We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval and minisum approval voting rules, we provide a dichotomous result, resolving the parameterized complexity of this problem for all subsets of five natural parameters considered (by showing either FPT or W[1]-hardness for all subsets of parameters). For the approval voting rule, we resolve the parameterized complexity of this problem for all subsets of parameters except one. We also study approximation algorithms for this problem. We show that there does not exist any alpha(.) factor approximation algorithm for approval and net approval voting rules, for any computable function alpha(.), unless P=NP. For the minisum voting rule, we provide a pseudopolynomial (1+eps) factor approximation algorithm.
In the minimax notion of distance, the problem of finding a closest string in the presence of outliers was initiated in @cite_29 , and was further refined in @cite_30 . The work in @cite_30 demonstrates hardness of approximation for variations involving outliers. With the minisum notion of distance, the study of finding the best string with respect to outliers was studied in @cite_7 . In contrast with the minimax rule, this work shows a PTAS for .
@cite_30 @cite_29 @cite_7
2180828828 2030393641 2049696330
Many problems in bioinformatics are about finding strings that approximately represent a collection of given strings. We look at more general problems where some input strings can be classified as outliers. The Close to Most Strings problem is, given a set S of the same-length strings, and a parameter d, find a string x that maximizes the number of ''non-outliers'' within Hamming distance d of x. We prove that this problem has no polynomial-time approximation scheme (PTAS) unless NP has randomized polynomial-time algorithms, correcting a decade-old erroneous proof made previously in the literature. The Most Strings with Few Bad Columns problem is to find a maximum-size subset of input strings so that the number of non-identical positions is at most k; we show it has no PTAS unless P=NP. We also observe Closest to k Strings has no efficient PTAS (EPTAS) unless a parameterized complexity hierarchy collapses. In sum, outliers help model problems associated with using biological data, but we show the problem of finding an approximate solution is computationally difficult. Background Given n strings s1, …, sn each of length l and a nonnegative integer d, the CLOSEST STRING problem asks to find a center string s such that none of the input strings has Hamming distance greater than d from s. Finding a common pattern in many – but not necessarily all – input strings is an important task that plays a role in many applications in bioinformatics. RNA splicing is a cellular process driven by the interaction between numerous regulatory sequences and binding sites, however, such interactions have been primarily explored by laboratory methods since computational tools largely ignore the relationship between different splicing elements. Current computational methods identify either splice sites or other regulatory sequences, such as enhancers and silencers. We present a novel approach for characterizing co-occurring relationships between splice site motifs and splicing enhancers. Our approach relies on an efficient algorithm for approximately solving Consensus Sequence with Outliers, an NP-complete string clustering problem. In particular, we give an algorithm for this problem that outputs near-optimal solutions in polynomial time. To our knowledge, this is the first formulation and computational attempt for detecting co-occurring sequence elements in RNA sequence data. Further, we demonstrate that SeeSite is capable of showing that certain ESEs are preferentially associated with weaker splice sites, and that there exists a co-occurrence relationship with splice site motifs.