A beer graph is an undirected graph G, in which each edge has a positive weight and some vertices have a beer store. A beer path between two vertices u and v in G is any path in G between u and v that visits at least one beer store. We show that any outerplanar beer graph G with n vertices can be preprocessed in O(n) time into a data structure of size O(n), such that for any two query vertices u and v, (i) the weight of the shortest beer path between u and v can be reported in $$O(\alpha (n))$$ time (where $$\alpha (n)$$ is the inverse Ackermann function), and (ii) the shortest beer path between u and v can be reported in O(L) time, where L is the number of vertices on this path. Note that the running time for (ii) does not depend on the number of vertices of G. Both results are optimal, even when G is a beer tree (i.e., a beer graph whose underlying graph is a tree).

A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D1 and D2 are disjoint if their intersection is empty (i.e., no interval of D1 intersects any interval of D2). There can be three different relations between two disjoint 2-intervals; namely, preceding (<), nested (⊏) and crossing (≬). Two 2-intervals D1 and D2 are called R-comparable for some R∈{<,⊏,≬}, if either D1RD2 or D2RD1. A set D of disjoint 2-intervals is R-comparable, for some R⊆{<,⊏,≬} and R≠∅, if every pair of 2-intervals in D are R-comparable for some R∈R. Given a set of 2-intervals and some R⊆{<,⊏,≬}, the objective of the 2-interval pattern problem is to find a largest subset of 2-intervals that is R-comparable. The 2-interval pattern problem is known to be W[1]-hard when |R|=3 and NP-hard when |R|=2 (except for R={<,⊏}, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing that it is W[1]-hard for both R={⊏,≬} and R={<,≬} (when parameterized by the size of an optimal solution). This answers the open question posed by Vialette ((2008) [22]).

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any $ε>0$, there exists a $(2+ε)$-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first $O(1)$-approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a $(2+ε)$-approximation for the problem with "diagonal-anchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).

Consider k robots initially located at the centroid of an equilateral triangle T of sides of length one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T. Each robot can move anywhere in T independently of other robots with maximum speed one. The objective is to minimize the evacuation time, which is defined as the time required for all k robots to reach the exit. We consider the face-to-face communication model for the robots: a robot can communicate with another robot only when they meet in T. In this paper, we give upper and lower bounds for the face-to-face evacuation time by k robots. We show that for any k, any algorithm for evacuating k >= 1 robots from T requires at least sqrt(3) time. This bound is asymptotically optimal, as we show that a straightforward strategy of evacuation by k robots gives an upper bound of sqrt(3) + 3/k. For k = 3, 4, 5, 6, we show significant improvements on the obvious upper bound by giving algorithms with evacuation times of 2.0887, 1.9816, 1.876, and 1.827, respectively. For k = 2 robots, we give a lower bound of 1 + 2/sqrt(3) ~= 2.154, and an algorithm with upper bound of 2.3367 on the evacuation time.

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.