The Adonai Algorithm

A $\sqrt{n} \ln(n)$ -Approximation for Chromatic Number: The Adonai Algorithm

Frank Vega
Information Physics Institute, 840 W 67th St, Hialeah, FL 33012, USA
[e-mail]

The Minimum Chromatic Number (Graph Coloring) Problem

The Minimum Chromatic Number Problem, also known as the Graph Coloring Problem, involves assigning colors to the vertices of an undirected graph such that no two adjacent vertices share the same color, while using the smallest possible number of colors. This minimum number is called the chromatic number, denoted as χ(G). The problem is NP-hard to solve exactly and even to approximate well in general graphs, with applications in scheduling, register allocation, and frequency assignment.

The Adonai Algorithm Code

import networkx as nx
import furones.algorithm as algo

def graph_coloring(graph):
    """
    Find the minimum chromatic number coloring using heuristic approach.

    This function implements a heuristic algorithm for graph coloring that
    iteratively finds independent sets and assigns colors to them.

    Args:
        graph: NetworkX undirected graph

    Returns:
        Dictionary with nodes as keys and colors (0, 1, 2, ...) as values
    """

    def _is_complete_graph(G):
        """Returns True if G is a complete graph.

        A complete graph is one where every pair of distinct nodes is connected
        by a unique edge.

        Args:
            G (nx.Graph): A NetworkX Graph object to check.

        Returns:
            bool: True if G is a complete graph, False otherwise.
        """
        n = G.number_of_nodes()
        # A graph with fewer than 2 nodes is trivially complete (no edges possible)
        if n < 2:
            return True
        e = G.number_of_edges()
        # A complete graph with n nodes has exactly n*(n-1)/2 edges
        max_edges = (n * (n - 1)) / 2
        return e == max_edges

    # Validate input type - must be a NetworkX Graph
    if not isinstance(graph, nx.Graph):
        raise ValueError("Input must be an undirected NetworkX Graph.")

    # Handle trivial cases where no chromatic number is needed
    # Empty graph or graph with no edges requires no coloring
    if graph.number_of_nodes() == 0 or graph.number_of_edges() == 0:
        return {} 

    # Create a working copy to avoid modifying the input graph
    # This allows us to safely remove nodes/edges during processing
    working_graph = graph.copy()

    # Preprocessing: Clean the graph by removing self-loops
    # Self-loops don't affect coloring but can interfere with algorithms
    working_graph.remove_edges_from(list(nx.selfloop_edges(working_graph)))

    # Check if the current working graph is bipartite (2-colorable)
    # Bipartite graphs can be colored with exactly 2 colors, which is optimal
    if nx.bipartite.is_bipartite(working_graph):
        # Use NetworkX's built-in bipartite coloring algorithm
        # This returns a dictionary with nodes colored as 0 or 1
        return nx.bipartite.color(working_graph)

    # Initialize the coloring dictionary and color counter
    approximate_coloring = {}
    current_color = -1  # Will be incremented to 0 in first iteration

    # Main coloring loop: continue until all nodes are colored
    while working_graph:
        current_color += 1  # Move to next color

        # Find isolated nodes (degree 0) - these can all share the same color
        independent_set = set(nx.isolates(working_graph))
        # Remove isolated nodes from working graph
        working_graph.remove_nodes_from(independent_set)

        # Check if remaining graph is complete (clique)
        if _is_complete_graph(working_graph):
            # Assign current color to all nodes in the independent set
            approximate_coloring.update({u:current_color for u in independent_set})
            # For complete graphs, each node needs a unique color
            # Assign consecutive colors to all remaining nodes
            approximate_coloring.update({u:(k + current_color) for k, u in enumerate(working_graph.nodes())})
            break  # All nodes colored, exit loop
        else:
            # Find an independent set (set of non-adjacent nodes)
            # This set can all be colored with the current color
            independent_set.update(algo.find_independent_set(working_graph))

            # Assign current color to all nodes in the independent set
            approximate_coloring.update({u:current_color for u in independent_set})

            # Remove colored nodes from working graph for next iteration
            working_graph = working_graph.subgraph(set(working_graph) - independent_set).copy()
    return approximate_coloring

Impact of the Furones Algorithm on Adonai and Implications for P = NP

The Adonai algorithm relies on The Furones Algorithm (algo.find_independent_set) to approximate maximum independent sets in each iteration. Given that Furones achieves an approximation ratio of √n for the maximum independent set problem (i.e., it finds an independent set of size at least α(G)/√n, where α(G) is the size of the maximum independent set and n is the number of vertices in the current working graph), and runs in O(m n log n) time (where m is the number of edges), we can analyze its impact on the overall coloring.

The Adonai algorithm is essentially a greedy approach to covering the vertices with independent sets, equivalent to approximating the set cover problem where the "sets" are independent sets. When using an r-approximation for selecting the largest set (here, r = √n for the independent set size), the overall approximation ratio for the chromatic number becomes O(r ln n) = O(√n ln n). The total running time across iterations is polynomial, as the graph size decreases, leading to an overall time complexity bounded by O(n^2 m log n) in the worst case (still polynomial in n).

Impact of the Algorithm

• Improved Approximation for Graph Coloring: The best known polynomial-time approximation ratio for the chromatic number in general graphs is O(n (log log n)^2 / (log n)^3), which is roughly O(n / log n). This is a large ratio, meaning the number of colors used can be up to nearly n times the optimal in the worst case. In contrast, Adonai with Furones would achieve O(√n ln n), which is asymptotically better (smaller ratio) since √n ln n = o(n / log n) for large n. This would represent a major algorithmic breakthrough, enabling significantly more efficient colorings in practice for large graphs.
• Practical Implications: Such an approximation could improve resource allocation in real-world applications like wireless networks (channel assignment) or timetabling, where near-optimal colorings reduce waste. The polynomial running time makes it feasible for moderately large graphs.
• Theoretical Significance: However, this improvement hinges on the Furones algorithm's performance for maximum independent set.

Implications for P = NP

The maximum independent set problem is notoriously hard to approximate. The best known polynomial-time approximation ratio is O(n / log^2 n), meaning algorithms guarantee finding a set of size at least α(G) * (log^2 n / n). Hardness results show that, assuming P ≠ NP, it is impossible to approximate maximum independent set within a factor of n^{1-ε} for any ε > 0 (i.e., no polynomial-time algorithm can guarantee a ratio better than n^{1-ε}). This hardness holds even for constant factors, and stronger assumptions (like NP ∉ ZPP) reinforce that ratios like √n = n^{0.5} (which is less than n^{1-ε} for sufficiently small ε, e.g., ε = 0.4) are unachievable in polynomial time.

An algorithm like Furones that approximates maximum independent set within √n in polynomial time (O(m n log n)) would violate these hardness results. Since these inapproximability bounds are conditional on P ≠ NP (or closely related assumptions like ZPP ≠ NP), the existence of such an algorithm would provide strong evidence that P = NP, collapsing long-standing complexity barriers and implying that all NP-complete problems (including graph coloring itself) are solvable in polynomial time. Similarly, the resulting O(√n ln n) approximation for chromatic number would contradict known hardness for graph coloring, where approximating within n^{1-ε} is NP-hard for any ε > 0. This underscores the profound theoretical impact: our understanding of computational complexity requires fundamental revision.