Scalable Audience Reach Estimation in Real-time Online Advertising

Online advertising has been introduced as one of the most efficient methods of advertising throughout the recent years. Yet, advertisers are concerned about the efficiency of their online advertising campaigns and consequently, would like to restrict their ad impressions to certain websites and/or certain groups of audience. These restrictions, known as targeting criteria, limit the reachability for better performance. This trade-off between reachability and performance illustrates a need for a forecasting system that can quickly predict/estimate (with good accuracy) this trade-off. Designing such a system is challenging due to (a) the huge amount of data to process, and, (b) the need for fast and accurate estimates. In this paper, we propose a distributed fault tolerant system that can generate such estimates fast with good accuracy. The main idea is to keep a small representative sample in memory across multiple machines and formulate the forecasting problem as queries against the sample. The key challenge is to find the best strata across the past data, perform multivariate stratified sampling while ensuring fuzzy fall-back to cover the small minorities. Our results show a significant improvement over the uniform and simple stratified sampling strategies which are currently widely used in the industry

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