Maximizing Revenues for Online-Dial-a-Ride

In the classic Dial-a-Ride Problem, a server travels in some metric space to serve requests for rides. Each request has a source, destination, and release time. We study a variation of this problem where each request also has a revenue that is earned if the request is satisfied. The goal is to serve requests within a time limit such that the total revenue is maximized. We first prove that the version of this problem where edges in the input graph have varying weights is NP-complete. We also prove that no algorithm can be competitive for this problem. We therefore consider the version where edges in the graph have unit weight and develop a 2-competitive algorithm for this problem

On the complexity of strongly connected components in directed hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

Approximate solution of NP optimization problems

AbstractThis paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility

A novel structure-based encoding for machine-learning applied to the inference of SH3 domain specificity

MOTIVATION: Unravelling the rules underlying protein-protein and protein-ligand interactions is a crucial step in understanding cell machinery. Peptide recognition modules (PRMs) are globular protein domains which focus their binding targets on short protein sequences and play a key role in the frame of protein-protein interactions. High-throughput techniques permit the whole proteome scanning of each domain, but they are characterized by a high incidence of false positives. In this context, there is a pressing need for the development of in silico experiments to validate experimental results and of computational tools for the inference of domain-peptide interactions. RESULTS: We focused on the SH3 domain family and developed a machine-learning approach for inferring interaction specificity. SH3 domains are well-studied PRMs which typically bind proline-rich short sequences characterized by the PxxP consensus. The binding information is known to be held in the conformation of the domain surface and in the short sequence of the peptide. Our method relies on interaction data from high-throughput techniques and benefits from the integration of sequence and structure data of the interacting partners. Here, we propose a novel encoding technique aimed at representing binding information on the basis of the domain-peptide contact residues in complexes of known structure. Remarkably, the new encoding requires few variables to represent an interaction, thus avoiding the 'curse of dimension'. Our results display an accuracy >90% in detecting new binders of known SH3 domains, thus outperforming neural models on standard binary encodings, profile methods and recent statistical predictors. The method, moreover, shows a generalization capability, inferring specificity of unknown SH3 domains displaying some degree of similarity with the known data

A novel approach to represent and compare RNA secondary structures

Structural information is crucial in ribonucleic acid (RNA) analysis and functional annotation; nevertheless, how to include such structural data is still a debated problem. Dot-bracket notation is the most common and simple representation for RNA secondary structures but its simplicity leads also to ambiguity requiring further processing steps to dissolve. Here we present BEAR (Brand nEw Alphabet for RNA), a new context-aware structural encoding represented by a string of characters. Each character in BEAR encodes for a specific secondary structure element (loop, stem, bulge and internal loop) with specific length. Furthermore, exploiting this informative and yet simple encoding in multiple alignments of related RNAs, we captured how much structural variation is tolerated in RNA families and convert it into transition rates among secondary structure elements. This allowed us to compute a substitution matrix for secondary structure elements called MBR (Matrix of BEAR-encoded RNA secondary structures), of which we tested the ability in aligning RNA secondary structures. We propose BEAR and the MBR as powerful resources for the RNA secondary structure analysis, comparison and classification, motif finding and phylogeny

A neural strategy for the inference of SH3 domain-peptide interaction specificity

The SH3 domain family is one of the most representative and widely studied cases of so-called Peptide Recognition Modules (PRM). The polyproline II motif PxxP that generally characterizes its ligands does not reflect the complex interaction spectrum of the over 1500 different SH3 domains, and the requirement of a more refined knowledge of their specificity implies the setting up of appropriate experimental and theoretical strategies. Due to the limitations of the current technology for peptide synthesis, several experimental high-throughput approaches have been devised to elucidate protein-protein interaction mechanisms. Such approaches can rely on and take advantage of computational techniques, such as regular expressions or position specific scoring matrices (PSSMs) to pre-process entire proteomes in the search for putative SH3 targets. In this regard, a reliable inference methodology to be used for reducing the sequence space of putative binding peptides represents a valuable support for molecular and cellular biologists

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure

On Approximating Restricted Cycle Covers

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change