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Hadrien Cambazard

Publications and source records attributed to Hadrien Cambazard.

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A note about a transition of Ratliff and Rosenthal's order picking algorithm for rectangular warehouses

In the order picking problem, a picker has to collect a number of products in a warehouse with a minimum length tour. Ratliff and Rosenthal gave a linear algorithm solving the order picking problem in the case where the warehouse has two cross aisles. Their algorithm allow the tour to double cross an entire aisle. We prove that, in rectangular warehouses, there always exists a minimum length tour which doesn't double cross an aisle.

math.OC

Logistic regression to boost exoplanet detection performances

Direct imaging of exoplanets requires to separate the background noise from the exoplanet signals. Statistical methods have been recently proposed to avoid subtracting any signal of interest as opposed to initial self-subtracting methods based on Angular Differential Imaging (ADI). However, unless conservative thresholds are chosen to claim for a detection, such approaches tend to produce a list of candidates that include many false positives. Choosing high, conservative, thresholds leads to miss the faintest planets. We extend a statistical framework with a logistic regression to filter the list of candidates. Features with physical/optical meaning (in two wavelengths) are used, leading to a very fast and pragmatic approach. The overall method requires a simple edge detection (image processing) and clustering algorithm to work with sub-images. To estimate its efficiency, we apply our approach to targets observed with the ESO/SPHERE high contrast imager, that were previously used as tests for blind surveys. Experimental results with injected signals show that either the number of false detections is considerably reduced or faint exoplanets that would otherwise not be detected can be sometimes found. Typically, on the blind tests performed, we are now able to detect around 50% more of the injected planets with an SNR below 5, and with a very low number of additional candidates.

astro-ph.EP

Lower and upper bounds for the joint batching, routing and sequencing problem

Warehouses are nowadays the scene of complex logistic problems integrating different decision layers. This paper addresses the Joint Order Batching, Picker Routing and Sequencing Problem with Deadlines (JOBPRSP-D) in rectangular warehouses. To tackle the problem an exponential linear programming formulation is proposed. It is solved with a column generation heuristic able to provide valid lower and upper bounds on the optimal value. We start by showing that the JOBPRSP-D is related to the bin packing problem rather than the scheduling problem. We take advantage of this aspect to derive a number of valid inequalities that enhance the resolution of the master problem. The proposed algorithm is evaluated on publicly available data-sets. It is able to optimally solve instances with up to 18 orders in few minutes. It is also able to prove optimality or to provide high-quality lower bounds on larger instances with 100 orders. To the best of our knowledge this is the first paper that provides optimality guarantee on large size instances for the JOBPRSP-D, thus the results can be used to assert the quality of heuristics proposed for the same problem.

math.OC

Innovative ideas for teaching supports: Application to Graph theory

Teaching graph theory with the most adequate tools requires time and ideas. We present how an open community of teachers shares contents and ideas on an innovative platform. The objective is to get the students autonomous in their training with activities that give them immediate feedback on their understanding. Beyond learning, the very large collection of exercises of various levels can also be used to evaluate the student's level. The proposed activities can be algorithm's code in classical programming languages (e.g. Java, Python) that the student can test with predefined tests proposed by the teacher or collections of generated questions.

cs.DM

Arc-consistency with linear programming reduced costs (applied to stable set in chordal graphs)

A well known technique to reduce the search space in integer programming is known as variable fixing or reduced cost strengthening. The reduced costs given by an optimal dual solution of the linear relaxation can be used to strengthen the bounds of the variables but this filtering is incomplete. We show how reduced costs can be used to achieve Arc-Consistency (AC), i.e a complete filtering, of a global constraint with a cost variable and an assignment cost for each value. We assume that an ideal Integer Linear Programming (ILP) formulation is available i.e the convex hull of the characteristic vectors of the supports is known. A detailed analysis of reduced cost based filtering is proposed. We characterize arc-consistency based on complementary slackness i.e completeness of reasoning as opposed to only optimality. We also give a simple sufficient condition allowing a set of dual solutions to ensure arc-consistency through reduced costs. In practice, when the constraint has a such an ideal ILP, n dual solutions are always enough to achieve AC (where n is the number of variables of the global constraint). It extends the work presented in (German et al., 2017) for satisfaction problems and in (Claus et al., 2020) for the specific case of the minimum weighted alldifferent constraint. Our analysis is illustrated on constraints related to the assignment and shortest path problem and also demonstrated on the weighted stable set problem in chordal graphs. A novel AC algorithm is proposed in this latter case based on reduced costs.

math.OC

Dealing with elementary paths in the Kidney Exchange Problem

We study an elementary path problem which appears in the pricing step of a column generation scheme solving the kidney exchange problem. The latter aims at finding exchanges of donations in a pool of patients and donors of kidney transplantations. Informally, the problem is to determine a set of cycles and chains of limited length maximizing a medical benefit in a directed graph. The cycle formulation, a large-scale model of the problem restricted to cycles of donation, is efficiently solved via branch-and-price. When including chains of donation however, the pricing subproblem becomes NP-hard. This article proposes a new complete column generation scheme that takes into account these chains initiated by altruistic donors. The development of non-exact dynamic approaches for the pricing problem, the NG-route relaxation and the color coding heuristic, leads to an efficient column generation process.

cs.DS

An integer programming formulation using convex polygons for the convex partition problem

A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition based on the minimum number of internal faces. The problem has been shown to be NP-Hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.

cs.CG

Tree search algorithms for the Sequential Ordering Problem

We present a study of several generic tree search techniques applied to the Sequential Ordering Problem. This study enables us to propose a simple and competitive tree search algorithm. It consists of an iterative Beam Search algorithm that favors search over inference and integrates dynamic programming inspired cuts. It proves optimality on half of the SOPLIB instances and finds new best known solutions on 6 among 7 open instances of the benchmark in a small amount of time.

cs.DM

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.

cs.DS

A global constraint for the capacitated single-item lot-sizing problem

The goal of this paper is to set a constraint programming framework to solve lot-sizing problems. More specifically, we consider a single-item lot-sizing problem with time-varying lower and upper bounds for production and inventory. The cost structure includes time-varying holding costs, unitary production costs and setup costs. We establish a new lower bound for this problem by using a subtle time decomposition. We formulate this NP-hard problem as a global constraint and show that bound consistency can be achieved in pseudo-polynomial time and when not including the costs, in polynomial time. We develop filtering rules based on existing dynamic programming algorithms, exploiting the above mentioned time decomposition for difficult instances. In a numerical study, we compare several formulations of the problem: mixed integer linear programming, constraint programming and dynamic programming. We show that our global constraint is able to find solutions, unlike the decomposed constraint programming model and that constraint programming can be competitive, in particular when adding combinatorial side constraints.

math.OC

Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde.

cs.DS

Bin Packing with Linear Usage Costs

Bin packing is a well studied problem involved in many applications. The classical bin packing problem is about minimising the number of bins and ignores how the bins are utilised. We focus in this paper, on a variant of bin packing that is at the heart of efficient management of data centres. In this context, servers can be viewed as bins and virtual machines as items. The efficient management of a data-centre involves minimising energy costs while ensuring service quality. The assignment of virtual machines on servers and how these servers are utilised has a huge impact on the energy consumption. We focus on a bin packing problem where linear costs are associated to the use of bins to model the energy consumption. We study lower bounds based on Linear Programming and extend the bin packing global constraint with cost information.

cs.DS

A Combinatorial Optimisation Approach to Designing Dual-Parented Long-Reach Passive Optical Networks

We present an application focused on the design of resilient long-reach passive optical networks. We specifically consider dual-parented networks whereby each customer must be connected to two metro sites via local exchange sites. An important property of such a placement is resilience to single metro node failure. The objective of the application is to determine the optimal position of a set of metro nodes such that the total optical fibre length is minimized. We prove that this problem is NP-Complete. We present two alternative combinatorial optimisation approaches to finding an optimal metro node placement using: a mixed integer linear programming (MIP) formulation of the problem; and, a hybrid approach that uses clustering as a preprocessing step. We consider a detailed case-study based on a network for Ireland. The hybrid approach scales well and finds solutions that are close to optimal, with a runtime that is two orders-of-magnitude better than the MIP model.

cs.AI