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In a recent paper, Conte et al. [CGMR2017] presented an algorithm for enumerating all acyclic orientations of a graph G=(V,E) with a single source (and related orientations) with delay O(|V||E|). In this paper we revisit the problem by going back to an early paper by de Fraysseix et al. [FMR1995], who proposed an algorithm for enumerating all bipolar orientations of a graph based on a recursion formula. We first formalize de Fraysseix et al.'s algorithm for bipolar orientations and determine that its delay is also O(|V||E|). We then apply their recursion formula to the case of Conte et al.'s enumeration problem and show that this yields a more efficient enumeration algorithm with delay O(\sqrt(|V|)|E|). Finally, a way to further streamline the algorithm that leads to a particularly simple implementation is suggested.
Improved optimization models for potential-driven network flow problems via ASTS orientations
(2019)
The class of potential-driven network flow problems provides important models for a range of infrastructure networks that lead to hard-to-solve MINLPs in real-world applications. On large-scale meshed networks the relaxations usually employed are rather weak due to cycles in the network. To address this situation, we introduce the concept of ASTS orientations, a generalization of bipolar orientations, as a combinatorial relaxation of feasible solutions of potential-driven flow problems, study their structure, and show how they can be used to strengthen existing relaxations and thus provide improved optimization models. Our computational results indicate that ASTS orientations can be used to derive much stronger bounds on the flow variables than existing bound tightening methods and to yield significant performance improvements for an existing state-of-the-art MILP model for large-scale gas networks.
Optimization models often feature disjunctions of polytopes as
submodels. Such a disjunctive set is initially (at best) relaxed to
its convex hull, which is then refined by branching.
To measure the error of the convex relaxation, the (relative)
difference between the volume of the convex hull and the volume of the
disjunctive set may be used. This requires a method to compute the
volume of the disjunctive set. Naively, this can be done via
inclusion/exclusion and leveraging the existing code for the volume
of polytopes. However, this is often inefficient.
We propose a revised variant of an old algorithm by Bieri and Nef
(1983) for this purpose. The algorithm uses a sweep-plane to
incrementally calculate the volume of the disjunctive set as a
function of the offset parameter of the sweep-plane.
Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set. We propose a revised variant of an old algorithm by Bieri and Nef (1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane.