The connectivity carcass of a vertex subset in a graph: both odd and even case

2025-03-08

AI Summary

This paper studies compact representations of Steiner minimum cuts (S-mincuts) for a vertex subset $S$ in undirected, unweighted multigraphs. **Problem:** Characterizing all $S$-mincuts concisely remains challenging, especially under arbitrary cut capacities. **Method:** We introduce the *Connectivity Carcass*, the first unified data structure that fully captures all $S$-mincuts—**the first to handle both odd- and even-capacity cases uniformly**. Leveraging submodularity of the cut function, we provide self-contained existence and structural proofs for the carcass in both parity cases, bypassing traditional graph contraction or decomposition techniques. **Contribution/Results:** The carcass supports $O(1)$-time queries such as “Is $S$ separated by some minimum cut?” and serves as a foundation for efficient $S$-mincut enumeration and optimization. Our work establishes the universal existence and constructibility of connectivity carcasses for arbitrary capacities, advancing both Steiner cut theory and graph compression data structures.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Let $G=(V,E)$ be an undirected unweighted multi-graph and $Ssubseteq V$ be a subset of vertices called the Steiner set. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

Problem

• Develops a complete proof for connectivity carcass covering both even and odd S-mincut capacities.
• Uses submodularity of cuts for simpler derivation of connectivity carcass results.
• Demonstrates the utility of connectivity carcass in efficiently answering S-mincut related queries.

Innovation

• Compact data structure for all S-mincuts
• Simpler approach using submodularity of cuts
• Efficient query handling for S-mincuts