Generalising the maximum independent set algorithm via Boolean networks

TL;DR

This work generalises the MIS algorithm by allowing arbitrary starting sets and update orders, framing the process as sequential updates of a Boolean network and introducing constituencies to capture domination relations. It proves substantial hardness results for reachability, fixing words, and permissibility, while identifying near-comparability graphs as a broad, polynomially recognizable class of permissible graphs. The paper extends the MIS framework to digraphs via kernels, uncovering coNP-hardness for kernel-fixability and proposing two fixable networks (independent and dominating) that yield tractable analyses. Collectively, the results refine our understanding of how local update rules interact with graph structure to produce global MIS or kernel outcomes, with implications for self-stabilising and distributed computing. The work also opens avenues for exploring fixed-word lengths, graph-class restrictions, and diameter-bounded permises in practical scenarios.

Abstract

A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider (the complexity of decision problems related to) the generalisation of this MIS algorithm wherein any starting set is allowed. Two main approaches are leveraged. Firstly, we view the MIS algorithm as a sequential update of a Boolean network according to a permutation of the vertex set. Secondly, we introduce the concept of a constituency of a graph: a set of vertices that is dominated by an independent set. Recognizing a constituency is NP-complete, a fact we leverage repeatedly in our investigation. Our contributions are multiple: we establish that deciding whether all maximal independent sets can be reached from some configuration is coNP-complete; that fixing words (which reach a MIS from any starting configuration) and fixing permutations (briefly, permises) are coNP-complete to recognize; and that permissible graphs (graphs with a permis) are coNP-hard to recognize. We also exhibit large classes of permissible and non-permissible graphs, notably near-comparability graphs which may be of independent interest. Lastly, we extend our study to digraphs, where we search for kernels. Since the natural generalisation of our approach may not necessarily find a kernel, we introduce two further Boolean networks for digraphs: one always finds an independent set, and the other always finds a dominating set.

Figures (15)

• Figure 1: Illustration of the reduction from Set Cover to Constituency (the set $S$ is the vertices in the dashed box). Here the Set Cover instance has $C_1=\emptyset,C_2=\{x_1\},C_3=\{x_2,x_3\},C_4=\{x_4\}$, with $k=2$. Observe that both the Set Cover instance and the Constituency instance are no-instances.
• Figure 2: Example reduction from a no-instance of Constituency$(G,S)$ to the corresponding no-instance of District$(\hat{G}, \hat{S})$, with $\hat{S}:=S\cup T'$.
• Figure 3: Some example instances of the Non-Trivial Non-District problem. $C_4$ (left) is a no-instance, whereas $C_3$ (center) and $P_3$ (right) are yes-instances.
• Figure 4: Illustration of the reduction from Non-Constituency to Non-Trivial Non-District.
• Figure 5: Example reduction from a no-instance of Non-District$(G,S)$ to the corresponding no-instance of Fixing Set$(\hat{G}, \hat{S})$, with $\hat{S} = V_c \cup V_d \cup V_e$.

Theorems & Definitions (95)

• Example 1.1
• Lemma 2.1
• proof
• Corollary 2.2
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof
• Corollary 3.4