Dynamic Parameterized Problems

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740^k*n^{O(1)} (polynomial space) and 1.1277^k*n^{O(1)} (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667^k*n^{O(1)} running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2^k*n^{O(1)} running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture

Lossy Kernels for Graph Contraction Problems

We study some well-known graph contraction problems in the recently introduced framework of lossy kernelization. In classical kernelization, given an instance (I,k) of a parameterized problem, we are interested in obtaining (in polynomial time) an equivalent instance (I\u27,k\u27) of the same problem whose size is bounded by a function in k. This notion however has a major limitation. Given an approximate solution to the instance (I\u27,k\u27), we can say nothing about the original instance (I,k). To handle this issue, among others, the framework of lossy kernelization was introduced. In this framework, for a constant alpha, given an instance (I,k) we obtain an instance (I\u27,k\u27) of the same problem such that, for every c>1, any c-approximate solution to (I\u27,k\u27) can be turned into a (c*alpha)-approximate solution to the original instance (I, k) in polynomial time. Naturally, we are interested in a polynomial time algorithm for this task, and further require that |I\u27| + k\u27 = k^{O(1)}. Akin to the notion of polynomial time approximation schemes in approximation algorithms, a parameterized problem is said to admit a polynomial size approximate kernelization scheme (PSAKS) if it admits a polynomial size alpha-approximate kernel for every approximation parameter alpha > 1. In this work, we design PSAKSs for Tree Contraction, Star Contraction, Out-Tree Contraction and Cactus Contraction problems. These problems do not admit polynomial kernels, and we show that each of them admit a PSAKS with running time k^{f(alpha)}|I|^{O(1)} that returns an instance of size k^{g(alpha)} where f(alpha) and g(alpha) are constants depending on alpha

On the Parameterized Complexity of Simultaneous Deletion Problems

For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, ldots, F_alpha)-Deletion problem. In the latter problem, we are given an alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 leq i leq alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = ldots = F_alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and alpha, and can be solved in O(2^{O(alpha k)}n^{O(1)}) time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, ldots, F_alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i\u27s is the class of bipartite graphs and the rest (if any) are forests. We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = ldots = F_alpha is the family of all forests then the problem is fixed-parameter tractable parameterized by k and alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard

Mechanoaccumulative elements of the mammalian actin cytoskeleton

To change shape, divide, form junctions, and migrate, cells reorganize their cytoskeletons in response to changing mechanical environments [1-4]. Actin cytoskeletal elements, including myosin II motors and actin crosslinkers, structurally remodel and activate signaling pathways in response to imposed stresses [5-9]. Recent studies demonstrate the importance of force-dependent structural rearrangement of α-catenin in adherens junctions [10] and vinculin's molecular clutch mechanism in focal adhesions [11]. However, the complete landscape of cytoskeletal mechanoresponsive proteins and the mechanisms by which these elements sense and respond to force remain to be elucidated. To find mechanosensitive elements in mammalian cells, we examined protein relocalization in response to controlled external stresses applied to individual cells. Here, we show that non-muscle myosin II, α-actinin, and filamin accumulate to mechanically stressed regions in cells from diverse lineages. Using reaction-diffusion models for force-sensitive binding, we successfully predicted which mammalian α-actinin and filamin paralogs would be mechanoaccumulative. Furthermore, a Goldilocks zone must exist for each protein where the actin-binding affinity must be optimal for accumulation. In addition, we leveraged genetic mutants to gain a molecular understanding of the mechanisms of α-actinin and filamin catch-bonding behavior. Two distinct modes of mechanoaccumulation can be observed: a fast, diffusion-based accumulation and a slower, myosin II-dependent cortical flow phase that acts on proteins with specific binding lifetimes. Finally, we uncovered cell-type and cell-cycle-stage-specific control of the mechanosensation of myosin IIB, but not myosin IIA or IIC. Overall, these mechanoaccumulative mechanisms drive the cell's response to physical perturbation during proper tissue development and disease

In Vivo Antidiabetic Potential Of Niosome Naringin Nanoconjugate In Streptozotocin Induced Diabetic Mice

Diabetes mellitus remains a significant global health concern, necessitating the exploration of novel therapeutic strategies. In this study, we investigated the in vivo antidiabetic potential of Niosome Naringin Nanoconjugate, a nanoformulation designed to enhance the bioavailability and therapeutic efficacy of naringin, a naturally occurring flavonoid with reported antidiabetic properties. Male Wistar rats were induced with diabetes mellitus using streptozotocin and treated with Niosome Naringin Nanoconjugate orally for a specified duration. The effects of the nanoconjugate on glucose tolerance, fasting blood glucose levels, insulin resistance, and lipid profile were evaluated. Our results demonstrate that Niosome Naringin Nanoconjugate significantly improved glucose tolerance, reduced fasting blood glucose levels, and ameliorated insulin resistance compared to diabetic control groups. Additionally, the nanoconjugate exhibited enhanced bioavailability and prolonged circulation time, suggesting potential for sustained therapeutic effects. These findings highlight the promising antidiabetic efficacy of Niosome Naringin Nanoconjugate and underscore its potential as a novel therapeutic agent for the management of diabetes mellitus. Further investigations, including long-term safety assessments and clinical trials, are warranted to fully elucidate its clinical applicability. Overall, this study contributes valuable insights into the development of innovative nanomedicine approaches for diabetes management, with Niosome Naringin Nanoconjugate emerging as a promising candidate for future therapeutic interventions