arXiv subjects
Publications and source records attributed to Sheikh Shakil Akhtar.
We study two related problems on simple, un-directed graphs: Edge Deletion to Cactus and Spanning Tree to Cactus. Edge Deletion to Cactus has been known to be NP-hard on general graphs at least since 1988. We show improved exact algorithms for the former and a polynomial time algorithm for the latter.
We study "space efficient" FPT algorithms for graph problems with limited memory. Let n be the size of the input graph and k be the parameter. We present algorithms that run in time f(k)*poly(n) and use g(k)*polylog(n) working space, where f and g are functions of k alone, for k-Path, MaxLeaf SubTree and Multicut in Trees. These algorithms are motivated by big-data settings where very large problem instances must be solved, and using poly(n) memory is prohibitively expensive. They are also theoretically interesting, since most of the standard methods tools, such as deleting a large set of vertices or edges, are unavailable, and we must a develop different way to tackle them.
A typical goal of research in combinatorial optimization is to come up with fast algorithms that find optimal solutions to a computational problem. The process that takes a real-world problem and extracts a clean mathematical abstraction of it often throws out a lot of "side information" which is deemed irrelevant. However, the discarded information could be of real significance to the end-user of the algorithm's output. All solutions of the same cost are not necessarily of equal impact in the real-world; some solutions may be much more desirable than others, even at the expense of additional increase in cost. If the impact, positive or negative, is mostly felt by some specific (minority) subgroups of the population, the population at large will be largely unaware of it. In this work we ask the question of finding solutions to combinatorial optimization problems that are "unbiased" with respect to a collection of specified subgroups of the total population.