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On modeling NP-Complete problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers"

Moustapha Diaby, Mark Karwan, Lei Sun

TL;DR

This paper clarifies that the nonexistence of a polynomial-sized extended formulation for the $TSP$ polytope does not preclude solving the $TSP$ with a polynomial-sized linear program. By exploiting a linear assignment problem (LAP) perspective via the $TSP$ Assignment Graph (TSPAG) and introducing auxiliary variables to encode tour costs, the authors show that an EF of the LAP polytope can enable a polynomial-sized LP to solve the $TSP$ without requiring the traditional travel-leg variables or a projection to the $TSP$ polytope. They provide a motivating example with a specific EF of the LAP and develop general theorems indicating that, under suitable cost attachments, LPs over EF-extensions of $\\\mathcal{A}_{n}$ can correctly recover the optimal $TSP$ tour. The results imply that escaping the barriers relies on cost modeling via LAP-based extensions rather than relying on projections to $\\\mathcal{P}_{n}$, with potential applicability to other NP-Complete problems modeled as LAP variants.

Abstract

In view of the extended formulations (EFs) developments (e.g. "Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf [2015]. Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2"), we focus in this paper on the question of whether it is possible to model an NP-Complete problem as a polynomial-sized linear program. For the sake of simplicity of exposition, the discussions are focused on the TSP. We show that a finding that there exists no polynomial-sized extended formulation of "the TSP polytope" does not (necessarily) imply that it is "impossible" for a polynomial-sized linear program to solve the TSP optimization problem. We show that under appropriate conditions the TSP optimization problem can be solved without recourse to the traditional city-to-city ("travel leg") variables, thereby side-stepping/"escaping from" "the TSP polytope" and hence, the barriers. Some illustrative examples are discussed.

On modeling NP-Complete problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers"

TL;DR

This paper clarifies that the nonexistence of a polynomial-sized extended formulation for the polytope does not preclude solving the with a polynomial-sized linear program. By exploiting a linear assignment problem (LAP) perspective via the Assignment Graph (TSPAG) and introducing auxiliary variables to encode tour costs, the authors show that an EF of the LAP polytope can enable a polynomial-sized LP to solve the without requiring the traditional travel-leg variables or a projection to the polytope. They provide a motivating example with a specific EF of the LAP and develop general theorems indicating that, under suitable cost attachments, LPs over EF-extensions of can correctly recover the optimal tour. The results imply that escaping the barriers relies on cost modeling via LAP-based extensions rather than relying on projections to , with potential applicability to other NP-Complete problems modeled as LAP variants.

Abstract

In view of the extended formulations (EFs) developments (e.g. "Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf [2015]. Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2"), we focus in this paper on the question of whether it is possible to model an NP-Complete problem as a polynomial-sized linear program. For the sake of simplicity of exposition, the discussions are focused on the TSP. We show that a finding that there exists no polynomial-sized extended formulation of "the TSP polytope" does not (necessarily) imply that it is "impossible" for a polynomial-sized linear program to solve the TSP optimization problem. We show that under appropriate conditions the TSP optimization problem can be solved without recourse to the traditional city-to-city ("travel leg") variables, thereby side-stepping/"escaping from" "the TSP polytope" and hence, the barriers. Some illustrative examples are discussed.
Paper Structure (13 sections, 10 theorems, 40 equations, 4 figures)

This paper contains 13 sections, 10 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions, definitions, and notations
  4. Alternate representations of TSP tours
  5. “The TSP polytope”
  6. LAP solution representation of TSP tours
  7. Overview of the barriers
  8. Basic results
  9. Inferences and Misconceptions
  10. Side-stepping of the barriers
  11. A motivating illustrative example
  12. General results
  13. Conclusions

Key Result

Theorem 6

“ The TSP polytope does not have a symmetric EF of subexponential size$.$”

Figures (4)

  • Figure 1: Illustration of the TSP Graph
  • Figure 2: Illustration of the TSP Assignment Graph (TSPAG)
  • Figure 3: Illustration of a Hamiltonian Cycle
  • Figure 4: Illustration of TSP paths

Theorems & Definitions (19)

  • Definition 2: “ TSP paths”
  • Remark 3
  • Definition 4: Extended Formulation (EF); Yannakakis [1991]
  • Definition 5: “ Symmetric Polytope"; Yannakakis [1991]
  • Theorem 6: Barrier #1; Yannakakis [1991]
  • Theorem 7: Barrier #2; Fiorini et al. [2015]
  • Claim 8: Reasoning
  • Remark 9: Misconceptions
  • Lemma 10
  • Lemma 11
  • ...and 9 more