There is a Hyper-Greedoid lurking behind every Graphical Accessible Computational Search Problem solvable in Polynomial Time: $P \not= NP$

TL;DR

The paper develops a greedoid-inspired framework, introducing Hyper-greedoid structures to characterize graphical search problems defined on contraction-minors of isthmus-less graphs. It formalizes Graphical Search Problems $\Pi(G[X],\gamma)$ with solutions, sub-solutions, and closures, illustrating connections to classic graph problems such as $\text{MISP}$ and $\text{HCP}$. The central result proves that polynomial-time solvability is equivalent to augmentability ($M2'$) of all non-basis feasible sets, using a reduction to the Maximal Independent Set problem to bridge to $\mathcal{P}$- and $\mathcal{NP}$-complete problems; the Hamiltonian Cycle Problem serves as the NP-complete example that violates augmentability, supporting $\mathcal{P} \neq \mathcal{NP}$. The work thus links a new combinatorial structure to fundamental complexity questions and proposes a polynomial-time algorithmic paradigm (Generalised Greedy Algorithm) under augmentability, with broad implications for understanding why certain graphical problems resist efficient solutions.

Abstract

Consider $G[X]$, where $G$ is a connected, isthmus-less and labelled graph, and $X$ is the edge-set or the vertex-set of the graph $G$. A Graphical Search Problem (GSP), denoted $Π(G[X],γ)$, consists of finding $Y$, where $Y \subseteq X$ and $Y$ satisfies the predicate $γ$ in $G$. The subset $Y$ is a solution of the problem $Π(G[X],γ)$. A sub-solution of $Π(G[X],γ)$ is a subset $Y'$ such that $Y'$ is not a solution of $Π(G[X],γ)$, but $Y'$ is a solution of the problem $Π(H[X'],γ)$, where $X' \subset X$ and $H[X']$ is a contraction-minor of $G[X]$. Solutions and sub-solutions are the feasible sets of $Π(G[X],γ)$. Let $\mathfrak{I}$ be the family of all the feasible sets of $Π(G[X],γ)$. A Hyper-greedoid is a set system $(X, \mathfrak{I})$ satisfying the following axioms. A1: Accessibility: if $I \in \mathfrak{I}$, there is an element $x \in I$ such that $I-x \in \mathfrak{I}$ A2: Augmentability: If $I$ is a sub-solution, there is a polynomial time function $κ: \mathfrak{I} \rightarrow \mathfrak{I}$ and there is a element $x \in X-κ(I)$ such that $κ(I) \cup x \in \mathfrak{I}$. That is, every sub-solution can be augmented using a polynomial time algorithm akin to Edmond Augmenting Path Algorithm. Given a graph $G$, the GSP MISP consists of finding an independent set of vertices of $G$. MISP satisfies axioms A1 and A2. Using the P-completeness of the Decision Problem associated to MISP, we prove that every GSP that satisfies A1 is solvable in Polynomial Time if and only if it satisfies A2. On the other hand, let HCP be the GSP that consists of finding a Hamiltonian cycle of the graph $G$. HCP satisfies A1, but does not satisfies A2. Since the Decision Problem associated with HCP is NP-Complete, we get $P \not = NP$.

Figures (15)

• Figure 1: The set $\{ v_{_{1}}, v_{_{3}}, v_{_{4}}\}$ is a maximal independent set of the graph $G/e= G/A$. The vertex $v_{_{2}}$ can be added to it to obtain the independent set $\{ v_{_{1}}, v_{_{2}}, v_{_{3}}, v_{_{4}}\}$ by re-inserting the edge $e$.
• Figure 2: Suppose that $A= \{e\}$ so that $G= G/(A-e)$. The set $\{ v_{_{1}}, v_{_{3}}, v_{_{4}}\}$ is a maximal independent set of the graph $G/A$. The vertex $v_{_{2}}$ can not be added to it since $v_{_{2}}$ is adjacent to $v_{_{1}}$ in $G/(A-e)$.
• Figure 3: The maximal independent set $\{ v_{_{1}}, v_{_{3}}, v_{_{4}}\}$ obtained from the independent set $\{ v_{_{1}}, v_{_{2}}, v_{_{3}}, v_{_{4}}\}$ by the contraction of the edge $e$.
• Figure 4: The hamiltonian cycle $\{ e_{_{2}}, e_{_{3}}, e_{_{4}}, e_{_{5}}\}$ obtained from the hamiltonian cycle $\{ e_{_{1}}, e_{_{2}}, e_{_{3}}, e_{_{4}}, e_{_{5}}\}$ by the contraction of the edge $e_{_{1}}$.
• Figure 5: A graph where a Hamiltonian circuit exists.

Theorems & Definitions (24)

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• Lemma 1
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