Composing dynamic programming tree-decomposition-based algorithms

Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[V_i] \in \mathcal{H}_i$, $G[V_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We show that, in some known cases, the obtained running times are comparable to those of the best know algorithms

Ruling out FPT algorithms for Weighted Coloring on forests

Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i) = \max_{v \in S_i} w(v)$ and the weight of a coloring $c$ as $w(c) = \sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair $(G,w)$, denoted by $\sigma(G,w)$, as the minimum weight of a proper coloring of $G$. For a positive integer $r$, they also defined $\sigma(G,w;r)$ as the minimum of $w(c)$ among all proper $r$-colorings $c$ of $G$. The complexity of determining $\sigma(G,w)$ when $G$ is a tree was open for almost 20 years, until Ara\'ujo et al. [SIAM J. Discrete Math., 2014] recently proved that the problem cannot be solved in time $n^{o(\log n)}$ on $n$-vertex trees unless the Exponential Time Hypothesis (ETH) fails. The objective of this article is to provide hardness results for computing $\sigma(G,w)$ and $\sigma(G,w;r)$ when $G$ is a tree or a forest, relying on complexity assumptions weaker than the ETH. Namely, we study the problem from the viewpoint of parameterized complexity, and we assume the weaker hypothesis $FPT \neq W[1]$. Building on the techniques of Ara\'ujo et al., we prove that when $G$ is a forest, computing $\sigma(G,w)$ is $W[1]$-hard parameterized by the size of a largest connected component of $G$, and that computing $\sigma(G,w;r)$ is $W[2]$-hard parameterized by $r$. Our results rule out the existence of $FPT$ algorithms for computing these invariants on trees or forests for many natural choices of the parameter.Comment: 14 pages, 4 figure

Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth

For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most k such that G-S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F}, the smallest function f_F such that F-M-DELETION can be solved in time f_F(tw)n^{O(1)} on n-vertex graphs. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds: - f_{K_4}(w) = 2^{Theta(wlog w)}. - f_{C_i}(w) = 2^{Theta(w)} for every i4. - f_{P_i}(w) = 2^{Theta(w)} for every i5. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove essentially the same set of results holds

Contraction-Bidimensionality of Geometric Intersection Graphs

Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Gamma_k. A graph class G has the SQGC property if every graph G in G has treewidth O(bcg(G)c) for some 1 <= c < 2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects

Exclusive Radiative Decays of B Mesons

We present within the Standard Model the exclusive radiative decays B -> K*/rho gamma and B_(s/d) -> gamma gamma in QCD factorization based on the heavy-quark limit m_b >> Lambda_QCD. For the decays with a vector meson in the final state we give results complete to next-to-leading order in QCD.Comment: 4 pages, contributed to QCD 02: High-Energy Physics International Conference in Quantum Chromodynamics, Montpellier, France, 2-9 July 200