Overview
The complexity of exact separation for comb inequalities, which function as critical cutting planes within the symmetric Travelling Salesman Problem (TSP) context, has represented a persistent inquiry in polyhedral combinatorics. This research presents a reduction from the 3-Satisfiability (3-SAT) problem, establishing that determining whether a comb inequality is violated is NP-complete. Consequently, the associated separation problem is proven to be NP-hard.
Research Context
Comb inequalities are integral to the formulation and solution of the symmetric travelling salesman problem, particularly in the realm of polyhedral combinatorics, where they serve as cutting planes. The computational challenge of their exact separation has been a long-standing question within this field.
Approach
The methodology employed involves constructing a reduction from 3-SAT. This reduction generates a graph where specific structural elements encode Boolean relations and logical variable consistency. Key components of this graph construction include:
- Six-vertex ladder gadgets: These are utilized to encode Boolean relations between variables.
- Cubic graphs: These structures enforce consistency among the various occurrences of each logical variable throughout the formula.
For a propositional logic formula characterized by $v$ variables and $m$ clauses, the graph constructed through this reduction consists of $40v+70m+30$ vertices. This constructed graph also exhibits a linear number of positive edges.
Findings
The reduction from 3-SAT unequivocally demonstrates that:
- Deciding if a comb inequality is violated is NP-complete.
- The corresponding separation problem is NP-hard.
These findings hold under specific conditions, including scenarios where:
- The input vector is part of the subtour elimination polytope.
- Every edge value is restricted to zero, one half, or one.
- The support graph is nonplanar with a maximum degree of four.
Further, the reduction also establishes hardness even when specific structural constraints are applied to the comb, specifically when every permitted tooth possesses either two or four vertices, with half of the tooth being situated within the handle structure.
Why This Matters
The established NP-hardness of separating comb inequalities has implications for approximation strategies and optimization efforts. This complexity result offers insight into the challenge of approximating the maximum comb violation and the optimization over the comb relaxation. The research also clarifies that the hardness of separation for comb inequalities does not automatically extend to larger families of inequalities.