Linear-Delay Enumeration for Minimal Steiner Problems

TL;DR

This paper introduces efficient algorithms for enumerating minimal Steiner trees and related structures in graphs, significantly improving enumeration delay and extending applicability to various Steiner problems.

Contribution

It presents a linear-delay, polynomial-space enumeration algorithm for minimal Steiner trees and extends the approach to other Steiner variants, also analyzing the complexity of group Steiner trees.

Findings

Linear-delay enumeration algorithm for minimal Steiner trees

Extension of algorithms to Steiner forests and terminal Steiner trees

Complexity result for minimal group Steiner tree enumeration

Abstract

Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv, Inf. Syst. 2008] pointed out the problem of enumerating $K$-fragments is of great importance in a keyword search on data graphs. In a graph-theoretic term, the problem corresponds to enumerating minimal Steiner trees in (directed) graphs. In this paper, we propose a linear-delay and polynomial-space algorithm for enumerating all minimal Steiner trees, improving on a previous result in [Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be extended to other Steiner problems, such as minimal Steiner forests, minimal terminal Steiner trees, and minimal directed Steiner trees. As another variant of the minimal Steiner tree enumeration problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay and exponential-space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs.