Extended Formulation Lower Bounds via Hypergraph Coloring?

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate relaxation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product formulations for the metric capacitated facility location problem, a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned

Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at $\Omega(n)$ levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question of \cite{LiS13, AnBS13}. Here, $n$ denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities \cite{AardalPW95} has also an unbounded gap. This disproves a long-standing conjecture of \cite{LeviSS12}. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599

Factors Influencing Electric Vehicle Penetration in the EU by 2030: A Model-Based Policy Assessment

The European Commission (EC) has set ambitious CO"sub"2"/sub" emission reduction objectives for the transport sector by 2050. In this context, most decarbonisation scenarios for transport foresee large market penetration of electric vehicles in 2030 and 2050. The emergence of electrified car mobility is, however, uncertain due to various barriers such as battery costs, range anxiety and dependence on battery recharging networks. Those barriers need to be addressed in the 2020–2030 decade, as this is key to achieving electrification at a large scale in the longer term. The paper explores the uncertainties prevailing in the first decade and the mix of policies to overcome the barriers by quantifying a series of sensitivity analysis scenarios of the evolution of the car markets in the EU Member States and the impacts of each barrier individually. The model used is PRIMES-TREMOVE, which has been developed by E3MLab and constitutes a detailed energy-economic model for the transport sector. Based on model results, the paper assesses the market, energy, emission and cost impacts of various CO"sub"2"/sub" car standards, infrastructure development plans with different geographic coverage and a range of battery cost reductions driven by learning and mass industrial production. The assessment draws on the comparison of 29 sensitivity scenarios for the EU, which show that removing the barriers in the decade 2020–2030 is important for electrification emergence. The results show that difficult policy dilemmas exist between adopting stringent standards and infrastructure of wide coverage to push technology and market development and adverse effects on costs, in case the high cost of batteries persists. However, if the pace of battery cost reductions is fast, a weak policy for standards and infrastructure is not cost-effective and sub-optimal. These policies are shown to have impacts on the competition between pure electric and plug-in hybrid vehicles. Drivers that facilitate electrification also favour the uptake of the former technology, the latter being a reasonable choice only in case the barriers persist and obstruct electrification. Document type: Articl

Extended Formulation Lower Bounds via Hypergraph Coloring?

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate extended formulation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product relaxations for the metric capacitated facility location problem (CFL), a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned. Our lower bound implies an unbounded integrality gap for CFL at Theta(N) levels of the universal SA hierarchy which is independent of the starting relaxation; we only require that the starting relaxation has size 2 degrees((N)), where N is the number of facilities in the instance. This proof yields the first such tradeoff for an SA procedure that is independent of the initial relaxation

Simulating the Evolution of Business Models for Electricity Recharging Infrastructure Development by 2030: A Case Study for Greece

It is widely accepted that the market uptake of electric vehicles is essential for the decarbonisation of transport. However, scaling up the roll out of electric vehicles (EV) is challenging considering the lack of charging infrastructure. The latter is, currently, developing in an uneven way across the EU countries. A charging infrastructure with wide coverage addresses range limitations but requires high investment with uncertain returns during the early years of deployment. The aim of this paper is to assess how different policy options affect EV penetration and the involvement of private sector in infrastructure deployment. We propose a mathematical programming model of the decision problem and the interaction between the actors of EV charging ecosystem and apply it to the case of Greece from the time period until 2030. Greece represents a typical example of a country with ambitious targets for EV penetration by 2030 (10% of the total stock) with limited effort made until now. The results indicate that it is challenging to engage private investors in the early years, even using subsidies; thus, publicly financed infrastructure deployment is important for the first years. In the mid-term, subsidization on the costs of charging points is necessary to positively influence the uptake of private investments. These are mainly attracted from 2025 onwards, after a critical mass of EVs and infrastructure has been deployed