Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Complexity of Combinatorial Optimization on Fixed Structures

Nimrod Megiddo

TL;DR

This paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is fixed, and demonstrated that in some cases fixing the structure makes the problem easier, whereas in general the problem remains NP-complete.

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.

On the Complexity of Combinatorial Optimization on Fixed Structures

TL;DR

This paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is fixed, and demonstrated that in some cases fixing the structure makes the problem easier, whereas in general the problem remains NP-complete.

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.

Paper Structure

This paper contains 13 sections, 10 theorems, 60 equations, 2 figures, 2 tables.

Table of Contents

  1. . Introduction
  2. . Improved time bounds for special classes
  3. Maximal feasible subsets
  4. Dynamic-Programming bounds
  5. . NP-completeness
  6. An optimization model for 1-in-3-SAT
  7. Variables
  8. Constraints
  9. Converting to unique-choice constraints
  10. Aggregating constraints
  11. The reduction
  12. An example
  13. . Other problems

Key Result

Proposition 3.2

A CNF formula with at most three literals per clause has a satisfying assignment with exactly one literal per clause if and only if the constraints (constr:eq), (constr:ineq, (const:Ci), (constr:tij) and (constr:tijbar) are satisfied

Figures (2)

  • Figure 1: Digits of the weight coefficients
  • Figure 2: Digits of the value coefficient

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 7 more