Reoptimization of Some Maximum Weight Induced Hereditary Subgraph Problems

The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Π, an optimal solution OPT for Π in I and an instance I′ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Π in I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph and max split subgraph under vertex insertions and deletion

The cyclic-routing UAV problem is PSPACE-complete

© 2015, Springer-Verlag Berlin Heidelberg. Consider a finite set of targets, with each target assigned a relative deadline, and each pair of targets assigned a fixed transit flight time. Given a flock of identical UAVs, can one ensure that every target is repeatedly visited by some UAV at intervals of duration at most the target’s relative deadline? The Cyclic-Routing UAV Problem (cr-uav) is the question of whether this task has a solution. This problem can straightforwardly be solved in PSPACE by modelling it as a network of timed automata. The special case of there being a single UAV is claimed to be NP-complete in the literature. In this paper, we show that the cr-uav Problem is in fact PSPACE-complete even in the single-UAV case

The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality

In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened β-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-costs are restricted to the values 1 and 1+ for an arbitrary small > 0), Steiner tree reoptimization still is NP-hard for several different types of local modifications, and even APX-hard for some of them.As for the upper bounds, for some local modifications, we design lineartime (1/2+β)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (β = 1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2β-approximation algorithm for the classical Steiner tree problem on such instances, which might be of independent interest since it improves over the previously best known ratio for any β < 1/2 + ln(3)/4 0.775

On the advice complexity of the online L(2,1)-coloring problem on paths and cycles

In an L(2,1)-coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L(2,1)-coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance two among the previously revealed vertices. For this, we restrict our attention to paths and cycles. Offline, paths can easily be colored within the range {0,…,4} of colors. We prove that, considering deterministic algorithms in an online setting, the range {0,…,6} is necessary and sufficient while a simple greedy strategy needs range {0,…,7}. Advice complexity is a recently developed framework to measure the complexity of online problems. The idea is to measure how many bits of advice about the yet unknown parts of the input an online algorithm needs to compute a solution of a certain quality. We show a sharp threshold on the advice complexity of the online L(2,1)-coloring problem on paths and cycles. While achieving color range {0,…,6} does not need any advice, improving over this requires a number of advice bits that is linear in the size of the input. Thus, the L(2,1)-coloring problem is the first known example of an online problem for which sublinear advice does not help. We further use our advice complexity results to prove that no randomized online algorithm can achieve a better expected competitive ratio than (1−δ)5/4, for any δ > 0

Improved Approximations for TSP with Simple Precedence Constraints

International audienc

Building Phylogeny with Minimal Absent Words

International audienc