Improved Approximation Algorithm for Graph Burning on Trees

Given a graph $G=(V,E)$, the problem of \gb{} is to find a sequence of nodes from $V$, called burning sequence, in order to burn the whole graph. This is a discrete-step process, in each step an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A node that is burnt spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The \gb{} problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a $3$-approximation algorithm for general graphs and a $2$- approximation algorithm for trees. In this paper, we propose an approximation algorithm for trees that produces a burning sequence of length at most $\lfloor 1.75b(T) \rfloor + 1$, where $b(T)$ is length of the optimal burning sequence, also called the burning number of the tree $T$. In other words, we achieve an approximation factor of $(\lfloor 1.75b(T) \rfloor + 1)/b(T)$

Vertex partitioning problems on graphs with bounded tree width

In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The MATCHING CUT problem is the problem of deciding whether a given graph has a matching cut. Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an H-FREE COLORING if none of the color classes contain H as an induced subgraph. The H-FREE CHROMATIC NUMBER of G is the minimum number of colors required for an H-FREE COLORING of G. Both The MATCHING CUT problem and the H-FREE COLORING problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of f(||φ||,t)nO(1), where ||φ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f(||φ||,t) on ||φ|| can be as bad as a tower of exponentials. In this paper, we provide explicit combinatorial FPT algorithms for MATCHING CUT problem and H-FREE COLORING problem, parameterized by the tree-width of G. The single exponential FPT algorithm for the MATCHING CUT problem answers an open question posed by Kratsch and Le (2016). The techniques used in the paper are also used to provide an FPT algorithm for a variant of H-free coloring, where H is forbidden as a subgraph (not necessarily induced) in the color classes of G

Parameterized complexity of happy coloring problems

In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. Motivated by the computation of homophily in social networks, we consider the algorithmic aspects of the following MAXIMUM HAPPY EDGES ( k-MHE ) problem: given a partially k-colored graph G and an integer ℓ, find an extended full k-coloring of G making at least ℓ edges happy. When we want to make ℓ vertices happy on the same input, the problem is known as MAXIMUM HAPPY VERTICES ( k-MHV ). We perform an extensive study into the complexity of the problems, particularly from a parameterized viewpoint. For every k≥3, we prove both problems can be solved in time 2nnO(1). Moreover, by combining this result with a linear vertex kernel of size (k+ℓ) for k-MHE, we show that the edge-variant can be solved in time 2ℓnO(1). In contrast, we prove that the vertex-variant remains W[1]-hard for the natural parameter ℓ. However, the problem does admit a kernel with O(k2ℓ2) vertices for the combined parameter k+ℓ. From a structural perspective, we show both problems are fixed-parameter tractable for treewidth and neighborhood diversity, which can both be seen as sparsity and density measures of a graph. Finally, we extend the known [Formula presented]-completeness results of the problems by showing they remain hard on bipartite graphs and split graphs. On the positive side, we show the vertex-variant can be solved optimally in polynomial-time for cographs