[Paper Review] Tight MIP formulations for bounded length cyclic sequences

Thomas Kalinowski, Tomas Lidén+1 more|arXiv (Cornell University)|Sep 6, 2018

Scheduling and Optimization Algorithms12 references3 citations

TL;DR

This paper presents tight mixed-integer programming (MIP) formulations for bounded-length cyclic binary sequences, modeling periodic schedules with minimum and maximum on/off interval lengths. It introduces a compact extended network flow formulation and derives facet-defining valid inequalities in the state and start-up variable space, using disjunctive programming to construct an extended formulation for the convex hull, ensuring integrality and tightness for cyclic scheduling problems.

ABSTRACT

We study cyclic binary strings with bounds on the lengths of the intervals of consecutive ones and zeros. This is motivated by scheduling problems where such binary strings can be used to represent the state (on/off) of a machine. In this context the bounds correspond to minimum and maximum lengths of on- or off-intervals, and cyclic strings can be used to model periodic schedules. Extending results for non-cyclic strings is not straight forward. We present a non-trivial tight compact extended network flow formulation, as well as valid inequalities in the space of the state and start-up variables some of which are shown to be facet-defining. Applying a result from disjunctive programming, we also convert the extended network flow formulation into an extended formulation over the space of the state and start-up variables.

Motivation & Objective

• To develop tight MIP formulations for cyclic binary sequences with bounded on- and off-interval lengths.
• To address the challenge of extending non-cyclic formulations to cyclic settings where wraparound intervals violate standard integrality.
• To derive valid inequalities in the space of state and start-up variables that are facet-defining under constant bounds.
• To construct an extended formulation for the convex hull using disjunctive programming techniques.
• To ensure integrality and tightness of the formulation despite cyclic dependencies.

Proposed method

• Adapts the extended network flow formulation from [13] to cyclic sequences by exploiting a disjunction to enlarge the network and restore integrality.
• Defines state variables $ y_t $ and start-up variables $ z_t $ to model on/off states and transitions.
• Introduces a path-based formulation using auxiliary variables $ \lambda $ and constraints indexed by $ \tau \in T_i $ to model wraparound intervals.
• Applies disjunctive programming to combine multiple polytopes $ \hat{P}^{(i,\tau)} $, each corresponding to a possible start of an on- or off-interval.
• Uses projection $ f $ to map extended formulations back to the original $ y $ and $ z $ variables.
• Derives a convex hull formulation via a convex combination of $ \hat{P}^{(i,\tau)}(1) $, ensuring integrality and tightness.

Experimental results

Research questions

• RQ1 Can a compact extended network flow formulation be constructed for cyclic binary sequences with bounded interval lengths, ensuring integrality?
• RQ2 Are the valid inequalities in the state and start-up variable space facet-defining when bounds are constant over time?
• RQ3 Does the disjunctive programming approach yield an extended formulation that describes the convex hull of the feasible set $ Z(n,\bm{\alpha},\bm{\beta},\bm{\gamma},\bm{\delta}) $?
• RQ4 Is the projection of the extended formulation $ Q $ onto the original variables equal to the original polytope $ P $, i.e., $ \operatorname{proj}_{y,z}(Q) = P $?
• RQ5 What is the complexity of the convex hull in terms of number of facets, and is it polynomial?

Key findings

• A non-trivial extended network flow formulation is constructed that yields an integral polytope by exploiting a disjunctive structure to enlarge the network.
• The formulation is tight and compact, extending the non-cyclic approach from [13] to cyclic settings.
• For constant bounds, sufficient conditions are provided under which valid inequalities in the $ y $-$ z $ space are facet-defining.
• An extended formulation for the convex hull is derived via disjunctive programming, combining multiple path-based polytopes $ \hat{P}^{(i,\tau)} $ with a convex combination over $ \lambda $.
• The projection of the extended formulation $ Q $ onto the original variables equals the original polytope $ P $, as verified in small cases with polymake.
• The method ensures that the convex hull of the feasible set is described exactly using a polynomial-sized extended formulation.