Parameterized Algorithms on Perfect Graphs for deletion to (r,ℓ)(r,\ell)-graphs

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. The class of $(r, \ell)$ graphs generalizes $r$-colourable graphs (when $\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \ell)$-graph is \NP-hard even when $r \geq 3$ or $\ell \geq 3$ in general graphs. When $r$ and $\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\sc Chromatic Number} problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by $r$ and $\ell$. I.e. there is an f(r+\ell) \cdot n^{\Oh(1)} algorithm on perfect graphs on $n$ vertices where $f$ is some (exponential) function of $r$ and $\ell$. In this paper, we consider the parameterized complexity of the following problem, which we call {\sc Vertex Partization}. Given a perfect graph $G$ and positive integers $r,\ell,k$ decide whether there exists a set $S\subseteq V(G)$ of size at most $k$ such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. We obtain the following results: \begin{enumerate} \item {\sc Vertex Partization} on perfect graphs is FPT when parameterized by $k+r+\ell$. \item The problem does not admit any polynomial sized kernel when parameterized by $k+r+\ell$. In other words, in polynomial time, the input graph can not be compressed to an equivalent instance of size polynomial in $k+r+\ell$. In fact, our result holds even when $k=0$. \item When $r,\ell$ are universal constants, then {\sc Vertex Partization} on perfect graphs, parameterized by $k$, has a polynomial sized kernel. \end{enumerate

Improved Analysis of Highest-Degree Branching for Feedback Vertex Set

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in O(3.460^k n) time, where k is the solution size. This improves the previous best O(3.619^k n)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf. Process. Lett., 2014)

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances