On the Complexity of Combinatorial Optimization on Fixed Structures

TL;DR

This paper examines how fixing the structure in combinatorial optimization problems affects their computational complexity, showing that it can sometimes simplify the problem but often remains NP-complete.

Contribution

It provides a detailed analysis of the complexity of combinatorial optimization problems with fixed structures, highlighting cases where fixing the structure reduces complexity.

Findings

Fixing the structure can simplify some combinatorial problems.

Many fixed-structure problems remain NP-complete.

Structural constraints influence problem complexity.

Abstract

Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is fixed. It is demonstrated that in some cases fixing the structure makes the problem easier, whereas in general the problem remains NP-complete.