In this work, we consider d-{\sc Hyperedge Estimation} and d-{\sc Hyperedge Sample} problem in a hypergraph H(U(H),F(H)) in the query complexity framework, where U(H) denotes the set of vertices and F(H) denotes the set of hyperedges. The oracle access to the hypergraph is called {\sc Colorful Independence Oracle} ({\sc CID}), which takes d (non-empty) pairwise disjoint subsets of vertices A1,…,Ad⊆U(H) as input, and answers whether there exists a hyperedge in H having (exactly) one vertex in each Ai,i∈{1,2,…,d}. The problem of d-{\sc Hyperedge Estimation} and d-{\sc Hyperedge Sample} with {\sc CID} oracle access is important in its own right as a combinatorial problem. Also, Dell {\it{et al.}}~[SODA '20] established that {\em decision} vs {\em counting} complexities of a number of combinatorial optimization problems can be abstracted out as d-{\sc Hyperedge Estimation} problems with a {\sc CID} oracle access.
The main technical contribution of the paper is an algorithm that estimates m=∣F(H)∣ with m such that { Cdlogd−1n1≤mm≤Cdlogd−1n. by using at most Cdlogd+2n many {\sc CID} queries, where n denotes the number of vertices in the hypergraph H and Cd is a constant that depends only on d}. Our result coupled with the framework of Dell {\it{et al.}}~[SODA '21] implies improved bounds for a number of fundamental problems