Fast Dynamic Graph Algorithms for Parameterized Problems

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are $o(|G|)$, that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT). We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size $k$. These dynamic graphs achieve almost the best possible update time $O(\mathrm{poly}(k)\log n)$ and the query time $O(f(\mathrm{poly}(k),k))$, where $f(n,k)$ is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem. We also give a dynamic graph for Chromatic Number parameterized by the solution size of Cluster Vertex Deletion, and a dynamic graph for bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming the parameter is a constant, each dynamic graph can be updated in $O(\log n)$ time and can compute a solution in $O(1)$ time. These results are obtained by another approach.Comment: SWAT 2014 to appea

Revisiting the tree Constraint

International audienceThis paper revisits the tree constraint introduced in [2] which partitions the nodes of a n-nodes, m-arcs directed graph into a set of node-disjoint anti-arborescences for which only certain nodes can be tree roots. We introduce a new filtering algorithm that enforces generalized arc-consistency in O(n + m) time while the original filtering algorithm reaches O(nm) time. This result allows to tackle larger scale problems involving graph partitioning

Efficient Splitting and Merging Algorithms for Order Decomposable Problems

AbstractLet S be a set whose items are sorted with respect to d>1 total orders ≺1, …, ≺d, and which is subject to dynamic operations, such as insertions of a single item, deletions of a single item, split and concatenate operations performed according to any chosen order ≺i (1⩽i⩽d). This generalizes to dimension d>1 the notion of concatenable data structures, such as the 2-3-trees, which support splits and concatenates under a single total order. The main contribution of this paper is a general and novel technique for solving order decomposable problems on S which yields new and efficient concatenable data structures for dimension d>1. By using our technique we maintain S with the time bounds: O(logn) for the insertion or the deletion of a single item, where n is the number of items currently in S; n1−1/d for splits and concatenates along any order, and for rectangular range queries. The space required is O(n). We provide several applications of our technique. Namely, we present new multidimensional data structures implementing two-dimensional priority queues, two-dimensional search trees, and concatenable interval trees; these data structures allow us to improve many previously known results on decomposable problems under split and concatenate operations, such as membership query, minimum-weight item, range query, convex hulls, and Voronoi diagrams

Algoritmi e strutture dati (seconda edizione)

Questo libro offre un'introduzione allo studio degli algoritmi e delle strutture dati, cercando di conciliare comprensibilità, chiarezza di esposizione e rigore matematico. Particolare enfasi è posta sull'astrazione delle tecniche e delle metodologie generali di progetto e analisi di algoritmi, stimolandone la comprensione intuitiva dei principi fondamentali. Il libro è concepito soprattutto per corsi universitari delle facoltà di ingegneria e di scienze matematiche, fisiche e naturali. Il testo, pur essendo indipendente dalla scelta di un particolare linguaggio di programmazione, adotta un approccio orientato agli oggetti sia nella descrizione delle strutture dati che nello pseudocodice utilizzato per descrivere gli algoritmi. In tal modo, pur astraendo dai dettagli implementativi di basso livello, gli algoritmi presentati non risultano troppo distanti da una loro reale implementazione