Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomial cases

Miroslav Rada, Milan Hladík, Elif Garajová

TL;DR

The paper addresses the problem of determining whether a given solution is weakly optimal for an interval linear program (ILP). It debunks a claimed polynomial-time result, proving NP-hardness in general via a reduction from weak feasibility of interval systems, and introduces an algorithm based on orthant decomposition that reduces the task to solving $2^k$ linear systems, where $k$ is the number of equality constraints. It also shows that polynomial-time solvability is achieved when the ILP contains only inequality constraints, or when the number of equalities is small, clarifying the boundary between tractable and intractable cases. The work combines a strong duality-based characterization with a practical constructive approach to identify witness scenarios, advancing both theoretical understanding and algorithmic capability for ILPs.

Abstract

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomial cases

TL;DR

The paper addresses the problem of determining whether a given solution is weakly optimal for an interval linear program (ILP). It debunks a claimed polynomial-time result, proving NP-hardness in general via a reduction from weak feasibility of interval systems, and introduces an algorithm based on orthant decomposition that reduces the task to solving linear systems, where is the number of equality constraints. It also shows that polynomial-time solvability is achieved when the ILP contains only inequality constraints, or when the number of equalities is small, clarifying the boundary between tractable and intractable cases. The work combines a strong duality-based characterization with a practical constructive approach to identify witness scenarios, advancing both theoretical understanding and algorithmic capability for ILPs.

Abstract

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.

Paper Structure

This paper contains 13 sections, 7 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Notation, intervals and interval linear programming
  4. Feasibility and optimality in classical linear programming.
  5. Feasibility and optimality in interval linear programming.
  6. Related work
  7. Counterexample and the weakness in the former proof
  8. Auxiliary result -- strong duality in LP
  9. NP-hardness proof
  10. Algorithm for \ref{['enu:cwopt']}
  11. The simple case: inequality constraints
  12. The general case: equations are allowed
  13. Conclusions

Key Result

Lemma 1

Consider an interval linear program $\textsl{ILP}(D^\mathrm{p}) = \textsl{ILP}(\bm{A^{\mathrm{f}}}, \bm{A^{\mathrm{n}}}, \bm{B^{\mathrm{f}}}, \bm{B^{\mathrm{n}}}, \bm{a}, \bm{b}, \bm{c^{\mathrm{f}}}, \bm{c^{\mathrm{n}}})$. The solution $x = (x^{\mathrm{f}}, x^{\mathrm{n}})$ is a weakly optimal solut for some $y^{\mathrm{f}} , y^{\mathrm{n}}$ and $(A^{\mathrm{f}}, A^{\mathrm{n}}, B^{\mathrm{f}}, B^

Figures (2)

  • Figure 1: Illustration of Example \ref{['exa:illustration']}. The space of rows of the matrix $B^{\mathrm{f}}$ is depicted. For example, the line $B^{\mathrm{f}}_{2:4}x=\bm{b}_{2:4}$ contains pairs of coefficients $(B^{\mathrm{f}}_{2,1},B^{\mathrm{f}}_{2,2}), (B^{\mathrm{f}}_{3,1}, B^{\mathrm{f}}_{3,2}), (B^{\mathrm{f}}_{4,1},B^{\mathrm{f}}_{4,2})$ such that second to fourth constraint is satisfied as equality for the given $x$.
  • Figure 2: Illustration of Example \ref{['exa:illustration2']}. The space of rows of the matrix $A^{\mathrm{f}}$ is depicted. For example, the line $A^{\mathrm{f}}_{2}x=\bm{b}_{2}$ contains pairs of coefficients $(A^{\mathrm{f}}_{2,1},A^{\mathrm{f}}_{2,2})$ such that second constraint is satisfied.

Theorems & Definitions (23)

  • Definition 1: Interval linear programming
  • Remark 1: on notation
  • Remark 2
  • Definition 2: Weak feasibility
  • Definition 3: Weak optimality
  • Remark 3
  • Example 1
  • Lemma 1: Characterization of weak optimality using strong duality
  • proof : of Lemma \ref{['lem:characterization:weak:optimality']}
  • Remark 4
  • ...and 13 more