algorithms (ESA) 2021: 26:1–26:15
We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a \(3\)-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.
Casel, Katrin; Friedrich, Tobias; Issac, Davis; Klodt, Nicolas; Seifert, Lars; Zahn, Arthur A Color-blind 3-Approximation for Chromatic Correlation Clustering and Improved HeuristicsKnowledge Discovery and Data Mining (KDD) 2021: 882–891
Chromatic Correlation Clustering (CCC) models clustering of objects with categorical pairwise relationships. The model can be viewed as clustering the vertices of a graph with edge-labels (colors). Bonchi et al. [KDD 2012] introduced it as a natural generalization of the well studied problem Correlation Clustering (CC), motivated by real-world applications from data-mining, social networks and bioinformatics. We give theoretical as well as practical contributions to the study of CCC. Our main theoretical contribution is an alternative analysis of the famous Pivot algorithm for CC. We show that, when simply run color-blind, Pivot is also a linear time 3-approximation for CCC. The previous best theoretical results for CCC were a 4-approximation with a high-degree polynomial runtime and a linear time 11-approximation, both by Anava et al. [WWW 2015]. While this theoretical result justifies Pivot as a baseline comparison for other heuristics, its blunt color-blindness performs poorly in practice. We develop a color-sensitive, practical heuristic we call Greedy Expansion that empirically outperforms all heuristics proposed for CCC so far, both on real-world and synthetic instances. Further, we propose a novel generalization of CCC allowing for multi-labelled edges. We argue that it is more suitable for many of the real-world applications and extend our results to this model.