A framework for ETH-Tight algorithms and lower bounds in geometric intersection graphs

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2O(n1−1/d) for any fixed dimension d ≥ 2 for many well known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2Ω(n1−1/d) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs

European agricultural landscapes, common agricultural policy and ecosystem services: a review

Since the 1950s, intensification and scale enlargement of agriculture have changed agricultural landscapes across Europe. The intensification and scale enlargement of farming was initially driven by the large-scale application of synthetic fertilizers, mechanization and subsidies of the European Common Agricultural Policy (CAP). Then, after the 1990s, a further intensification and scale enlargement, and land abandonment in less favored areas was caused by globalization of commodity markets and CAP reforms. The landscape changes during the past six decades have changed the flows and values of ecosystem services. Here, we have reviewed the literature on agricultural policies and management, landscape structure and composition, and the contribution of ecosystem services to regional competitiveness. The objective was to define an analytical framework to determine and assess ecosystem services at the landscape scale. In contrast to natural ecosystems, ecosystem service flows and values in agricultural landscapes are often a result of interactions between agricultural management and ecological structures. We describe how land management by farmers and other land managers relates to landscape structure and composition. We also examine the influence of commodity markets and policies on the behavior of land managers. Additionally, we studied the influence of consumer demand on flows and values of the ecosystem services that originate from the agricultural landscape

Improved lower bounds for graph embedding problems

In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these cannot be solved in time $2^{o(|s|/\log{|s|})}$, unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in $2^{o(n/\log n)}$ time for Subgraph Isomorphism on graphs of pathwidth 1, Induced Subgraph Isomorphism on graphs of pathwidth 1, Graph Minor on graphs of pathwidth 1, Induced Graph Minor on graphs of pathwidth 1, Intervalizing 5-Colored Graphs on trees, and finding a tree or path decomposition with width at most $c$ with a minimum number of bags, for any fixed $c\geq 16$. $2^{\Theta(n/\log n)}$ appears to be the correct running time for many packing and embedding problems on restricted graph classes, and we think String Crafting and Orthogonal Vector Crafting form a useful framework for establishing lower bounds of this form

On the exact complexity of hamiltonian cycle and q-colouring in disk graphs

\u3cp\u3eWe study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in (formula presented) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no (formula presented)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Expo-nential Time Hypothesis, for any fixed q, q-Colouring does not admit a (formula presented)-time algorithm, even when restricted to unit disk graphs, and it is solvable in (formula presented)-time on disk graphs.\u3c/p\u3

Computing treewidth on the GPU

We present a parallel algorithm for computing the treewidth of a graph on a GPU. We implement this algorithm in OpenCL, and experimentally evaluate its performance. Our algorithm is based on an $O^*(2^{n})$-time algorithm that explores the elimination orderings of the graph using a Held-Karp like dynamic programming approach. We use Bloom filters to detect duplicate solutions. GPU programming presents unique challenges and constraints, such as constraints on the use of memory and the need to limit branch divergence. We experiment with various optimizations to see if it is possible to work around these issues. We achieve a very large speed up (up to $77\times$) compared to running the same algorithm on the CPU

Improved lower bounds for graph embedding problems

\u3cp\u3eIn this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these cannot be solved in time 2\u3csup\u3eo\u3c/sup\u3e \u3csup\u3e(\u3c/sup\u3e \u3csup\u3e|s|/\u3c/sup\u3e \u3csup\u3elog\u3c/sup\u3e \u3csup\u3e|s|\u3c/sup\u3e \u3csup\u3e)\u3c/sup\u3e, unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previ-ously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in 2\u3csup\u3eo\u3c/sup\u3e \u3csup\u3e(\u3c/sup\u3e \u3csup\u3en/\u3c/sup\u3e \u3csup\u3elog\u3c/sup\u3e \u3csup\u3en\u3c/sup\u3e \u3csup\u3e)\u3c/sup\u3etime for Subgraph Isomorphism on graphs of pathwidth 1, Induced Subgraph Isomorphism on graphs of pathwidth 1, Graph Minor on graphs of pathwidth 1, Induced Graph Minor on graphs of pathwidth 1, Intervalizing 5-Colored Graphs on trees, and finding a tree or path decomposition with width at most c with a minimum number of bags, for any fixed c ≥ 16. 2\u3csup\u3eΘ\u3c/sup\u3e \u3csup\u3e(\u3c/sup\u3e \u3csup\u3en/\u3c/sup\u3e \u3csup\u3elog\u3c/sup\u3e \u3csup\u3en\u3c/sup\u3e \u3csup\u3e)\u3c/sup\u3eappears to be the “correct” running time for many pack-ing and embedding problems on restricted graph classes, and we think String Crafting and Orthogonal Vector Crafting form a useful framework for establishing lower bounds of this form.\u3c/p\u3

On exploring always-connected temporal graphs of small pathwidth

\u3cp\u3eWe show that the TEMPORAL GRAPH EXPLORATION PROBLEM is NP-complete, even when the underlying graph has pathwidth 2 and at each time step, the current graph is connected.\u3c/p\u3

On exploring temporal graphs of small pathwidth

We show that the Temporal Graph Exploration Problem is NP-complete, even when the underlying graph has pathwidth 2 and at each time step, the current graph is connected

PSPACE-completeness of Bloxorz and of games with 2-Buttons

Bloxorz is an online puzzle game where players move a 1 by 1 by 2 block by tilting it on a subset of the two dimensional grid. Bloxorz features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the destination square. We show that the problem of deciding whether a given Bloxorz level is solvable is PSPACE-complete and that this remains so even when all trapdoors are initially closed or all trapdoors are initially open. We also answer an open question of Viglietta, showing that 2-buttons are sufficient for PSPACE-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a 1 by 1 by 1 cube, and variants with single-use tiles