Lower bounds for kernelization

\u3cp\u3eKernelization is the process of transforming the input of a combinatorial decision problem to an equivalent instance, with a guarantee on the size of the resulting instances as a function of a parameter. Recent techniques from the field of fixed parameter complexity and tractability allow to give lower bounds for such kernels. In particular, it is discussed how one can show for parameterized problems that these do not have polynomial kernels, under the assumption that coNP⊆ NP/poly.\u3c/p\u3

MSOL-definability equals recognizability for Halin graphs and bounded degree kk-outerplanar graphs

One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for a number of special cases in a stronger form. That is, we show that each recognizable property is definable in MSOL, i.e. the counting operation is not needed in our expressions. We give proofs for Halin graphs, bounded degree $k$-outerplanar graphs and some related graph classes. We furthermore show that the conjecture holds for any graph class that admits tree decompositions that can be defined in MSOL, thus providing a useful tool for future proofs

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

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