ILP Formulation Research Articles

Efficient branch-and-bound algorithms for finding triangle-constrained 2-clubs

In the Vertex Triangle 2-Club problem, we are given an undirected graph G and aim to find a maximum-vertex subgraph of G that has diameter at most 2 and in which every vertex is contained in at least ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} triangles in the subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies on an ILP formulation (Almeida and Brás in Comput Oper Res 111:258–270, 2019). In this work, we develop a combinatorial branch-and-bound algorithm that, coupled with a set of data reduction rules, outperforms the existing implementation and is able to find optimal solutions on sparse real-world graphs with more than 100,000 vertices in a few minutes. We also extend our algorithm to the Edge Triangle 2-Club problem where the triangle constraint is imposed on all edges of the subgraph.

Revisiting the complexity of and algorithms for the graph traversal edit distance and its variants

The graph traversal edit distance (GTED), introduced by Ebrahimpour Boroojeny et al. (2018), is an elegant distance measure defined as the minimum edit distance between strings reconstructed from Eulerian trails in two edge-labeled graphs. GTED can be used to infer evolutionary relationships between species by comparing de Bruijn graphs directly without the computationally costly and error-prone process of genome assembly. Ebrahimpour Boroojeny et al. (2018) propose two ILP formulations for GTED and claim that GTED is polynomially solvable because the linear programming relaxation of one of the ILPs always yields optimal integer solutions. The claim that GTED is polynomially solvable is contradictory to the complexity results of existing string-to-graph matching problems. We resolve this conflict in complexity results by proving that GTED is NP-complete and showing that the ILPs proposed by Ebrahimpour Boroojeny et al. do not solve GTED but instead solve for a lower bound of GTED and are not solvable in polynomial time. In addition, we provide the first two, correct ILP formulations of GTED and evaluate their empirical efficiency. These results provide solid algorithmic foundations for comparing genome graphs and point to the direction of heuristics. The source code to reproduce experimental results is available at https://github.com/Kingsford-Group/gtednewilp/.

A Bigram Based ILP Formulation for Break Minimization in Sports Scheduling Problems

A Bigram Based ILP Formulation for Break Minimization in Sports Scheduling Problems

Two is better than one? Order aggregation in a meal delivery scheduling problem

We address a single-machine scheduling problem motivated by a last-mile-delivery setting for a food company. Customers place orders, each characterized by a delivery point (customer location) and an ideal delivery time. An order is considered on time if it is delivered to the customer within a time window given by the ideal delivery time ±δ2, where δ is the same for all orders. A single courier (machine) is in charge of delivery to all customers. Orders are either delivered individually, or two orders can be aggregated in a single courier trip. All trips start and end at the restaurant, so no routing decisions are needed. The problem is to schedule courier trips so that the number of late orders is minimum. We show that the problem with order aggregation is NP-hard and propose a combinatorial branch and bound algorithm for its solution. The algorithm performance is assessed through a computational study on instances derived by a real-life application and on randomly generated instances. The behavior of the combinatorial algorithm is compared with that of the best ILP formulation known for the problem. Through another set of computational experiments, we also show that an appropriate choice of design parameters allows to apply the algorithm to a dynamic context, with orders arriving over time.

Cycle selections

We introduce the following cycle selection problem which is motivated by an application to kidney exchange problems. Given a directed graph G = ( V , A ) , a cycle selection is a subset of arcs B ⊆ A forming a union of directed cycles. A related optimization problem, the Maximum Weighted Cycle Selection problem can be defined as follows: given a weight w i , j ∈ R for all arcs ( i , j ) ∈ A , find a cycle selection B which maximizes w ( B ) . We prove that this problem is strongly NP-hard. Next, we focus on cycle selections in complete directed graphs. We provide several ILP formulations of the problem: an arc formulation featuring an exponential number of constraints which can be separated in polynomial time, four extended compact formulations, and an extended non compact formulation. We investigate the relative strength of these formulations. We concentrate on the arc formulation and on the description of the associated cycle selection polytope. We prove that this polytope is full-dimensional, and that all the inequalities used in the arc formulation are facet-defining. Furthermore, we describe three new classes of facet-defining inequalities and a class of valid inequalities. We also consider the consequences of including additional constraints on the cardinality of a selection or on the length of the associated cycles.

Using Simple Incentives to Improve Two-Sided Fairness in Ridesharing Systems

State-of-the-art order dispatching algorithms for ridesharing batch passenger requests and allocate them to a fleet of vehicles in a centralized manner, optimizing over the estimated values of each vehicle-passenger matching using integer linear programming. Using good estimates of future value, such ILP-based approaches are able to significantly increase the service rates (percentage of requests served) for a fixed fleet of vehicles. However, such approaches that focus solely on maximizing efficiency can lead to disparities for both drivers (e.g., income inequality) and passengers (e.g., inequality of service for different groups). Existing approaches that consider fairness only do it for naive assignment policies, require extensive training or look at only single-sided fairness. We propose a simple incentive-based fairness scheme (SI) that can be implemented online as a part of this ILP formulation that allows us to improve fairness over a variety of fairness metrics. Deriving from a lens of variance minimization, we describe how these fairness incentives can be formulated for two distinct use cases for passenger groups (SIP) and driver (SID) fairness. We show that under mild conditions, our approach can guarantee an improvement in the chosen metric for the worst-off individual. We also show empirically that our Simple Incentives approach significantly outperforms prior art, despite requiring no retraining; indeed, it often leads to a large improvement over the state-of-the-art fairness-aware approach in both overall service rate and fairness.

GeoSS: Geographic Segmentation Security Barriers for Virtual Emotion Detection With Discriminative Priorities in Intelligent Cooperative Vehicular System

A development of the integrated vehicular system through ground and aerial cooperation using intelligent mobile robots and smart UAVs is required to support various applications including intelligent transportation, secure service, surveillance reinforcement. Also, it is highly anticipated that the applicability of virtual emotion is expanded continuously in smart systems. In this paper, we introduce a geographic segmentation security barrier system for the purpose of secure surveillance through virtual emotion detection in intelligent cooperative vehicular area with mobile robots and UAVs. The proposed system is necessary to provide discriminative priorities in the requested cooperative vehicular area consisting of mobile robots and UAVs with rapid construction. Moreover, with ILP formulation, we formally defined a problem whose goal is to maximize a total number of geographic segmentation security barriers such that the completed establishment of those barriers is done by a specific group of mobile robots and UAVs within the requested various districts where they are divided according to differential security priorities. To solve the problem, we devise novel schemes to return the maximum number of geographic segmentation security barriers so that it pursues the maximization of system lifetime consequently. Then, the proposed approaches are implemented through expansive simulations and their performances based on numerical results are evaluated clearly.

Checking Phylogenetics Decisiveness in Theory and in Practice.

Suppose we aim to build a phylogeny for a set of taxa X using information from a collection of loci, where each locus offers information for only a fraction of the taxa. The question is whether, based solely on the pattern of data availability, called a taxon coverage pattern, one can determine if the data suffices to construct a reliable phylogeny. The problem can be expressed combinatorially as follows. Let us call a taxon coverage pattern decisive if, for any binary phylogenetic tree T for X, the collection of phylogenetic trees obtained by restricting T to the subset of X covered by each locus uniquely determines T. Here we relate the problem of checking whether a taxon coverage pattern is decisive to a hypergraph coloring problem. Using this connection, we (1) show that checking decisiveness is co-NP complete; (2) obtain lower bounds on the amount of coverage needed to achieve decisiveness; (3) devise an exact algorithm for decisiveness; (4) develop problem reduction rules, and use them to obtain efficient algorithms for inputs with few loci; and (5) devise Boolean satisfiability (SAT) and integer linear programming formulations (ILP) of decisiveness that allow us to analyze data sets that arise in practice. For data sets that are not decisive, we use our SAT and ILP formulations to obtain decisive subsets of the data.

Algorithmic aspects of Roman {3}-domination in graphs

Let G be a simple, undirected graph. A function g : V(G) → {0, 1, 2, 3} having the property that ∑v∈NG(u)g(v)≥3, if g(u) = 0, and ∑v∈NG(u)g(v)≥2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V)=∑v∈Vg(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ{R3}(G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. We propose a 3(1 + ln(Δ − 1))-approximation algorithm for the MR3DP, where Δ is the maximum degree of G and show that the MR3DP problem cannot be approximated within (1 − ε)ln|V| for any ε > 0 unless NP ⊆ DTIME(|V|O(loglog|V|)). Next, we show that the MR3DP problem is APX-complete for graphs with maximum degree 4. We also show that the domination and Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, an ILP formulation for MR3DP is proposed.

Partial coverage optimization under network connectivity constraints in heterogeneous sensor networks

The maximum network lifetime problem (MLP) in heterogeneous wireless sensor networks under connectivity and coverage constraints is addressed in this paper. Two main variants of the studied problem are considered. The first variant is α-coverage where a portion ((1 - α) percent) of the targets are allowed to be left uncovered. The second one is called β-coverage or β-constraint where each target has a minimum coverage rate β during the network’s lifetime service. When the two variants are considered at the same time, the problem is called αβ−Connected Maximum Lifetime Problem (αβ−CMLP) where we consider both global (whole targets) resp. local (individual target) monitoring level thresholds to improve the coverage quality of the deployed WSN. Unlike earlier works devoted to only α coverage, we deal with both local (α) and global (β) coverage leveling thresholds under network connectivity constraint. One approach to optimize the network’s lifetime is to divide the sensor nodes into Non-Disjoint subsets of sensors, or cover sets, and to schedule these covers with variable activation time periods, so that the global time lifespan of the network is optimized. To this end, we provide both exact and heuristic approaches in this study. First, a novel mathematical Mixed Integer Linear Programming (MILP) is presented to solve the αβ-coverage with network connectivity requirement to optimality. Unfortunately, due to the NP-Completeness of the addressed problem, the MILP’s resolution becomes impracticable for large optimization problems. To remedy this and to cope with large instances, we propose a new exact approach based on column generation able to achieve optimal solutions in reasonable time. In addition, since the CG’s subproblem resolution is also an NP-Hard problem, a new dedicated Heuristic (DH) is designed to solve the CG’s subproblem in polynomial time complexity. Moreover, we propose an exact ILP formulation for the CG’s subproblem if the DH heuristic fails to compute an attractive solution in each iteration of the CG computation process. Finally, several experiments are performed to evaluate the performances of our proposals.

Improved solution of the Budget constrained Fuel Treatment Scheduling problem and extensions

This paper considers the budget constrained fuel treatment scheduling (BFTS) problem where, in the context of wildfire mitigation, the goal is to inhibit the potential of fire spread in a landscape by proper fuel treatment activities. Given a time horizon represented by consecutive unit periods, the landscape is divided into cells and represented as a grid graph where each cell has a fuel age that increases over time and becomes old if no treatment is applied in the meantime: this induces a potential high fire risk whenever two contiguous cells are old. Cells fuel ages can be reset to zero under appropriate fuel treatments but there is a limited budget for treatment in each period. The problem calls for finding a suitable selection of cells to be treated so as to minimize the presence of old contiguous cells over the whole time horizon. We prove that problem BFTS is strongly NP-complete on paths and thus on grid graphs and show that no polynomial time approximation algorithm exists unless P=NP. We provide an enhanced integer linear programming formulation of the problem with respect to the relevant literature that shows up to be efficiently solved by an ILP solver on reasonably large size instances. Finally, we consider a harder periodic variant of the problem with the aim of finding a cyclic treatment plan with cycles of length T and propose a matheuristic approach capable of efficiently tackling those instances where an ILP solver applied to the ILP formulation runs into difficulties.

The potential of family-free rearrangements towards gene orthology inference.

Recently, we proposed an efficient ILP formulation [Rubert DP, Martinez FV, Braga MDV, Natural family-free genomic distance, Algorithms Mol Biol16:4, 2021] for exactly computing the rearrangement distance of two genomes in a family-free setting. In such a setting, neither prior classification of genes into families, nor further restrictions on the genomes are imposed. Given two genomes, the mentioned ILP computes an optimal matching of the genes taking into account simultaneously local mutations, given by gene similarities, and large-scale genome rearrangements. Here, we explore the potential of using this ILP for inferring groups of orthologs across several species. More precisely, given a set of genomes, our method first computes all pairwise optimal gene matchings, which are then integrated into gene families in the second step. Our approach is implemented into a pipeline incorporating the pre-computation of gene similarities. It can be downloaded from gitlab.ub.uni-bielefeld.de/gi/FFGC. We obtained promising results with experiments on both simulated and real data.

Compact ILP formulations for a class of solutions to berth allocation and quay crane scheduling problems

Compact ILP formulations for a class of solutions to berth allocation and quay crane scheduling problems

Constructing small genome graphs via string compression.

MotivationThe size of a genome graph—the space required to store the nodes, node labels and edges—affects the efficiency of operations performed on it. For example, the time complexity to align a sequence to a graph without a graph index depends on the total number of characters in the node labels and the number of edges in the graph. This raises the need for approaches to construct space-efficient genome graphs.ResultsWe point out similarities in the string encoding mechanisms of genome graphs and the external pointer macro (EPM) compression model. We present a pair of linear-time algorithms that transform between genome graphs and EPM-compressed forms. The algorithms result in an upper bound on the size of the genome graph constructed in terms of an optimal EPM compression. To further reduce the size of the genome graph, we propose the source assignment problem that optimizes over the equivalent choices during compression and introduce an ILP formulation that solves that problem optimally. As a proof-of-concept, we introduce RLZ-Graph, a genome graph constructed based on the relative Lempel–Ziv algorithm. Using RLZ-Graph, across all human chromosomes, we are able to reduce the disk space to store a genome graph on average by 40.7% compared to colored compacted de Bruijn graphs constructed by Bifrost under the default settings. The RLZ-Graph scales well in terms of running time and graph sizes with an increasing number of human genome sequences compared to Bifrost and variation graphs produced by VGtoolkit.AvailabilityThe RLZ-Graph software is available at: https://github.com/Kingsford-Group/rlzgraph.Supplementary information Supplementary data are available at Bioinformatics online.

Joint optimization of primary and backup controller placement and availability link upgrade in SDN networks

In Software-Defined Networking (SDN), the control and data planes are decoupled, leading to a more programmable and efficient network management. In this paper, the controller placement problem in SDN is addressed, jointly with the problem of exploring a high-availability tree subgraph, in order to support delay and availability requirements between the switches and the controllers. We consider that each switch connects to a primary and to a backup controller. We formulate the joint optimization model as an integer linear programming model (ILP), and propose a heuristic method when the exact model becomes impractical. Furthermore, we compare two ILP formulations, and we also compare the controller redundancy solutions with those considering path redundancy alone.

Natural family-free genomic distance

BackgroundA classical problem in comparative genomics is to compute the rearrangement distance, that is the minimum number of large-scale rearrangements required to transform a given genome into another given genome. The traditional approaches in this area are family-based, i.e., require the classification of DNA fragments of both genomes into families. Furthermore, the most elementary family-based models, which are able to compute distances in polynomial time, restrict the families to occur at most once in each genome. In contrast, the distance computation in models that allow multifamilies (i.e., families with multiple occurrences) is NP-hard. Very recently, Bohnenkämper et al. (J Comput Biol 28:410–431, 2021) proposed an ILP formulation for computing the genomic distance of genomes with multifamilies, allowing structural rearrangements, represented by the generic double cut and join (DCJ) operation, and content-modifying insertions and deletions of DNA segments. This ILP is very efficient, but must maximize a matching of the genes in each multifamily, in order to prevent the free lunch artifact that would otherwise let empty or almost empty matchings give smaller distances.ResultsIn this paper, we adopt the alternative family-free setting that, instead of family classification, simply uses the pairwise similarities between DNA fragments of both genomes to compute their rearrangement distance. We adapted the ILP mentioned above and developed a model in which pairwise similarities are used to assign weights to both matched and unmatched genes, so that an optimal solution does not necessarily maximize the matching. Our model then results in a natural family-free genomic distance, that takes into consideration all given genes, without prior classification into families, and has a search space composed of matchings of any size. In spite of its bigger search space, our ILP seems to be boosted by a reduction of the number of co-optimal solutions due to the weights. Indeed, it converged faster than the original one by Bohnenkämper et al. for instances with the same number of multiple connections. We can handle not only bacterial genomes, but also fungi and insects, or sets of chromosomes of mammals and plants. In a comparison study of six fruit fly genomes, we obtained accurate results.

Compact integer linear programming formulations for the temporal bin packing problem with fire-ups

In this article we examine a specific version of the temporal bin packing problem (TBPP) that occurs in job-to-server scheduling. The TBPP represents a generalization of the well-known bin packing problem (BPP) with respect to an additional time dimension, and it requires to find the minimum number of bins (servers) to accommodate a given list of items (jobs) at any instant of time. In addition to this goal, recent publications suggest to also incorporate the number of fire-ups of the respective servers as an important factor in sustainable and energy-efficient overall operation. Addressing both these objectives by a weighted sum method typically increases the modeling complexity, thus leading to challenging ILP formulations, two of which have already been described in the literature. It has been shown that the parameter used to weight the two objectives strongly influences the applicability of heuristics and, therefore, the difficulty of the overall problem, so that only favorable choices can be handled so far. Even for these tailored scenarios, the current approaches already fail to compute an exact solution of many moderately-sized instances in reasonable times. For this reason, we propose various new preprocessing techniques to strengthen the LP bound and/or to reduce the numbers of variables or constraints appearing in the existing ILP formulations as well as one alternative modeling approach for the problem under consideration. Based on numerical tests with differently characterized sets of benchmark instances, the new and improved formulations are shown to lead (on average) to better performances (than compact state-of-the-art models) in terms of instances solved to optimality and computation times.

A Virtual Emotion Detection System With Maximum Cumulative Accuracy in Two-Way Enabled Multi Domain IoT Environment

In this letter, we introduce a multiplex cumulative virtual emotion detection system for two-way enabled multi domain IoT environment, which avoids detection holes in all IoT domains. Formally, we define a problem whose objective is to maximize cumulative accuracy of detection holes on condition that perpendicular barriers or horizontal barriers are constructed in every domain area. After presenting the defined problem with ILP formulation, we devise a novel scheme to solve the problem. Moreover, we evaluate the obtained results through extensive experiments.

Integer linear programming formulations for the minimum connectivity inference problem and model reduction principles

The minimum connectivity inference (MCI) problem represents an NP-hard generalization of the well-known minimum spanning tree problem. Given a set of vertices and a finite collection of subsets (of this vertex set), the MCI problem requires to find an edge set of minimal cardinality so that the vertices of each subset are connected. Although the problem under consideration has appeared in different application-oriented scientific contexts in the last decades, (efficient) approaches to its exact solution have hardly been addressed in the literature. Currently, even the most promising ILP formulation (an improved flow-based model) can only deal with rather small instances in reasonable times. In order to also tackle practically relevant problem sizes, our contribution is twofold: At first, we propose several new modeling frameworks for the MCI problem and investigate their theoretical properties as well as their computational behavior. Moreover, we introduce the concepts of simple model reduction and generalized model reduction which can be applied to reduce the numbers of variables and/or constraints in the various formulations. Based on extensive numerical experiments, the practical advantages of these principles are validated.

On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering

Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classical work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is, subgraphs of diameter at most two. This naturally leads to the two NP-hard problems $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ (the editing operations are edge insertions and edge deletions) and $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$ (the editing operations are vertex deletions). Answering an open question from the literature, we show that $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ is W[2]-hard with respect to the number of edge modifications, thus contrasting the fixed-parameter tractability result for the classical $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ problem (considering cliques instead of 2-clubs). Then, focusing on $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem kernel when parameterized by the number of vertex deletions. Nevertheless, we develop several effective data reduction and pruning rules, resulting in a competitive solver, outperforming a standard ILP formulation in most instances of an established biological test data set.