Network conduciveness with application to the graph-coloring and independent-set optimization transitions

Area of Science:

• Graph theory
• Combinatorial optimization
• Network science

Background:

• Considers NP-hard problems of finding chromatic and independence numbers in undirected graphs.
• Identifies performance jumps in solving these problems on incrementally denser graphs at critical edge values.

Purpose of the Study:

• Introduce and apply the concept of network conduciveness to explain these performance jumps.
• Analyze the solution space network for graph coloring and independent set problems.

Main Methods:

• Define network conduciveness as a measure of agent movement between network portions.
• Examine the conduciveness of the solution space network between non-optimal and optimal solutions.
• Correlate conduciveness changes with performance jumps at critical graph density transitions.

Main Results:

• Network conduciveness quantifies how easily optimal solutions are reached.
• Solution space networks become significantly more conducive to optimal solutions post-transition.
• Conversely, conduciveness towards non-optimal solutions decreases after transitions.

Conclusions:

• Network conduciveness offers a framework for understanding performance jumps in graph coloring and independent set problems.
• This concept may aid in clarifying NP-hardness issues.
• Potential applications in other network theory domains are suggested.