The interval constrained 3-coloring problem

In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known that it is NP-complete, if the number of colors is part of the input and that the problem is solvable in polynomial time, if the number of colors is at most 2. We also show that it is hard to satisfy almost all of the constraints for a feasible instance.Comment: minor revisio

An Exact Algorithm for Robust Network Design

Modern life heavily relies on communication networks that operate efficiently. A crucial issue for the design of communication networks is robustness with respect to traffic fluctuations, since they often lead to congestion and traffic bottlenecks. In this paper, we address an NP-hard single commodity robust network design problem, where the traffic demands change over time. For k different times of the day, we are given for each node the amount of single-commodity flow it wants to send or to receive. The task is to determine the minimum-cost edge capacities such that the flow can be routed integrally through the net at all times. We present an exact branch-and-cut algorithm, based on a decomposition into biconnected network components, a clever primal heuristic for generating feasible solutions from the linear-programming relaxation, and a general cutting-plane separation routine that is based on projection and lifting. By presenting extensive experimental results on realistic instances from the literature, we show that a suitable combination of these algorithmic components can solve most of these instances to optimality. Furthermore, cutting-plane separation considerably improves the algorithmic performance

Single-Sink Fractionally Subadditive Network Design

We study a generalization of the Steiner tree problem, where we are given a weighted network G together with a collection of k subsets of its vertices and a root r. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for k=2, and give a 3/2-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known O(log n)-approximation. Despite these obstacles, we conjecture that this problem should have an O(1)-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path

Node Connectivity Augmentation of Highly Connected Graphs

Node-connectivity augmentation is a fundamental network design problem. We are given a $k$-node connected graph $G$ together with an additional set of links, and the goal is to add a cheap subset of links to $G$ to make it $(k+1)$-node connected. In this work, we characterize completely the computational complexity status of the problem, by showing hardness for all values of $k$ which were not addressed previously in the literature. We then focus on $k$-node connectivity augmentation for $k=n-4$, which corresponds to the highest value of $k$ for which the problem is NP-hard. We improve over the previously best known approximation bounds for this problem, by developing a $\frac{3}{2}$-approximation algorithm for the weighted setting, and a $\frac{4}{3}$-approximation algorithm for the unweighted setting

The phytochelatin synthase from Nitella mucronata (Charophyta) plays a role in the homeostatic control of iron(II)/(III)

Although some charophytes (sister group to land plants) have been shown to synthesize phytochelatins (PCs) in response to cadmium (Cd), the functional characterization of their phytochelatin synthase (PCS) is still completely lacking. To investigate the metal response and the presence of PCS in charophytes, we focused on the species Nitella mucronata. A 40 kDa immunoreactive PCS band was revealed in mono-dimensional western blot by using a polyclonal antibody against Arabidopsis thaliana PCS1. In two-dimensional western blot, the putative PCS showed various spots with acidic isoelectric points, presumably originated by post-translational modifications. Given the PCS constitutive expression in N. mucronata, we tested its possible involvement in the homeostasis of metallic micronutrients, using physiological concentrations of iron (Fe) and zinc (Zn), and verified its role in the detoxification of a non-essential metal, such as Cd. Neither in vivo nor in vitro exposure to Zn resulted in PCS activation and PC significant biosynthesis, while Fe(II)/(III) and Cd were able to activate the PCS in vitro, as well as to induce PC accumulation in vivo. While Cd toxicity was evident from electron microscopy observations, the normal morphology of cells and organelles following Fe treatments was preserved. The overall results support a function of PCS and PCs in managing Fe homeostasis in the carophyte N. mucronata

Approximating Weighted Tree Augmentation via Chvatal-Gomory Cuts

The weighted tree augmentation problem (\WTAP) is a fundamental network design problem. We are given an undirected tree $G = (V,E)$ with $n = |V|$ nodes, an additional set of edges $L$ called \emph{links} and a cost vector $c \in \R^L_{\geq 1}$. The goal is to choose a minimum cost subset $S \subseteq L$ such that $G = (V, E \cup S)$ is $2$-edge-connected. In the unweighted case, that is, when we have $c_\ell = 1$ for all $\ell \in L$, the problem is called the tree augmentation problem (\TAP). Both problems are known to be \APX-hard, and the best known approximation factors are $2$ for \WTAP by (Frederickson and J\'aJ\'a, '81) and $\tfrac{3}{2}$ for \TAP due to (Kortsarz and Nutov, TALG '16). Adjashvili (SODA~'17) recently presented an \approx 1.96418+\eps-approximation algorithm for \WTAP\ for the case where all link costs are bounded by a constant. This is the first approximation with a better guarantee than $2$ that does not require restrictions on the structure of the tree or the links. In this paper, we improve Adjiashvili's approximation to a \tfrac{3}{2}+\eps-approximation for \WTAP under the bounded cost assumption. We achieve this by introducing a strong \LP that combines \zhcgcuts for the standard \LP for the problem with bundle constraints from Adjiashvili. We show that our \LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved performance guarantee of \tfrac{3}{2}+\eps, which is asymptotically on par with the result by Kortsarz and Nutov. Our result also is the best-known \LP-relative approximation algorithm for \TAP.Non UBCUnreviewedAuthor affiliation: University of WaterlooResearche

Exponentiality of the exchange algorithm for finding another room-partitioning

Let T be a triangulated surface given by the list of vertex-triples of its triangles, called rooms. A room-partitioning for T is a subset R of the rooms such that each vertex of T is in exactly one room in R. Given a room-partitioning R for T, the exchange algorithm walks from room to room until it finds a second different room-partitioning R'. In fact, this algorithm generalizes the Lemke-Howson algorithm for finding a Nash equilibrium for two-person games. In this paper, we show that the running time of the exchange algorithm is not polynomial relative to the number of rooms, by constructing a sequence of (planar) instances, in which the algorithm walks from room to room an exponential number of times. We also show a similar result for the problem of finding a second perfect matching in Eulerian graphs. (C) 2012 Elsevier B.V. All rights reserved