[Paper Review] An Algorithm Computing the Core of a Konig-Egervary Graph

Vadim E. Levit, Eugen Mǎndrescu|arXiv (Cornell University)|Feb 6, 2011

Advanced Graph Theory Research14 references3 citations

TL;DR

This paper presents a polynomial-time algorithm to compute the core of a König-Egerváry graph—the intersection of all maximum independent sets—using matching size changes upon vertex removal. It introduces both sequential and parallel algorithms with time complexities of $O(mn^{3/2})$ and $O(m oot n o)$ respectively, enabling efficient identification of vertices common to all maximum independent sets in this graph class.

ABSTRACT

A set S of vertices is independent in a graph G if no two vertices from S are adjacent, and alpha(G) is the cardinality of a maximum independent set of G. G is called a Konig-Egervary graph if its order equals alpha(G)+mu(G), where mu(G) denotes the size of a maximum matching. By core(G) we mean the intersection of all maximum independent sets of G. To decide whether core(G) is empty is known to be NP-hard. In this paper, we present some polynomial time algorithms finding core(G) of a Konig-Egervary graph G.

Motivation & Objective

• To develop a polynomial-time algorithm for computing the core of a König-Egerváry graph, defined as the intersection of all maximum independent sets.
• To address the NP-hardness of core computation in general graphs by restricting the problem to König-Egerváry graphs, where it becomes tractable.
• To provide both sequential and parallel algorithmic solutions with provable time complexity bounds.
• To enable efficient recognition of König-Egerváry graphs with a unique maximum independent set via core computation.
• To generalize prior results on bipartite and perfect graphs by extending efficient core computation to non-bipartite König-Egerváry graphs.

Proposed method

• Leverage the key characterization: a vertex $v$ belongs to the core if and only if $\mu(G) = \mu(G - v)$, i.e., removing $v$ does not reduce the size of a maximum matching.
• Use the fact that $G$ is a König-Egerváry graph if and only if $\alpha(G) + \mu(G) = n$, and that $G - v$ remains König-Egerváry if $\mu(G) = \mu(G - v)$.
• For each vertex $v$, compute $\mu(G - v)$ and compare it to $\mu(G)$; if equal, $v \in \mathrm{core}(G)$.
• Apply maximum matching algorithms (e.g., Dinic’s or Hopcroft-Karp) with $O(m\sqrt{n})$ complexity per vertex, enabling scalable computation.
• Design a parallel version by evaluating $\mu(G - v)$ for all $v$ in parallel using $n$ processors, reducing total time to $O(m\sqrt{n})$.
• Optimize for special cases: when $G$ has a perfect matching, use the condition $v \in \mathrm{core}(G) \iff G - v \text{ is not König-Egerváry}$, simplifying computation.

Experimental results

Research questions

• RQ1 Can the core of a König-Egerváry graph be computed in polynomial time, despite the NP-hardness of the general core computation problem?
• RQ2 What structural properties of König-Egerváry graphs allow for efficient core computation via matching size analysis?
• RQ3 How can sequential and parallel algorithms be designed to compute the core with optimal time complexity?
• RQ4 Under what conditions does a König-Egerváry graph have a unique maximum independent set, and how can this be detected via core computation?
• RQ5 Can the core computation be optimized for special subclasses, such as bipartite graphs or graphs with perfect matchings?

Key findings

• The core of a König-Egerváry graph can be computed in $O(mn^{3/2})$ time using a sequential algorithm based on matching size comparisons.
• A parallel algorithm with $n$ processors computes the core in $O(m\sqrt{n})$ time, significantly improving scalability.
• For bipartite König-Egerváry graphs, the core is computed by checking $\mu(G) = \mu(G - v)$ for each vertex, with a sequential complexity of $O(mn\sqrt{n})$.
• When $G$ has a perfect matching, $v \in \mathrm{core}(G)$ if and only if $G - v$ is not a König-Egerváry graph, enabling a more efficient algorithm.
• The algorithm provides a constructive proof that a bipartite König-Egerváry graph with a perfect matching has an empty core, confirming Proposition 1(ii).
• The core computation enables polynomial-time recognition of König-Egerváry graphs with a unique maximum independent set, as such graphs have a core that is a maximal independent set.