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

Parameterized Complexity of Perfectly Matched Sets

For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(?k)? n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^?(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs

MYCOLOGICAL PROFILE AND ANTIFUNGAL SENSITIVITY OF INFECTIVE KERATITIS IN A TERTIARY CARE HOSPITAL OF SOUTHERN ODISHA

Background: Infective keratitis is the second major cause of blindness next to cataract. Mycotic keratitis is an important ophthalmologic problem especially in developing countries including India. Fungal infection involving cornea is a fatal condition which needs early diagnosis and treatment to save the patient's eye. Though studies on mycotic keratitis have been reported from different part of India, to the best of our knowledge this study showing antifungal susceptibility is the first to be reported from Southern Odisha. Objective: The purpose of this study was to study epidemiological characteristics, predisposing factors, fungal etiology and antifungal susceptibility of common fungal isolates in infective keratitis cases. Materials and Methods: A prospective study was conducted from November 2015 to October2017 in the Department of Microbiology and Ophthalmology M.K.C.G Medical College and Hospital. Relevant information was recorded using standard proforma of keratitis cases. Corneal scrapings were collected under strict aseptic conditions and subjected to10% KOH mount, Gram stain and culture. Identification of fungal agents were done as per standard microbiological procedures. An antifungal sensitivity test was done by microbroth dilutions as per CLSI reference method. Results: Over a period of two years 149 patients of infective keratitis were evaluated. Microbiological diagnosis of mycotic keratitis was established in 39 (26.17%) cases. Filamentous fungi were isolated more often than yeasts. The most frequently encountered filamentous fungi and yeasts were Aspergillus spp. 14(35.89%) and Candida albicans 7 (17.94%) respectively. Males were more commonly affected and were mostly in the age group of 46-60 years. Ocular trauma due to vegetative matter was the most common predisposing factor. Natamycin was the most effective antifungal against filamentous fungi and amphotericin B was most effective for Candida albicans. Conclusion: Because of serious consequences of mycotic keratitis, it is very important to know the exact etiological agents and effective antifungals to save the eye of the patients. So laboratory confirmation should be undertaken and fungal infection should be ruled out before prescribing antimicrobial agents

Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems

For a positive integer c, a graph G is said to be c-closed if every pair of non-adjacent vertices in G have at most c-1 neighbours in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by Fox et al. [ICALP `18 and SICOMP `20]. Koana et al. [ESA `20] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set and (Threshold) Dominating Set, and showed that each of these problems admits a polynomial kernel, either w.r.t. the parameter k+c or w.r.t. the parameter k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable w.r.t. the parameter k+cl(G), whereas Partial Dominating Set, parameterized by k is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs, unless NP ? co-NP/poly

A Polynomial Kernel for Bipartite Permutation Vertex Deletion

In a permutation graph, vertices represent the elements of a permutation, and edges represent pairs of elements that are reversed by the permutation. In the Permutation Vertex Deletion problem, given an undirected graph G and an integer k, the objective is to test whether there exists a vertex subset S ? V(G) such that |S| ? k and G-S is a permutation graph. The parameterized complexity of Permutation Vertex Deletion is a well-known open problem. Bo?yk et al. [IPEC 2020] initiated a study towards this problem by requiring that G-S be a bipartite permutation graph (a permutation graph that is bipartite). They called this the Bipartite Permutation Vertex Deletion (BPVD) problem. They showed that the problem admits a factor 9-approximation algorithm as well as a fixed parameter tractable (FPT) algorithm running in time ?(9^k |V(G)|?). And they posed the question {whether BPVD admits a polynomial kernel.} We resolve this question in the affirmative by designing a polynomial kernel for BPVD. In particular, we obtain the following: Given an instance (G,k) of BPVD, in polynomial time we obtain an equivalent instance (G\u27,k\u27) of BPVD such that k\u27 ? k, and |V(G\u27)|+|E(G\u27)| ? k^{?(1)}

EFFECT OF EDUCATIONAL INTERVENTION MEASURES ON KNOWLEDGE ABOUT RABIES AND ITS PREVENTIVE MEASURES AMONG FINAL YEAR NURSING STUDENTS OF A TERTIARY CARE HOSPITAL IN CENTRAL INDIA

Background: Rabies continues to be a major public health challenge with around 55,000 deaths every year. Amongst the health care providers nursing personnel are often the first point of contact and hence need to be well trained in the management of rabies cases. Methods: The present study was an educational intervention study conducted among 100 final year nursing students of a Medical College Hospital to assess the knowledge regarding rabies and its transmission, first aid measures undertaken, and pre and post exposure prophylaxis measures employed to prevent the infection. Results: 66% of the students knew about the signs and symptoms of the disease, post intervention this increased to 87%. Knowledge regarding animal bites which transmit rabies improved by 86 % mode of transmission by 49 % and first aid measures undertaken following an animal bite by 12%. 15% of the students knew about the correct site and route of PEP; post intervention 91% knew about it, 87% increase was observed as regards the dose of vaccine to be administered and 73% students correctly knew about the PEP schedule post educational intervention. Knowledge regarding groups / individuals who need to receive pre-exposure prophylaxis increased by 33% and that of the schedule of pre-exposure prophylaxis by 53%. The mean pre-intervention score was 6.95 and mean post-intervention score was 13.51; the results being statistically significant. Conclusion: Substantial improvement in knowledge about the disease was noted amongst the nursing students following the educational intervention session

Study of Ethno-Medicinal Wild Edible Leafy Vegetables Used by Local Tribes in District Jashpur, Chhattisgarh.

The leaves of numerous cultivated and wild plants are used as vegetables in India. They are incredibly simple to grow and offer a very high preventive food value. The natural vegetation of Chhattisgarh plays a crucial role in the economy and way of life of the tribal and ethnic communities. In remote areas of Chhattisgarh, leafy vegetables are crucial to the nutritional needs of the native and tribal population. In addition to providing a substantial amount of food, leafy vegetables also significantly contribute to the population's nutrition all year long. An extensive survey of the leafy vegetables consumed by the tribal community in different parts of the Sanna district of Jashpur Chhattisgarh was conducted as part of the current investigation. According to this study, the state of Chhattisgarh uses roughly 36 species of plants as a source of leafy vegetables. There are 35 leafy plants used in ethnomedicine. they were dispersed across many life forms. They included 4 species of trees, 3 species of shrubs, 23 species of herbs, and 5 species of climbers, but it was astonishing to see during the survey that nearly 35 plant species were discovered to be employed in various parts of Sanna. Numerous vegetables are also used by the locals as a source of herbal treatments for ailments like arthritis, jaundice, colds and coughs, fever, headaches, bronchial asthma, ulcers, skin problems, and other

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the linear programming dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT and ATT are fixed-parameter tractable, they can be solved in 2^{O(k log k)} n^{O(1)} time and 2^{O(k)} n^{O(1)} time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2^{o(sqrt{k})} n^{O(1)} time under the Exponential-Time Hypothesis