Compact Integer Programming Formulation Research Articles

Social classroom seating assignment problems

AbstractUniversity students benefit academically, personally and professionally from an expansion of their in‐class social network. To facilitate this, we present a novel and broadly‐applicable optimization approach that exposes individuals to as many as possible peers that they do not know. This novel class of “social seating assignment problems” is parameterized by the social network, the physical seating structure and “tie potentials” representing the likelihood for two neighbors to connect. The resulting problem is NP‐hard and belongs to assignment problems with an elaborate objective that depends on the relation of both seats and individuals assigned to them. We develop compact integer programming formulations and strengthen them with valid inequalities to improve performance. In parallel, we suggest fast heuristics that are guided by network centrality measures. Finally, we present the necessary modeling techniques to integrate practically relevant instructor preferences and special student needs. Combining the above, we experiment on a set of 320 realistic instances with up to 200 students forming both sparse and dense social networks and for both rectangular and circular classrooms. For sparse or small instances we present optimality gaps under 1.3% within a few minutes, whereas in larger or denser cases the algorithmic performance decreases. We also evaluate our approach from both a quantitative and a qualitative perspective on three actual classes with more than 70 students. A network‐analysis‐based comparison of a priori and a posteriori networks shows that 40% of opportunities led to new connections, while feedback from both instructor and student is particularly favorable for our method.

Integer optimization models and algorithms for the multi-period non-shareable resource allocation problem

The resource allocation problem (RAP) determines a solution to optimally allocate limited resources to several activities or tasks. In this study, we propose a novel resource allocation problem referred to as multi-period non-shareable resource allocation problem (MNRAP), which is motivated by the characteristics of resources considered in the stem cell culture process for producing stem cell therapeutics. A resource considered in the MNRAP has the following three characteristics: (i) resource consumption required to perform an activity and available resource capacity may change over time; (ii) multiple activities cannot share one resource; and (iii) resource requirements can be satisfied through the combination of different types of resources. The MNRAP selects some of the given activities to maximize the overall profit under limited resources with these characteristics. To address this problem, pattern-based integer programming formulations based on the concept of resource patterns are proposed. These formulations attempt to overcome the limitations of a compact integer programming formulation, the utilization of which is challenging for large-scale problems owing to their complexity. Further, based on a branch-and-price approach to solving pattern-based formulations, effective heuristic algorithms are proposed to provide high-quality solutions for large instances. Moreover, through computational experiments on a wide range of instances, including real-world instances, the superiority of the proposed formulations and heuristic algorithms is demonstrated.

Min–max optimization of node‐targeted attacks in service networks

AbstractThis article considers resilience of service networks that are composed of service and control nodes to node‐targeted attacks. Two complementary problems of selecting attacked nodes and placing control nodes reflect the interaction between the network operator and the network attacker. This interaction can be analyzed within the framework of game theory. Considering the limited performance of the previously introduced iterative solution algorithms based on non‐compact problem models, new compact integer programming formulations of the node attack optimization problem are proposed, which are based on the notion of pseudo‐components and on a bilevel model. The efficiency of the new formulations is illustrated by the numerical study that uses two reference networks (medium‐size and large‐size), and a wide range of the sizes of attacks and controllers placements.

Dock assignment and truck scheduling problem; consideration of multiple scenarios with resource allocation constraints

The notion of ‘resource’ plays an important role in the overall efficiency and performance of most cross-docks. The processing time can often be described in terms of the resources allocated to different trucks. Conversely, for a given processing time, different combinations of resources can be prescribed. We study the problem of truck scheduling and dock assignment in the presence of resource constraints. In the absence of a closed-form (or well-defined) linear formulation describing the processing times as a function of resources, expert’ knowledge has been mobilized to enable modelling of the problem as an integer linear model. Two cases are taken into account: In the first one, the expert believes in his/her estimation of the processing time for every truck and only proposes a different combination of resources for his/her estimation, while in the second one the expert proposes a limited number of resource deployment scenarios for serving trucks, each of which has a different combination of resources and different processing times. We propose a novel compact integer programming formulation for the problem, which is particularly designed with an embedded structure that can be exploited in dual decomposition techniques with a remarkably computationally efficient column generation approach in this case. The case in which a scenario with invariant processing time is considered and modelled as a special case of the proposed model. Since a direct application of commercial solvers such as CPLEX to solve instances of this problem is not realistic, we propose a branch-and-price framework and, moreover, several classes of valid inequalities. Our extensive computational experiments confirm that the proposed exact solution framework is very efficient and viable in solving real-size instances of the practice and in a reasonable amount of time.

A branch-and-price algorithm for the Minimum Sum Coloring Problem

A proper coloring of a given graph is an assignment of a positive integer number (color) to each vertex such that two adjacent vertices receive different colors. This paper studies the Minimum Sum Coloring Problem (MSCP), which asks for finding a proper coloring while minimizing the sum of the colors assigned to the vertices. We propose the first branch-and-price algorithm to solve the MSCP to proven optimality. The newly developed exact approach is based on an Integer Programming (IP) formulation with an exponential number of variables which is tackled by column generation. We present extensive computational experiments, on synthetic and benchmark DIMACS graphs from the literature, to compare the performance of our newly developed branch-and-price algorithm against three compact IP formulations. On synthetic graphs, our algorithm outperforms the compact formulations in terms of: (i) number of solved instances, (ii) running times and (iii) exit gaps obtained when optimality is not achieved. For the DIMACS instances, our algorithm is competitive with the best compact formulation and provides very strong dual bounds.

New compact integer programming formulations for the multi-trip vehicle routing problem with time windows

We study two integer programming (IP) models for the multi-trip vehicle routing problem with time windows, service-dependent loading times, and limited trip duration (MTVRPTW-SDLT). Our first two-index formulation model represents vehicle returns to the depot in a graph with multiple copies of the depot node. The second two-index formulation model has just one depot node per vehicle, but includes a parallel arc for each pair of customer nodes representing an intermediate vehicle return to the depot. We compare these formulations against three-index formulations available in the literature over a set of benchmark instances. Results show that our models outperform existing formulations. We also adapt the proposed formulations for a relaxation of the MTVRPTW-SDLT without trip duration limit (LT). Our computational results also suggest that our models also improve their performance on MTVRPTW-SD instances.

Computing maxmin edge length triangulations

In 1991, Edelsbrunner and Tan gave an $O(n^2)$ algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate MELT within any polynomial factor. While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of non-trivial size. A straightforward IP based on pairwise disjointness of the $\Theta(n^2)$ segments between the n points has $\Theta(n^4)$ constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small $n$. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to $n = 200$. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets.

On the minimum-cost $$\lambda $$ λ -edge-connected k-subgraph problem

In this paper, we propose several integer programming (IP) formulations to exactly solve the minimum-cost $$\lambda $$ -edge-connected k-subgraph problem, or the $$(k,\lambda )$$ -subgraph problem, based on its graph properties. Special cases of this problem include the well-known k-minimum spanning tree problem (if $$\lambda =1$$ ), $$\lambda $$ -edge-connected spanning subgraph problem (if $$k=|V|$$ ) and k-clique problem (if $$\lambda = k-1$$ and there are exact k vertices in the subgraph). As a generalization of k-minimum spanning tree and a case of the $$(k,\lambda )$$ -subgraph problem, the (k, 2)-subgraph problem is studied, and some special graph properties are proved to find stronger and more compact IP formulations. Additionally, we study the valid inequalities for these IP formulations. Numerical experiments are performed to compare proposed IP formulations and inequalities.

Improved compact formulations for a wide class of graph partitioning problems in sparse graphs

Given an undirected connected graph G=(V,E) where |V|=n and |E|=m, we consider a wide class of graph partitioning problems, which includes as special cases several versions classically considered in the literature. These problems are to find a partition of the nodes in V into clusters satisfying several generic constraints (of set function type) on the clusters, in order to minimize the number (or the total weight) of the edges whose end-nodes do not belong to the same cluster. Partitions of V are often modeled by using compact integer programming formulations containing O(n3) triangle inequalities. The latter is the same whatever the sparsity of graph G could be, i.e. its size does not depend on m. In this paper, we show that one can reduce the size of the integer programming formulation to O(nm) triangle inequalities. Moreover, it is shown that, when the additional constraints on the clusters satisfy some monotonicity property, the strength of the linear programming relaxation is preserved by this reduction. We present numerical experiments on two important special cases arising from applications to show the benefit in terms of computational efficiency of using the reduced formulation.

Erratum—Exact Algorithms for the Clustered Vehicle Routing Problem

The clustered vehicle routing problem (CluVRP) is a generalization of the capacitated vehicle routing problem (CVRP) were customers are partitioned into disjoint clusters. As in the CVRP, all the customers must be visited exactly once, but a vehicle visiting one customer in a cluster must visit all the remaining customers therein before leaving it. Hence, when the first and last customer to be visited in a cluster C are p and q, finding the optimal sequence to visit the remaining customers in the cluster is equal to finding the minimum cost Hamiltonian path with p and q as endpoints. Battarra et al. (2014) presented a new compact integer programming formulation for the CluVRP. This formulation exploits the special substructure of the clusters such that only inter-cluster connections have to be determined. The authors show that this formulation provides a stronger linear relaxation than the CluVRP formulation based on the tradional CVRP two-index formulation. In the sequence of propositions that show the equivalence of the new formulation with the traditional CluVRP formulation, an erroneous notation was introduced in Battarra et al. (2014). In particular, this appears in the proof of Lemma 2, which is rewritten here in correct form. We emphasize that the incorrect notation in the proof does not affect the overall validity of Lemma 2, hence has no effect on the correctness of the paper and on the results reported therein. Let „4S5 denote the set of edges connecting the vertices in a vertex set S ⊆ V to those outside the set, and the set of edges inside a vertex set S by E4S5. The decision variables x ij and x ∗∗ ij are equal to the number of times a vehicle traverses edge 4i1 j5 for a solution of the traditional and the new formulation, respectively. The values of x ij are determined through the following transformation from x ij . For each inter-cluster edge 4i1 j5 ∈ Ē, set x ij = x ∗∗ ij . Initialize x ij = 0 for all intra-cluster edges 4i1 j5 ∈ E. For all 4p1 q5 ∈ E and for all 4i1 j5 ∈ P4p1q5, increment x ij by x ∗∗ pq . For the details of the CluVRP formulations, we refer the reader to Battarra et al. (2014). Lemma 2. Given a set of customer vertices S ⊂ V \809, with C being the minimal set of clusters covering S, i.e., ∀i ∈ S, ∃C ∈ C: i ∈C, the inequality ∑

Scheduling Promotion Vehicles to Boost Profits

In addition to setting price discounts, retailers need to decide how to schedule promotion vehicles, such as flyers and TV commercials. Unlike the promotion pricing problem that received great attention from both academics and practitioners, the promotion vehicle scheduling problem was largely overlooked, and our goal is to study this problem both theoretically and in practice. We model the problem of scheduling promotion vehicles to maximize profits as a nonlinear bipartite matching-type problem, where promotion vehicles should be assigned to time periods, subject to capacity constraints. Our modeling approach is motivated and calibrated using actual data in collaboration with Oracle Retail, leading us to introduce and study a class of models for which the boost effects of promotion vehicles on demand are multiplicative. From a technical perspective, we prove that the general setting considered is computationally intractable. Nevertheless, we develop approximation algorithms and propose a compact integer programming formulation. In particular, we show how to obtain a (1 − ε)-approximation using an integer program of polynomial size, and investigate the performance of a greedy procedure, both analytically and computationally. We also discuss an extension that includes cross-term effects to capture the cannibalization aspect of using several vehicles simultaneously. From a practical perspective, we test our methods on actual data through a case study, and quantify the impact of our models. Under our model assumptions and for a particular item considered in our case study, we show that a rigorous optimization approach to the promotion vehicle scheduling problem allows the retailer to increase its profit by 2% to 9%. The online appendix is available at https://doi.org/10.1287/mnsc.2017.2926 . This paper was accepted by Yinyu Ye, optimization.

On connected dominating sets of restricted diameter

A connected dominating set (CDS) is commonly used to model a virtual backbone of a wireless network. To bound the distance that information must travel through the network, we explicitly restrict the diameter of a CDS to be no more than s leading to the concept of a dominating s-club. We prove that for any fixed positive integer s it is NP-complete to determine if a graph has a dominating s-club, even when the graph has diameter s+1. As a special case it is NP-complete to determine if a graph of diameter two has a dominating clique. We then propose a compact integer programming formulation for the related minimization problem, enhance the approach with variable fixing rules and valid inequalities, and present computational results.

Energy efficient network-flow-based algorithm for multiuser multicarrier systems

The resource allocation problem of minimising transmission power with per user rate constraint is studied for energy efficiency of multiuser multicarrier systems. The authors introduce an algorithm that deploys a flow-based decomposition strategy, called the network-flow-based algorithm (NFBA), to circumvent the NP-hard complexity of the resource allocation problem. The authors prove global optimality of the algorithm for the case of a flat-fading channel. For the general frequency selective channel, the flow size is adjusted adaptively to minimise the transmission power. Also, a compact integer programming formulation is developed to provide tight lower and upper bounds of the minimum power. Numerical results show that the authors' proposed NFBA achieves near-optimal performance with polynomial complexity even for large-scale systems.

Scheduling jobs with step-deterioration; Minimizing makespan on a single- and multi-machine

Certain maintenance procedures fail to perform prior to a pre-specified deadline and need extra time for successful accomplishment. Processing times under this specific deterioriation are represented by a step function. We study the problem of makespan minimization of step-wise deteriorating jobs. We introduce compact integer programming formulations. We then focus on heuristic methods, since even the case of single machine single-step deterioration is NP-complete. These methods are extended to settings of multi-machine and multi-step deterioration. Our numerical tests indicate that the proposed algorithms are highly accurate and efficient in the context of a numerical study discussed in this paper.