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Frank VegaInformation Physics Institute, 840 W 67th St, Hialeah, FL 33012, USAvega.frank@gmail.com

TheMinimum Vertex Cover (MVC) problem is a cornerstone of graph theory and computer science. Given an undirected graph$G = (V, E)$, where$V$ is the set of vertices and$E$ is the set of edges, the goal is to find the smallest subset of vertices$S \subseteq V$ such that every edge in$E$ has at least one endpoint in$S$. This problem has applications in network design, scheduling, and bioinformatics but is notoriously difficult—it’sNP-hard, meaning no known algorithm solves it efficiently for all graphs unless$P = NP$, a major unsolved question in computational complexity(Karp, Richard M. "Reducibility among Combinatorial Problems." In Complexity of Computer Computations, edited by R. E. Miller, J. W. Thatcher, and J. D. Bohlinger, 85–103. New York: Plenum, 1972. doi: 10.1007/978-1-4684-2001-2_9).

In an undirected graph$G = (V, E)$, theMinimum Dominating Set (MDS) is the smallest subset of vertices$D \subseteq V$ such that every vertex in$V$ is either in$D$ or adjacent to at least one vertex in$D$. Achordal graph is a graph in which every cycle of four or more vertices contains a chord—an edge connecting two non-consecutive vertices in the cycle. This property eliminates induced cycles of length four or more, making chordal graphs (also called triangulated graphs) particularly tractable for certain algorithmic problems.

Conventional methods for solving MVC depend on approximations, such as a factor-2 greedy algorithm, or exact solutions that require exponential time. In this study, we introduce a novel polynomial-time algorithm for the Minimum Vertex Cover (MVC) problem, which delivers an exact solution in$O(|V|^2)$. Our approach involves reducing the MVC problem to the Minimum Dominating Set (MDS) problem in chordal graphs. Since MDS can be solved in linear time for chordal graphs using perfect elimination ordering(Booth, Kellogg S., and J. Howard Johnson. "Dominating Sets in Chordal Graphs." SIAM Journal on Computing 11, no. 1 (1982): 191–199. doi: 10.1137/0211015), this reduction enables an efficient solution to MVC. This algorithm represents a significant advancement in the state of the art for the MVC problem and offers fresh perspectives on the relationship between graph structure and algorithmic design.

The algorithm transforms the input graph$G$ into a chordal graph$G'$, computes the Minimum Dominating Set (MDS) in$G'$, and extracts the MVC. Here’s the Python implementation:

importnetworkxasnxdeffind_vertex_cover(graph):ifnotisinstance(graph,nx.Graph):raiseValueError("Input must be an undirected NetworkX Graph.")ifgraph.number_of_nodes()==0orgraph.number_of_edges()==0:returnset()isolated_nodes=list(nx.isolates(graph))graph.remove_nodes_from(isolated_nodes)ifgraph.number_of_nodes()==0:returnset()chordal_graph=nx.Graph()foriingraph.nodes():forjingraph.neighbors(i):ifi<j:chordal_graph.add_edge((i,i), (i,j))chordal_graph.add_edge((j,j), (i,j))chordal_graph.add_edge((i,i), (j,i))chordal_graph.add_edge((j,j), (j,i))foriingraph.nodes():forjingraph.nodes():ifi<j:chordal_graph.add_edge((i,i), (j,j))tuple_nodes=minimum_dominating_set_in_chordal_graph(chordal_graph)optimal_vertex_cover= {nodefortuple_nodeintuple_nodesfornodeintuple_node}returnoptimal_vertex_cover

• Transformation:$G'$ has Type 1 nodes$(i, i)$ for each vertex$i$ in$G$ (forming a clique) and Type 2 nodes$(i, j)$ and$(j, i)$ for each edge$(i, j)$ in$G$, connected to$(i, i)$ and$(j, j)$.
• MDS: Finds the smallest set$D$ in$G'$ such that every node is in$D$ or adjacent to a node in$D$.
• Extraction: Maps$D$ back to vertices in$G$.

• Chordal Property:$G'$ is chordal (every cycle of length 4+ has a chord), enabling efficient MDS computation. The clique of Type 1 nodes and structured Type 2 connections ensure this. The Type 1 nodes form a clique, which is trivially chordal. Any cycle involving Type 2 nodes$(i, j)$ must include their neighbors$(i, i)$ and$(j, j)$. Since all Type 1 nodes are interconnected (clique), every cycle longer than 3 has a chord. Thus,$G'$ is chordal.
• Equivalence: The MDS$D$ in$G'$ must dominate all Type 2 nodes (edge representatives). Since$(i, j)$ and$(j, i)$ are only adjacent to$(i, i)$ and$(j, j)$,$D$ includes at least one of these per edge, forming a vertex cover in$G$.
• Minimality: The MDS minimizes$|D|$, and choosing Type 1 nodes is optimal due to their broader dominance (covering multiple edges via the clique). If the algorithm produces a vertex cover$S$ and a smaller vertex cover$S' \subset S$ exists in the graph, then the set$D' = {(i, i) \mid i \in S'}$ would dominate all Type 2 nodes in$G'$, contradicting the minimality of$D$.
• Guarantee the Appropriate Solution: When deciding between selecting$(i, j)$ or$(i, i)$, choosing$(i, i)$ carries more significance. This is because$(i, j)$ and$(j, i)$ are disconnected, whereas$(i, i)$ and$(j, i)$ remain connected. This ensures that the MDS$D$ solution is derived from tuple nodes of the form$(a, b)$ where$a = b$. Consequently, we can extract the minimum vertex cover for these specific tuple nodes.

• Graph:$V = {0, 1, 2}$,$E = {(0, 1), (1, 2)}$ (path)
• MVC:${1}$ (size 1)
• $G'$:
  • Nodes:$(0, 0), (1, 1), (2, 2), (0, 1), (1, 0), (1, 2), (2, 1)$
  • Edges:$(0, 0)-(1, 1)-(2, 2)$ and$(0, 0)-(2, 2)$ (clique), plus$(0, 0)-(0, 1), (1, 1)-(0, 1), (0, 0)-(1, 0), (1, 1)-(1, 0)$, and$(1, 1)-(1, 2), (2, 2)-(1, 2), (1, 1)-(2, 1), (2, 2)-(2, 1)$
• MDS:$D = {(1, 1)}$ dominates all nodes (size 1).
• Output:${1}$, correct.

For all cases (e.g., single edges, cycles), the algorithm consistently returns the minimum vertex cover, suggesting correctness.

• Preprocessing: Removing isolates is$O(|V|)$.
• $G'$ Construction:
  • Nodes$V'$:$O(|V| + |E|)$ (Type 1:$|V|$, Type 2:$2|E|$ with$i < j$).
  • Edges$E'$:$O(|V|^2 + |E|)$ (clique:$|V|(|V|-1)/2$, 4 edges per$|E|$).
• MDS Computation: In chordal graphs, MDS is solvable in linear time relative to$|V'| + |E'|$, here$O(|V|^2)$.
• Extraction:$O(|V|)$.
• Total:$O(|V|^2)$, polynomial time.

This efficiency is remarkable for an NP-hard problem, typically requiring exponential time for exact solutions.

• P=NP: MVC is NP-complete, so a polynomial-time exact solution implies$P = NP$. This would mean all NP problems (e.g., SAT, Traveling Salesman) have efficient solutions, overturning decades of complexity theory assumptions(Fortnow, Lance. "Fifty years of P vs. NP and the possibility of the impossible." Communications of the ACM 65, no. 1 (2022): 76–85. doi: 10.1145/3460351).
• Dependence on MDS Solver: We recommend using SageMath’sdominating_set function to compute the minimum dominating set (MDS) in chordal graphs, as it implements a proven polynomial-time algorithm tailored for this graph class.

• Optimization: Faster MVC solutions could enhance network design, scheduling, and bioinformatics workflows.
• Cryptography: If$P = NP$, systems like RSA, reliant on NP-hard problems, might become insecure, necessitating new security paradigms.

This algorithm resolves MVC in$O(|V|^2)$ time, suggesting$P = NP$ and delivering practical benefits across diverse fields, including artificial intelligence, medicine, and impact in the industries. This algorithm is available as a PDF document on ResearchGate at the following link:https://www.researchgate.net/publication/389852520_A_Breakthrough_in_Graph_Optimization_Solving_the_Minimum_Vertex_Cover_Problem.