90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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Keywords
- Branchwidth (1)
- Carvingwidth (1)
- Parameterized complexity (1)
- Periodic Event Scheduling Problem (1)
- Periodic timetabling (1)
- Treewidth (1)
- acyclic orientations (1)
- bipolar orientations (1)
- branch and bound (1)
- enumeration algorithm (1)
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- Mathematical Optimization (4) (remove)
Public transportation networks are typically operated with a periodic timetable. The Periodic Event Scheduling Problem (PESP) is the standard mathematical modelling tool for periodic timetabling. Since PESP can be solved in linear time on trees, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. To complete the picture, we present two pseudo-polynomial-time dynamic programming algorithms solving PESP on input networks with bounded tree- or branchwidth. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special -- but standard -- structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.
This paper investigates the estimation of the size of Branch-and-Bound (B&B) trees for solving mixed-integer programs. We first prove that the size of the B&B tree cannot be approximated within a factor of~2 for general binary programs, unless P equals NP. Second, we review measures of the progress of the B&B search, such as the gap, and propose a new measure, which we call leaf frequency.
We study two simple ways to transform these progress measures into B&B tree size estimates, either as a direct projection, or via double-exponential smoothing, a standard time-series forecasting technique. We then combine different progress measures and their trends into nontrivial estimates using Machine Learning techniques, which yields more precise estimates than any individual measure. The best method we have identified uses all individual measures as features of a random forest model.
In a large computational study, we train and validate all methods on the publicly available MIPLIB and Coral general purpose benchmark sets. On average, the best method estimates B&B tree sizes within a factor of 3 on the set of unseen test instances even during the early stage of the search, and improves in accuracy as the search progresses. It also achieves a factor 2 over the entire search on each out of six additional sets of homogeneous instances we have tested. All techniques are available in version 7 of the branch-and-cut framework SCIP.
The SCIP Optimization Suite provides a collection of software packages for
mathematical optimization centered around the constraint integer programming frame-
work SCIP. This paper discusses enhancements and extensions contained in version 7.0
of the SCIP Optimization Suite. The new version features the parallel presolving library
PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op-
timization problems and can be used stand-alone or integrated into SCIP via a presolver
plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im-
provements in the Benders’ decomposition solver of SCIP, user-defined decomposition
structures can be read, which are used by the automated Benders’ decomposition solver
and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation
that is used to predict the completion of the overall solving process and potentially
trigger restarts. Moreover, substantial performance improvements of the MIP core were
achieved by new developments in presolving, primal heuristics, branching rules, conflict
analysis, and symmetry handling. Last, not least, the report presents updates to other
components and extensions of the SCIP Optimization Suite, in particular, the LP solver
SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.
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.