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### Upper Bounding Rainbow Connection Number by Forest Number [article]

L. Sunil Chandran, Davis Issac, Juho Lauri, Erik Jan van Leeuwen
2020 arXiv   pre-print
A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G). A simple way to rainbow-connect a graph G is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of G. This proves that
more » ... (G) < |V(G)|-1. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of t(G) -1 for rc(G), where t(G) is the number of vertices in the largest induced tree of G? The answer turns out to be negative, as there are counter-examples that show that even c· t(G) is not an upper bound for rc(G)) for any given constant c. In this work we show that if we consider the forest number f(G), the number of vertices in a maximum induced forest of G, instead of t(G), then surprisingly we do get an upper bound. More specifically, we prove that rc(G) ≤ f(G) + 2. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound.

### Sampling in Space Restricted Settings [article]

Anup Bhattacharya, Davis Issac, Ragesh Jaiswal, Amit Kumar
2015 arXiv   pre-print
Space efficient algorithms play a central role in dealing with large amount of data. In such settings, one would like to analyse the large data using small amount of "working space". One of the key steps in many algorithms for analysing large data is to maintain a (or a small number) random sample from the data points. In this paper, we consider two space restricted settings -- (i) streaming model, where data arrives over time and one can use only a small amount of storage, and (ii) query
more » ... where we can structure the data in low space and answer sampling queries. In this paper, we prove the following results in above two settings: - In the streaming setting, we would like to maintain a random sample from the elements seen so far. We prove that one can maintain a random sample using O( n) random bits and O( n) space, where n is the number of elements seen so far. We can extend this to the case when elements have weights as well. - In the query model, there are n elements with weights w_1, ..., w_n (which are w-bit integers) and one would like to sample a random element with probability proportional to its weight. Bringmann and Larsen (STOC 2013) showed how to sample such an element using nw +1 space (whereas, the information theoretic lower bound is n w). We consider the approximate sampling problem, where we are given an error parameter ε, and the sampling probability of an element can be off by an ε factor. We give matching upper and lower bounds for this problem.

### Efficient Constructions for the Győri-Lovász Theorem on Almost Chordal Graphs [article]

Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, Ziena Zeif
2022 arXiv   pre-print
In the 1970s, Győri and Lovász showed that for a k-connected n-vertex graph, a given set of terminal vertices t_1, ..., t_k and natural numbers n_1, ..., n_k satisfying ∑_i=1^k n_i = n, a connected vertex partition S_1, ..., S_k satisfying t_i ∈ S_i and |S_i| = n_i exists. However, polynomial algorithms to actually compute such partitions are known so far only for k ≤ 4. This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the
more » ... ues of k. More precisely, we consider k-connected chordal graphs and a broader class of graphs related to them. For the first, we give an algorithm with O(n^2) running time that solves the problem exactly, and for the second, an algorithm with O(n^4) running time that deviates on at most one vertex from the given required vertex partition sizes.

### Connected k-partition of k-connected graphs and c-claw-free graphs [article]

Ralf Borndörfer, Katrin Casel, Davis Issac, Aikaterini Niklanovits, Stephan Schwartz, Ziena Zeif
2021 arXiv   pre-print
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. We consider Balanced Connected Partitions (BCP), where 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,
more » ... d present efficient (c-1)-approximation algorithms for both objectives. In particular, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs. In the 1970s Győri and Lovász showed for natural numbers w_1,...,w_k where ∑_i w_i is the vertex size, that if G is k-connected, then there exist a connected k-partition with part sizes w_1,...,w_k. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition T_1,..., T_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 w_i/3 or that each T_i has weight most 3w_i, respectively. Also, we present a both-side bounded version that produces a connected partition where each T_i has size at least w_i/3 and at most max({r,3}) w_i, where r ≥ 1 is the ratio between the largest and smallest value in w_1, ..., w_k. In particular for the balanced version, i.e. w_1=w_2=, ...,=w_k, this gives a partition with 1/3w_i ≤ w(T_i) ≤ 3w_i.

### Improved And Optimized Drug Repurposing For The SARS-CoV-2 Pandemic [article]

Davis Issac, Sarel Cohen, Moshik Hershcovitch, Martin Taraz, Otto Kißig, Daniel Waddington, Peter Chin, Tobias Friedrich
2022 bioRxiv   pre-print
The active global SARS-CoV-2 pandemic caused more than 426 million cases and 5.8 million deaths worldwide. The development of completely new drugs for such a novel disease is a challenging, time intensive process. Despite researchers around the world working on this task, no effective treatments have been developed yet. This emphasizes the importance of drug repurposing, where treatments are found among existing drugs that are meant for different diseases. A common approach to this is based on
more » ... emph{knowledge graphs}, that condense relationships between entities like drugs, diseases and genes. Graph neural networks (GNNs) can then be used for the task at hand by predicting links in such knowledge graphs. Expanding on state-of-the-art GNN research, Doshi {\sl et al.} recently developed the \drcov \ model. We further extend their work using additional output interpretation strategies. The best aggregation strategy derives a top-100 ranking of 8,070 candidate drugs, 32 of which are currently being tested in COVID-19-related clinical trials. Moreover, we present an alternative application for the model, the generation of additional candidates based on a given pre-selection of drug candidates using collaborative filtering. In addition, we improved the implementation of the \drcov \ model by significantly shortening the inference and pre-processing time by exploiting data-parallelism. As drug repurposing is a task that requires high computation and memory resources, we further accelerate the post-processing phase using a new emerging hardware --- we propose a new approach to leverage the use of high-capacity Non-Volatile Memory for aggregate drug ranking.

### Algorithms and Bounds for Very Strong Rainbow Coloring [chapter]

L. Sunil Chandran, Anita Das, Davis Issac, Erik Jan van Leeuwen
2018 Lecture Notes in Computer Science
A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (src(G)) of the graph. When proving upper bounds on src(G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path
more » ... ive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number (vsrc(G)). In this paper, we give upper bounds on vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on src(G) for these classes, showing that the study of vsrc(G) enables meaningful progress on bounding src(G). Then we study the complexity of the problem to compute vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for src(G). We also observe that deciding whether vsrc(G) = k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether vsrc(G) ≤ 3 nor to approximate vsrc(G) within a factor n 1−ε , unless P=NP.

### Balanced Crown Decomposition for Connectivity Constraints [article]

Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, Ziena Zeif
2021 arXiv   pre-print
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
more » ... nd partitioning problems. We illustrate this by deriving improved kernelization and approximation algorithms 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.

### Spanning Tree Congestion and Computation of Generalized Győri-Lovász Partition [article]

L. Sunil Chandran and Yun Kuen Cheung and Davis Issac
2018 arXiv   pre-print
We study a natural problem in graph sparsification, the Spanning Tree Congestion () problem. Informally, the problem seeks a spanning tree with no tree-edge routing too many of the original edges. The root of this problem dates back to at least 30 years ago, motivated by applications in network design, parallel computing and circuit design. Variants of the problem have also seen algorithmic applications as a preprocessing step of several important graph algorithms. For any general connected
more » ... h with n vertices and m edges, we show that its STC is at most O(√(mn)), which is asymptotically optimal since we also demonstrate graphs with STC at least Ω(√(mn)). We present a polynomial-time algorithm which computes a spanning tree with congestion O(√(mn)· n). We also present another algorithm for computing a spanning tree with congestion O(√(mn)); this algorithm runs in sub-exponential time when m = ω(n ^2 n). For achieving the above results, an important intermediate theorem is generalized Győri-Lovász theorem, for which Chen et al. gave a non-constructive proof. We give the first elementary and constructive proof by providing a local search algorithm with running time O^*( 4^n ), which is a key ingredient of the above-mentioned sub-exponential time algorithm. We discuss a few consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Θ(n) with high probability.

### Parameterized algorithms for identifying gene co-expression modules via weighted clique decomposition [article]

Madison Cooley, Casey S. Greene, Davis Issac, Milton Pividori, Blair D. Sullivan
2021 arXiv   pre-print
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
more » ... 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.

### Spanning Tree Congestion and Computation of Generalized Györi-Lovász Partition

L. Sunil Chandran, Yun Kuen Cheung, Davis Issac, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming
We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most O( √ mn), which is asymptotically optimal since we also demonstrate graphs with STC at least Ω( √ mn). We present a polynomial-time algorithm which computes a spanning tree with congestion O( √ mn • log n). We also
more » ... resent another algorithm for computing a spanning tree with congestion O( √ mn); this algorithm runs in sub-exponential time when m = ω(n log 2 n). For achieving the above results, an important intermediate theorem is generalized Győri-Lovász theorem. Chen et al. [8] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time O * (4 n ). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Θ(n) with high probability.

### Fixed-Parameter Tractability of the Weighted Edge Clique Partition Problem [article]

Andreas Emil Feldmann and Davis Issac and Ashutosh Rai
2020 arXiv   pre-print
The previously fastest algorithm known for ECP has a runtime of 2^O(k^2)n^O(1) [Issac, 2019].  ...

### Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs

Pinar Heggernes, Davis Issac, Juho Lauri, Paloma T. Lima, Erik Jan Van Leeuwen, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science
Issac, J. Lauri, P. T. Lima, and E.  ...  Issac, J. Lauri, P. T. Lima, and E. J. van Leeuwen 83:11 For our next result, we need to mention that every interval graph has a representation called an interval model.  ...

### Hadwiger's Conjecture for squares of 2-Trees [article]

L. Sunil Chandran, Davis Issac, Sanming Zhou
2019 arXiv   pre-print
Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split graphs. This observation implies that Hadwiger's conjecture for squares of chordal graphs is as difficult as the general case, since chordal graphs are a superclass of split graphs. Then we consider 2-trees which are a subclass of each of planar graphs,
more » ... erate graphs and chordal graphs. We prove that Hadwiger's conjecture is true for squares of 2-trees. We achieve this by proving the following stronger result: for any 2-tree T, its square T^2 has a clique minor of order χ(T^2) for which each branch set induces a path, where χ(T^2) is the chromatic number of T^2.

### Algorithms and Bounds for Very Strong Rainbow Coloring [article]

L. Sunil Chandran and Anita Das and Davis Issac and Erik Jan van Leeuwen
2018 arXiv   pre-print
A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number ((G)) of the graph. When proving upper bounds on (G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive
more » ... fferent colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number ((G)). In this paper, we give upper bounds on (G) for several graph classes, some of which are tight. These immediately imply new upper bounds on (G) for these classes, showing that the study of (G) enables meaningful progress on bounding (G). Then we study the complexity of the problem to compute (G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that (G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for (G). We also observe that deciding whether (G) = k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether (G) ≤ 3 nor to approximate (G) within a factor n^1-ε, unless P=NP.

### Online Ideology: A Comparison of Website Communication and Media Use

Cristina L. Byrne, Darin S. Nei, Jamie D. Barrett, Michael G. Hughes, Joshua L. Davis, Jennifer A. Griffith, Lauren N. Harkrider, Kimberly S. Hester, Amanda D. Angie, Issac C. Robledo, Shane Connelly, H. Dan O'Hair (+1 others)
2013 Journal of Computer-Mediated Communication
This study examined and compared the websites of ideological groups from a communications and media use perspective. Thirty-six websites with message boards categorized as either violent ideological, nonviolent ideological, or nonviolent nonideological were content coded for several distinguishing characteristics. The results indicated that group type was predicted by the type of information presented, the difficulty of becoming a member, and the amount of freedom members had on discussion
more » ... s. These findings suggest that characteristics of violent ideological group websites can be used to distinguish them from websites of both nonviolent ideological and nonideological groups. This study also provides a demonstration of a research methodology that can be used to naturally observe ideological groups via an online setting.
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