On the Hardness of Red-Blue Pebble Games

Red-blue pebble games model the computation cost of a two-level memory hierarchy. We present various hardness results in different red-blue pebbling variants, with a focus on the oneshot model. We first study the relationship between previously introduced red-blue pebble models (base, oneshot, nodel). We also analyze a new variant (compcost) to obtain a more realistic model of computation. We then prove that red-blue pebbling is NP-hard in all of these model variants. Furthermore, we show that in the oneshot model, a $\delta$-approximation algorithm for $\delta<2$ is only possible if the unique games conjecture is false. Finally, we show that greedy algorithms are not good candidates for approximation, since they can return significantly worse solutions than the optimum

On the Width of Complicated JSJ Decompositions

Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a "sufficiently complicated" JSJ decomposition of a 3-manifold enforces a "complicated structure" for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a linear lower bound on its treewidth tw (M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of M. We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth - previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds

On the Tractability of (k, i)-Coloring

In an undirected graph, a proper ( k, i )-coloring is an assign- ment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The ( k, i )-coloring problem is to compute the minimum number of colors required for a proper ( k, i )- coloring. This is a generalization of the classic graph colo ring problem. Majumdar et. al. [CALDAM 2017] studied this problem and show ed that the decision version of the ( k, i )-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They aske d if there exists an FPT algorithm with the size of the feedback vertex s et (FVS) as the parameter without using tree-width machinery. We ans wer this in positive by giving a parameterized algorithm with the size o f the FVS as the parameter. We also give a faster and simpler exact algo rithm for ( k, k − 1)-coloring, and make progress on the NP-completeness of sp ecific cases of ( k, i )-colorin

On the effectiveness of the incremental approach to minimal chordal edge modification

Because edge modification problems are computationally difficult for most target graph classes, considerable attention has been devoted to inclusion-minimal edge modifications, which are usually polynomial-time computable and which can serve as an approximation of minimum cardinality edge modifications, albeit with no guarantee on the cardinality of the resulting modification set. Over the past fifteen years, the primary design approach used for inclusion-minimal edge modification algorithms is based on a specific incremental scheme. Unfortunately, nothing guarantees that the set E of edge modifications of a graph G that can be obtained in this specific way spans all the inclusion-minimal edge modifications of G. Here, we focus on edge modification problems into the class of chordal graphs and we show that for this the set E may not even contain any solution of minimum size and may not even contain a solution close to the minimum; in fact, we show that it may not contain a solution better than within an Ω(n) factor of the minimum. These results show strong limitations on the use of the current favored algorithmic approach to inclusion-minimal edge modification in heuristics for computing a minimum cardinality edge modification. They suggest that further developments might be better using other approaches.publishedVersio

Excluding Surfaces as Minors in Graphs

We introduce an annotated extension of treewidth that measures the contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion provides a graph distance measure to some graph property $\mathcal{P}$: A vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the treewidth of $X$ in $G$ is at most $k$ and its removal gives a graph in $\mathcal{P}.$This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: $K_k$-minor free graphs are those that admit tree-decompositions whose torsos have $c_{k}$-treewidth modulators to some surface of Euler-genus $c_{k}.$ This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that $\Sigma$ is some particular surface and define a ``surface extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum $k$ for which $G$ admits a tree-decomposition whose torsos have a $k$-treewidth modulator to being embeddable in $\Sigma.$We identify a finite collection $\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_{\Sigma}$ precisely determines the asymptotic behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface $\Sigma.$ It follows that the collection $\mathfrak{D}_{\Sigma}$ bijectively corresponds to the ``surface obstructions'' for $\Sigma,$ i.e., surfaces that are minimally non-contained in $\Sigma.

On computing tree and path decompositions with metric constraints on the bags

We here investigate on the complexity of computing the \emph{tree-length} and the \emph{tree-breadth} of any graph $G$, that are respectively the best possible upper-bounds on the diameter and the radius of the bags in a tree decomposition of $G$. \emph{Path-length} and \emph{path-breadth} are similarly defined and studied for path decompositions. So far, it was already known that tree-length is NP-hard to compute. We here prove it is also the case for tree-breadth, path-length and path-breadth. Furthermore, we provide a more detailed analysis on the complexity of computing the tree-breadth. In particular, we show that graphs with tree-breadth one are in some sense the hardest instances for the problem of computing the tree-breadth. We give new properties of graphs with tree-breadth one. Then we use these properties in order to recognize in polynomial-time all graphs with tree-breadth one that are planar or bipartite graphs. On the way, we relate tree-breadth with the notion of \emph{$k$-good} tree decompositions (for $k=1$), that have been introduced in former work for routing. As a byproduct of the above relation, we prove that deciding on the existence of a $k$-good tree decomposition is NP-complete (even if $k=1$). All this answers open questions from the literature.Comment: 50 pages, 39 figure