Online Algorithms with Advice for the -search Problem

In the online search problem, a seller seeks to find the maximum price from a sequence of prices p1, p2,…, pn that is revealed in a piece-wise manner. The bound for all prices is well known in advance with m ≤ pί ≤ M. In the online k-search problem, the seller seeks to find the k maximum out of the n prices. In this paper, we present a tight bound of [Formula Presented] on the advice complexity of optimal online algorithms for online k-search. We also provide online algorithms with advice that use less than the required number of bits and compute the performance guarantee. Although it is natural to expect improvement due to the additional power of advice, we are interested to identify the relationship of additional information with respect to the improvement. We show that with 1 bit of advice, we can already surpass the quality of the best possible deterministic algorithm for online 2-search. We also provide a set of online algorithms, ALGί, that utilizes [Formula Presented] advice bits with a competitive ratio of (formula presented). We show that increasing the amount of advice improves the solution quality of the algorithm. Moreover, we compare the power of advice and randomization. We show that for some identified minimum number of advice bits, the lower bound on the competitive ratio of online algorithms with advice is better than any deterministic and randomized algorithm for online k-search

Reoptimization of the Consensus Pattern Problem under Pattern Length Modification

In Bioinformatics, finding conserved regions in genomic sequences remains to be a challenge not just because of the increasing size of genomic data collected but because of the hardness of the combinatorial model of the problem. One problem formulation is called the Consensus Pattern Problem (CPP). Given a set of t n-length strings S = {S1,..., St} defined over some constant size alphabet Σ and an integer l, where l ≤ n, the objective of CPP is to find an l-length string v and a set of l-length substrings si of each Si in S such that the total sum of d(si, v) is minimized for all 1 ≤ i ≤ t. Here d(x, y) denotes the Hamming distance between the two strings x and y. It is known that CPP is NP-hard i.e., unless P = NP, there is no polynomial-time algorithm that produces an optimal solution for CPP. In this study, we investigate a combinatorial setting called reoptimization in finding an approximate solution for this problem. We seek to identify whether a specific additional information can help in solving CPP. Specifically, we deal with the following reoptimization scenario. Suppose we have an optimal l-length consensus substring of a given set of sequences S. How can this information be beneficial in obtaining an (l + k)-length and (l – k)-length consensus for S? In this paper, we show that the reoptimization variant of the problem is still computationally hard even with k = 1. In response, we present four algorithms that make use of the given optimal solution – we prove that the first three algorithms produce solutions with quality that is bounded from above by an additive error that grows as the parameter k increases, while the fourth algorithm achieves a guaranteed approximation ratio. It has been shown that there is no efficient polynomial-time approximation scheme for CPP (Boucher 2015). In this paper, we show that we can save ( − ( + ) + ) ( ( + ) / ) steps in computation from the original running time of the known polynomial-time approximation scheme for CPP