data clustering engineers

Abstract: A graph is called a threshold graph if there is an assignment of real valued weights to its vertices such that two vertices have an edge between them if and only if the sum of the weights of the two vertices is above a certain `threshold value'. We will denote the class of all threshold graphs as $TH$. For a graph $G=(V,E)$, if for some integer $t\geq 0$, there are $t$ many threshold graphs $\{G_i=(V,E_i)\}_{i\in [t]}$ such that we have $E=E_1\cap E_2\cap \cdots \cap E_t$, then we say that the collection of threshold graphs $\{G_i=(V,E_i)\}_{i\in [t]}$ forms a $TH$-representation of $G$ of size $t$. \emph{The intersection dimension of $G$ with respect to threshold graphs} (denoted as $dim_{TH}(G)$) is the smallest non-negative integer $t$ such that there is a $TH$-representation of $G$ of size $t$. It is known that for a graph $G$ and integer $t>0$, the problem of deciding whether $dim_{TH}(G)\leq t$ is NP-complete even for $t=3$. In this paper, we study this problem in the framework of parameterized algorithms. We consider the vertex cover number and the cluster vertex deletion number as our parameters. We give a FPT algorithm for determining an optimal $TH$-representation of a graph when parameterized by its vertex cover number. Our main result is an efficient parameterized approximation scheme for this problem when parameterized by the cluster vertex deletion number of $G$.

Abstract: We study a semidefinite programming (SDP) relaxation for community recovery when the number of communities grows logarithmically. In the balanced stochastic block model with $n=km$ vertices, we consider the regime $k/\log m\toγ>0$, with edge probabilities $α\log m/m$ within communities and $β\log m/m$ across them, for fixed $α>β>0$. We derive the sharp asymptotic tightness boundary away from critical cases. When rare vertices cause tightness to fail while the bulk remains spectrally stable, the normalized matrix error of every near-optimal solution still vanishes. We prove matching high-probability exponents for this error and the normalized optimal objective gain, governed by the same local correction that determines tightness. Throughout this spectrally stable region, a single SDP solve followed by explicit rounding and refinement achieves exact community recovery above the information-theoretic threshold. This guarantee holds even when the planted community matrix is not an optimal solution to the SDP.