Exponential Lower Bounds for Polytopes in Combinatorial Optimization

Fiorini, Samuel; Massar, Serge; Pokutta, Sebastian; Tiwary, Hans Raj; Wolf, Ronald de

doi:10.1145/2716307

Cited by 120 publications

(116 citation statements)

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“…We will use their lower bound on the cut polytope below. Rothvoss [22] then showed such lower bounds on several other polytopes; we use his lower bound on the TSP polytope, improving that of Fiorini et al [8], to get the result above.…”

Section: Introductionmentioning

confidence: 74%

“…(Jerrum and Snir [11] already showed the permanent requires monotone circuits of size 2 Ω(n) over R and over the tropical (min, +) semi-ring.) Here, by building on Fiorini et al's [8] and Rothvoss's [22] extended formulation lower bound for the TSP polytope, we show that no such monotone reduction exists-over R, nor over the tropical semi-ring, nor over the Boolean semi-ring-by connecting monotone p-projections to extended formulations of polytopes.…”

Section: Introductionmentioning

confidence: 81%

“…But for more than 20 years, it was an open question of how to remove the condition of symmetry from Yannakakis's result. In a landmark result, Fiorini, Massar, Pokutta, Tiwary, and de Wolf [8] achieved this, by showing an exponential lower bound on arbitrary extended formulations of several different polytopes. We will use their lower bound on the cut polytope below.…”

Section: Introductionmentioning

confidence: 96%

“…We also use the same technique, this time in combination with the lower bound of Fiorini et al on extended formulations of the cut polytope [8], to show that the cut polynomials…”

Section: Introductionmentioning

confidence: 99%

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Grochow¹

2017

Theory of Comput.

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In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences.1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP R under monotone p-projections.2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs.As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.

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Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 96%

“…We also use the same technique, this time in combination with the lower bound of Fiorini et al on extended formulations of the cut polytope [8], to show that the cut polynomials…”

Section: Introductionmentioning

confidence: 99%

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Grochow¹

2017

Theory of Comput.

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“…The vehicle scheduling problem and the optimal allocation of resources are combinatorial optimization problems [4][5], the multi machine scheduling problem is also included. Sun F [6] studied the multiple machine scheduling problem based on fuzzy operation time.…”

Section: Introductionmentioning

confidence: 99%

Research on Weighing Strategy of Vehicle Entering Plant Based on Fuzzy Operation Time

Chen¹,

Wang²

2017

dtetr

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Abstract. In order to solve the problem of congestion and disorder in the process of weighing and handling of cement vehicles entering the plant, a new scheduling strategy based on fuzzy operation time is proposed. Firstly, the modeling and analysis of the problem, and then proposes a genetic algorithm to solve the problem. The algorithm was improved and the different algorithm and the improved algorithm optimization results was compared. The results show that the proposed scheme can effectively solve congestion problems and increase weighing efficiency of vehicle entering plant.

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References

2020

Integer Programming

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Exponential Lower Bounds for Polytopes in Combinatorial Optimization

Abstract: We provide a numerical refutation of the developments of Fiorini et al. (2015) * for modelswith disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models.

Cited by 120 publications

References 52 publications

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Research on Weighing Strategy of Vehicle Entering Plant Based on Fuzzy Operation Time

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