Uncapacitated flow-based extended formulations

Fiorini, Samuel; Pashkovich, Kanstantsin

doi:10.1007/s10107-015-0862-9

Cited by 5 publications

(2 citation statements)

References 31 publications

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“…We remark that it is possible that the semiorder polytope admits an extended formulation with size 2 Θ( n ) but no such extended formulation based on flows. Indeed, flow-based extended formulations are known to have strong limitations, see Fiorini & Pashkovich (2015).…”

Section: Optimalitymentioning

confidence: 99%

Extended formulations for order polytopes through network flows

Davis‐Stober

Doignon

Fiorini

et al. 2018

Journal of Mathematical Psychology

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Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathematical structure of these models. However, standard methods for characterizing the polytopes of such models are subject to a combinatorial explosion in complexity as the number of choice alternatives increases. Specifically, this is the case for random preference models based on linear, weak, semi- and interval orders. For these, a complete, linear description of the polytope is currently known only for, at most, 5–8 choice alternatives. We leverage the method of extended formulations to break through those boundaries. For each of the four types of preferences, we build an appropriate network, and show that the associated network flow polytope provides an extended formulation of the polytope of the choice model. This extended formulation has a simple linear description that is more parsimonious than descriptions obtained by standard methods for large numbers of choice alternatives. The result is a computationally less demanding way of testing the probabilistic choice model on data. We sketch how the latter interfaces with recent developments in contemporary statistics.

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Section: Optimalitymentioning

confidence: 99%

Extended formulations for order polytopes through network flows

Davis‐Stober

Doignon

Fiorini

et al. 2018

Journal of Mathematical Psychology

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“…Recently Lancia and Serafini provide a general framework to generate compact extended formulations for problems with exponentially many variables and/or constraints. Fiorini and Pashkovich discuss the “uncapacitated flow based extended formulations” for several polytopes.…”

Section: Extended Formulationmentioning

confidence: 99%

Two extended formulations for cardinality maximum flow network interdiction problem

Rad

Kakhki

2017

Networks

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We consider the maximum flow network interdiction problem in its cardinality case. There is an integer programming model for this problem by Wood (Math Comput Model 17 (1993), 1–18). Two types of valid inequalities have also been proposed (Altner et al., Oper Res Lett 38 (2010), 33–38 and Wood, Math Comput Model 17 (1993), 1–18) to strengthen the LP relaxation of the integer model. However, due to their combinatorial nature, the number of these inequalities are exponential. Here, we present an equivalent reformulation (extended formulation) for this problem which has a polynomial number of constraints. We also introduce new valid inequalities, and show that the corresponding reformulation of the LP relaxation of the integer model augmented with these inequalities, significantly decreases the integrality gap for a class of network interdiction problems with proven large integrality gaps. Numerical results for some benchmark as well as randomly generated instances are also reported. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 367–377 2017

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