Türker Bıyıkoğlu

A discrete nodal domain theorem for trees

Let G be a connected graph with n vertices and let x=(x1, ..., xn) be a real vector. A positive (negative) sign graph of the vector x is a maximal connected subgraph of G on vertices xi>0 (xi<0). For an eigenvalue of a generalized Laplacian of a tree: We characterize the maximal number of sign graphs of an eigenvector. We give an O(n2) time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an NP-complete problem. (authors abstract)

CAMPways: constrained alignment framework for the comparative analysis of a pair of metabolic pathways.

Motivation: Given a pair of metabolic pathways, an alignment of the pathways corresponds to a mapping between similar substructures of the pair. Successful alignments may provide useful applications in phylogenetic tree reconstruction, drug design and overall may enhance our understanding of cellular metabolism. Results: We consider the problem of providing one-to-many alignments of reactions in a pair of metabolic pathways. We first provide a constrained alignment framework applicable to the problem. We show that the constrained alignment problem even in a primitive setting is computationally intractable, which justifies efforts for designing efficient heuristics. We present our Constrained Alignment of Metabolic Pathways (CAMPways) algorithm designed for this purpose. Through extensive experiments involving a large pathway database, we demonstrate that when compared with a state-of-the-art alternative, the CAMPways algorithm provides better alignment results on metabolic networks as far as measures based on same-pathway inclusion and biochemical significance are concerned. The execution speed of our algorithm constitutes yet another important improvement over alternative algorithms. Availability: Open source codes, executable binary, useful scripts, all the experimental data and the results are freely available as part of the Supplementary Material at http://code.google.com/p/campways/. Contact: cesim@khas.edu.tr Supplementary information: Supplementary data are available at Bioinformatics online.

Nodal Domain Theorems and Bipartite Subgraphs

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. The number of strong nodal domains is shown not to exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

Castelnuovo-Mumford Regularity of Graphs

We present new combinatorial insights into the calculation of (Castelnuovo-Mumford) regularity of graphs.

Optimal and near-optimal partner selection algorithms in cooperative OFDMA

We obtain the jointly optimal power allocation and partner selection policies, that maximize the sum rate of a cooperative OFDMA system with mutually cooperating pairs of users. We show that the power allocation and partner selection steps can be performed sequentially, and the latter step can be formulated as a maximum weighted matching problem on an undirected graph, which can be solved in polynomial time. We further propose practical algorithms, and compare their performances to the optimal matching algorithm, and demonstrate that very simple and low complexity algorithms based on user-user and user-receiver distances may provide near-optimum rate performance. Moreover, we observe that algorithms that achieve superior sum-rate performance, surprisingly pair the cell edge users, with the strong users near the base station.

Enabling cooperation, resource allocation and receiver selection across cells: Complementary fractional frequency reuse

For a multi-cell multiple access channel, we develop a comprehensive cooperative communication framework: we propose a novel complementary fractional frequency reuse (FFR) strategy tailored specifically for pairwise user cooperation, also taking into account cell sectoring. This strategy allows the cell edge users not only to pool their resources and cooperate across cells, but also to choose the best receiver. We divide the users into cooperating inner and outer user pairs, and assign each pair orthogonal resources using OFDMA. We employ pairwise bidirectional cooperation based on block Markov superposition encoding among user pairs. We derive the achievable rates, while taking into account the geometry dependent interference at the users and the receiver. We find the jointly optimal power allocation, partner selection and receiver selection strategies that maximize the sum rate of the system. We then propose a heuristic matching algorithm, which operates based only on user and receiver locations. We compare the performance of our proposed strategies with several non-cooperative models, and demonstrate that the sum rate can nearly be doubled, while using the same resources.

Discovering cis-regulatory modules by optimizing barbecues

Gene expression in eukaryotic cells is regulated by a complex network of interactions, in which transcription factors and their binding sites on the genomic DNA play a determining role. As transcription factors rarely, if ever, act in isolation, binding sites of interacting factors are typically arranged in close proximity forming so-called cis-regulatory modules. Even when the individual binding sites are known, module discovery remains a hard combinatorial problem, which we formalize here as the Best Barbecue Problem. It asks for simultaneously stabbing a maximum number of differently colored intervals from K arrangements of colored intervals. This geometric problem turns out to be an elementary, yet previously unstudied combinatorial optimization problem of detecting common edges in a family of hypergraphs, a decision version of which we show here to be NP-complete. Due to its relevance in biological applications, we propose algorithmic variations that are suitable for the analysis of real data sets comprising either many sequences or many binding sites. Being based on set systems induced by interval arrangements, our problem setting generalizes to discovering patterns of co-localized itemsets in non-sequential objects that consist of corresponding arrangements or induce set systems of co-localized items. In fact, our optimization problem is a generalization of the popular concept of frequent itemset mining.