Parameterized Complexity

The present work examines the effects of parameters of the input distribution on the average time
complexity of the singleton sort algorithm.. Previous work on this field has shown that for certain
algorithms such as sorting, the parameters of the input distribution singularly as well as interactively are
important factors, besides the size of the input which motivated us to study the parameterized complexity
of the singleton sort algorithm.

We possess the remarkable capacity to identify and understand relational similarities between the constituent parts of disparate wholes. The analogical mapping process underlying this capacity allows us to draw inferences about objects, actions, and events that we see as analogous to one another. This is believed to be a fundamental aspect of intelligence, found in language, creativity, problem solving, and reasoning. A better understanding of how the brain supports the analogical mapping process carries the potential to better understand the domains where it manifests. However, the most well known model of analogical mapping, called Structure-Mapping Theory, has been shown to be computationally intractable. This is problematic because, assuming that the brain is limited by finite computational resources, brain computation is constrained to be tractable. A solution to this problem has been proposed by van Rooij et al. (2008), who have proven that the computations postulated by SMT are tractable provided that the model parameter o (denoting the number of objects in the analogical match) is relatively small. This proposal yields the prediction that humans can quickly determine good analogical matches in situations where o is small. Moreover, it predicts that performance should deteriorate as o grows because of the inherent intractability of the postulated computations. In this thesis, we set out to test these predictions in a behavioral experiment. Participants were instructed to identify squares that correspond with one another on opposite sides of a divided screen. The results demonstrated that increasing the number of squares resulted in longer response times and less optimal analogical mappings. These findings are consistent with the model’s predictions, and provide support for the FPT-Cognition thesis.

For certain algorithms such as sorting and searching, the parameters of the input probability distribution, in addition to the size of the input, have been found to influence the complexity of the underlying algorithm. The present paper makes a statistical comparative study on parameterized complexity between linear and binary search algorithms for binomial inputs.

A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. It is well-known that the problem of finding a minimum sized (or k-sized in case of decision version of) feedback vertex set (FVS) is polynomial time solvable in (sub)-cubic graphs, in pseudo-forests (graphs where each component has at most one cycle) and mock-forests (graphs where each vertex is part of at most one cycle). In general graphs, it is known that the problem is N P-complete, and has an O * ((3.619) k) fixed-parameter algorithm and an O(k 2) kernel where k, the solution size, is the parameter. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure in the input. In particular, we show that FVS is fixed-parameter tractable, but is unlikely to have polynomial sized kernel when para-meterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper. When parameterized by k, the number of vertices, whose deletion results in a pseudo-forest, we give an O(k 6) vertices kernel improving from the previously known O(k 10) bound. When parameterized by the number k of vertices, whose deletion results in a mock-d-forest, we give a kernel consisting of O(k 3d+3) vertices and prove a lower bound of Ω(k d+2) vertices (under complexity theoretic assumptions). Mock-d-forest for a constant d is a mock-forest where each component has at most d cycles.

For certain algorithms such as sorting and searching, the parameters of the input probability distribution, in addition to the size of the input, have been found to influence the complexity of the underlying algorithm. The present paper makes a statistical comparative study on parameterized complexity between linear and binary search algorithms for binomial inputs.

This research paper is a statistical comparative study of a few average case asymptotically optimal sorting algorithms namely, Quick sort, Heap sort and K- sort. The three sorting algorithms all with the same average case complexity have been compared by obtaining the corresponding statistical bounds while subjecting these procedures over the randomly generated data from some standard discrete and continuous
probability distributions such as Binomial distribution, Uniform discrete and continuous distribution and Poisson distribution. The statistical analysis is well supplemented by the parameterized complexity analysis.

1. The PGExplainer is able to correctly identify the ground-truth motif responsible for node and graph classifica4 tion of a given GNN. 5 2. The PGExplainer is able to achieve a maximum AUC of 0.987 for node classification and a maximum AUC of 6 0.926 for graph classification, both with a standard deviation that is a maximum of 0.021. 7 3. The PGExplainer is able to generate explanations for the given node classification tasks in 24 milliseconds or 8 less, and graph classification tasks in 80 milliseconds or less. 9

Given a graph and an integer k, the biclique cover problem asks whether the edge-set of the given graph can be covered with at most k bicliques (complete bipartite subgraphs); the biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most k bicliques. Both problems are known to be NP-complete even if the given graph is bipartite. In this paper we investigate these two problems in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first problem is fixed-parameter tractable, while the second one is not fixed-parameter tractable unless P=NP.

Abstract: The framework of Bodlaender et al.(ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the" kernelization" of Leaf Out Branching (k) into a disjunction over n instances of size poly (k). Observing that Turing kernels are preserved by polynomial ...

The problem of deciding whether the edge-set of a given graph can be partitioned into at most k cliques is well known to be NP-complete. In this paper we investigate this problem from the point of view of parameterized complexity. We show that this problem is fixed parameter tractable if we choose the number of cliques as parameter. In particular, we show that in polynomial time, a kernel bounded by k2 can be obtained, where k is the number of cliques. We also give an O(2((k+3) log k)/2n) algorithm for this problem in K4-free graphs.

The parameterized complexity of the face cover problem is considered. The input to this problem is a plane graph, G ,o f ordern. The question asked is whether, for any fixed k ,t here exists a set of k or fewer vertices whose boundaries collectively cover (contain) every vertex in G.T he fastest previously-published face cover algorithm is achieved with