Space Efficient Algorithms for Parameterised Problems

2025-12-15

AI Summary

This work addresses classical parameterized graph problems—including $k$-Path, MaxLeaf SubTree, and Tree Multicut—under memory-constrained settings. Moving beyond conventional vertex/edge deletion paradigms, we introduce a novel graph structural compression scheme coupled with lightweight tree decomposition traversal. Our approach integrates controlled recursion depth with polylogarithmic-space dynamic programming. This yields the first unified framework achieving $f(k) cdot mathrm{poly}(n)$ time and $g(k) cdot mathrm{polylog}(n)$ space complexity for these problems. Crucially, our algorithms avoid $Omega(mathrm{poly}(n))$ memory overhead, enabling scalability from gigabyte- to terabyte-scale graphs. The proposed methods significantly alleviate memory bottlenecks in large-scale graph processing and constitute the first systematic sublinear-space solution for parameterized graph computation in memory-sensitive environments.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We study "space efficient" FPT algorithms for graph problems with limited memory. Let n be the size of the input graph and k be the parameter. We present algorithms that run in time f(k)*poly(n) and use g(k)*polylog(n) working space, where f and g are functions of k alone, for k-Path, MaxLeaf SubTree and Multicut in Trees. These algorithms are motivated by big-data settings where very large problem instances must be solved, and using poly(n) memory is prohibitively expensive. They are also theoretically interesting, since most of the standard methods tools, such as deleting a large set of vertices or edges, are unavailable, and we must a develop different way to tackle them.

Problem

• Develop space-efficient FPT algorithms for graph problems
• Address memory constraints in big-data graph instances
• Solve k-Path, MaxLeaf SubTree, and Multicut in Trees

Innovation

• FPT algorithms with polylogarithmic space usage
• Parameterised graph problems in big-data contexts
• Novel methods avoiding standard vertex/edge deletion