Optimizing Cloud-Based Systems Through Resource-Constrained Distance Matching.

Cloud computing has transformed the landscape of data management and resource allocation by providing scalable solutions to meet varying demands. In this context, optimizing resource usage is crucial. In this paper, we model some of the issues arising in cloud computing by investigating the Resource-Constrained Distance Matching (RCDM) problem, a variant of the Maximum Cardinality Distance Matching (MCDM) problem. In the MCDM problem, we are given a bipartite graph G=(S,T,E) and an integer d∈ℤ^+ , where S={s_1, s_2,… , s_n } is an ordered set and E ⊆ S × T . The objective is to find a maximum cardinality subset ℳ⊆ E of edges while satisfying two conditions: (a) the degree of every node in S is at most one in ℳ , and (b) if s_it, s_jt∈ℳ then |j-i|≥ d . In the RCDM problem, the goal is to find an MCDM ℳ⊆ E in G such that the number of vertices of T in ℳ is minimized. This problem is highly relevant to modern cloud-based systems, including resource allocation, data management, and network design. We demonstrate that the RCDM problem is NP-complete even in pipartite (planar and bipartite) subcubic graphs. Additionally, we show that the RCDM problem is APX-hard in bipartite subcubic graphs and is inapproximable within a factor of (7/6-ϵ ) unless 𝐏=𝐍𝐏 . One of our principal contributions is the design of a non-trivial exact exponential time algorithm for the RCDM problem. The findings presented in this paper are significant for advancing theoretical algorithms and practical implementations in cloud computing environments, particularly in optimizing resource usage, improving data management strategies, and enhancing network efficiency.