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

Tolerant bipartiteness testing in dense graphs

Bipartite testing has been a central problem in the area of property testing since its inception in the seminal work of Goldreich, Goldwasser, and Ron. Though the non-tolerant version of bipartite testing has been extensively studied in the literature, the tolerant variant is not well understood. In this paper, we consider the following version of the tolerant bipartite testing problem: Given two parameters $\varepsilon, \delta \in (0,1)$, with $\delta > \varepsilon$, and access to the adjacency matrix of a graph $G$, we have to decide whether $G$ can be made bipartite by editing at most $\varepsilon n^2$ entries of the adjacency matrix of $G$, or we have to edit at least $\delta n^2$ entries of the adjacency matrix to make $G$ bipartite. In this paper, we prove that for $\delta=(2+\Omega(1))\varepsilon$, tolerant bipartite testing can be decided by performing $\widetilde{\mathcal{O}}\left({1}/{\varepsilon ^3}\right)$ many adjacency queries and in $2^{\widetilde{\mathcal{O}}(1/\varepsilon)}$ time complexity. This improves upon the state-of-the-art query and time complexities of this problem of $\widetilde{\mathcal{O}}\left({1}/{\varepsilon ^6}\right)$ and $2^{\widetilde{\mathcal{O}}(1/\varepsilon^2)}$, respectively, due to Alon, Fernandez de la Vega, Kannan and Karpinski, where $\widetilde{\mathcal{O}}(\cdot)$ hides a factor polynomial in $\log \left({1}/{\varepsilon}\right)$