arXiv subjects

Julian Yarkony

Publications and source records attributed to Julian Yarkony.

At least 19 recordsLinked to original sources

arXiv2024

A New Class of Compact Formulations for Vehicle Routing Problems

This paper introduces a novel compact mixed integer linear programming (MILP) formulation and a discretization discovery-based solution approach for the Vehicle Routing Problem with Time Windows (VRPTW). We aim to solve the optimization problem efficiently by constraining the linear programming (LP) solutions to use only flows corresponding to time and capacity-feasible routes that are locally elementary (prohibiting cycles of customers localized in space). We employ a discretization discovery algorithm to refine the LP relaxation iteratively. This iterative process alternates between two steps: (1) increasing time/capacity/elementarity enforcement to increase the LP objective, albeit at the expense of increased complexity (more variables and constraints), and (2) decreasing enforcement without decreasing the LP objective to reduce complexity. This iterative approach ensures we produce an LP relaxation that closely approximates the optimal MILP objective with minimal complexity, facilitating an efficient solution via an off-the-shelf MILP solver. The effectiveness of our method is demonstrated through empirical evaluations on classical VRPTW instances. We showcase the efficiency of solving the final MILP and multiple iterations of LP relaxations, highlighting the decreased integrality gap of the final LP relaxation. We believe that our approach holds promise for addressing a wide range of routing problems within and beyond the VRPTW domain.

math.OC↗

arXiv2023

Graph Master and Local Area Routes for Efficient Column Generation for the Capacitated Vehicle Routing Problem with Time Windows

In this research we consider the problem of accelerating the convergence of column generation (CG) for the weighted set cover formulation of the capacitated vehicle routing problem with time windows (CVRPTW). We adapt two new techniques, Local Area (LA) routes and Graph Master (GM) to these problems. LA-routes rely on pre-computing all lowest cost elementary sub-routes, called LA-arcs, where all customers but the final customer are localized in space. LA-routes are constructed by concatenating LA-arcs where the final customer in a given LA-arc is the first customer in the subsequent LA-arc. To construct the lowest reduced cost elementary route during the pricing step of CG we apply a Decremental State Space Relaxation/time window discretization method over time, remaining demand, and customers visited; where the edges in the associated pricing graph are LA-arcs. To accelerate the convergence of CG we use an enhanced GM approach. We map each route generated during pricing to a strict total ordering of all customers, that respects the ordering of customers in the route; and somewhat preserves spatial locality. Each such strict total ordering is then mapped to a multi-graph where each node is associated with a tuple of customer, capacity remaining, and time remaining. Nodes are connected by feasible LA-arcs when the first/last customers in the LA-arc are less/greater than each intermediate customer (and each other) with respect to the total order. The multi-graph of a given route can express that route and other "related" routes; and every path from source to sink describes a feasible elementary route. Solving optimization over the restricted master problem over all multi-graphs is done efficiently by constructing the relevant nodes/edges on demand.

math.OC↗

arXiv2023

Local Area Routes and Valid Inequalities for Efficient Vehicle Routing

In this research we introduce Local Area (LA) routes for improving the efficiency and tightness of column generation (CG) methods for solving vehicle routing problems (VRP). LA-routes rely on pre-computing the lowest cost elementary sub-route (called an LA-arc) for each tuple consisting of the following: (1) a (first) customer where the LA-arc begins, (2) a distant customer (from the first) where the LA-arc ends, and (3) a set of intermediate customers near the first customer. LA-routes are constructed by concatenating LA-arcs where the final customer in a given LA-arc is the first customer in the subsequent LA-arc. A Decremental State Space Relaxation method is applied over LA-routes to construct the lowest reduced cost elementary route during the pricing step of CG. LA-route based solvers can be used to efficiently tighten the standard set cover VRP using a variant of subset row inequalities, which do not alter the structure of pricing. We incorporate LA-arcs into a novel CG stabilization scheme. Specifically each column generated during pricing is mapped to an ordered list of customers consistent with that column. An LA-arc is consistent with an ordering if the first/last customer in the arc come before/after all other customers in the LA-arc in the associated ordering respectively. Each such ordering is then mapped to a multi-graph where nodes correspond to (customer/demand) and edges correspond to LA-arcs consistent with that ordering. Hence any path from source to sink on the multi-graph is a feasible elementary route. The ordering for a column places customers spatially nearby in nearby positions on the ordering so that routes can be generated so as to permit spatially nearby customers to be visited without traveling far away first. We solve the restricted master problem over these graphs, which has special structure allowing for fast solution.

math.OC↗

arXiv2022

Local Area Routes for Vehicle Routing Problems

We consider an approach for improving the efficiency of column generation (CG) methods for solving vehicle routing problems. We introduce Local Area (LA) route relaxations, an alternative/complement to the commonly used ng-route relaxations and Decremental State Space Relaxations (DSSR) inside of CG formulations. LA routes are a subset of ng-routes and a super-set of elementary routes. Normally, the pricing stage of CG must produce elementary routes, which are routes without repeated customers, using processes which can be computationally expensive. Non-elementary routes visit at least one customer more than once, creating a cycle. LA routes relax the constraint of being an elementary route in such a manner as to permit efficient pricing. LA routes are best understood in terms of ng-route relaxations. Ng-routes are routes which are permitted to have non-localized cycles in space; this means that at least one intermediate customer (called a breaker) in the cycle must consider the starting customer in the cycle to be spatially far away. LA routes are described using a set of special indexes corresponding to customers on the route ordered from the start to the end of the route. LA route relaxations further restrict the set of permitted cycles beyond that of ng-routes by additionally enforcing that the breaker must be a located at a special index where the set of special indexes is defined recursively as follows. The first special index in the route is at index 1 meaning that it is associated with the first customer in the route. The k'th special index corresponds to the first customer after the k-1'th special index, that is not considered to be a neighbor of (considered spatially far from) the customer located at the k-1'th special index. We demonstrate that LA route relaxations can significantly improve the computational speed of pricing when compared to the standard DSSR.

math.OC↗

arXiv2022

Principled Graph Management

Graph Generation is a recently introduced enhanced Column Generation algorithm for solving expanded Linear Programming relaxations of mixed integer linear programs without weakening the expanded relaxations which characterize these methods. To apply Graph Generation we must be able to map any given column generated during pricing to a small directed acyclic graph for which any path from source to sink describes a feasible column. This structure is easily satisfied for vehicle routing, crew scheduling and various logistics problems where pricing is a constrained shortest path problem. The construction of such graphs trades off the size/diversity of a subset of columns modeled by the graphs versus the additional computational time required to solve the problems induced by larger graphs. Graph Generation (GG) has two computational bottlenecks. The first is pricing. Pricing in GG and Column Generation (CG) is identical because of the structure of the problems solved. The second bottleneck is the restricted master problem (RMP), which is more computationally intensive in GG than in CG given the same number of columns generated. By design GG converges in fewer iterations than CG, and hence requires fewer calls to pricing. Therefore, when the computation time of GG is dominated by pricing, as opposed to solving the RMP, GG converges much faster than CG in terms of time. However GG need not converge faster than CG when the GG RMP, rather than pricing, dominates computation. In this paper we introduce Principled Graph Management (PGM), which is an algorithm to solve the GG RMP rapidly by exploiting its special structure. We demonstrate the effectiveness of PGM inside a GG solution to the classical Capacitated Vehicle Routing Problem. We demonstrate that PGM solves the GG RMP hundreds of times faster than the baseline solver and that the improvement in speed increases with problem size.

math.OC↗

arXiv2021

Graph Generation: A New Approach to Solving Expanded Linear Programming Relaxations

In this article we introduce Graph Generation, an enhanced Column Generation (CG) algorithm for solving expanded linear programming relaxations of mixed integer linear programs. To apply Graph Generation, we must be able to map any given column to a small directed acyclic graph for which any path from source to sink describes a feasible column. This structure is easily satisfied for vehicle routing and crew scheduling problems; and other such problems where pricing is a resource constrained shortest path problem. Such graphs are then added to the restricted master problem (RMP) when the corresponding column is generated during pricing. The use of Graph Generation does not weaken the linear programming relaxation being solved. At any given iteration of CG enhanced by Graph Generation; the technique permits the RMP to express a much wider set of columns than those generated during pricing, leading to faster convergence of CG. Graph Generation does not change the structure of the CG pricing problem. We show how the method can be applied in a general way, and then demonstrate the effectiveness of our approach on the classical Capacitated Vehicle Routing Problem.

math.OC↗

arXiv2021

Detour Dual Optimal Inequalities for Column Generation with Application to Routing and Location

We consider the problem of accelerating column generation (CG) for logistics optimization problems using vehicle routing as an example. Without loss of generality, we focus on the Capacitated Vehicle Routing Problem (CVRP) via the addition of a new class of dual optimal inequalities (DOI) that incorporate information about detours from the vehicle routes. These inequalities extend the Smooth-DOI recently introduced in the literature for the solution of certain classes of set-covering problems by CG. The Detour-DOI introduced in this article permit low cost swap operations between items on a given active route with items near to other items on that route to estimate (and bound) the values of the dual variables. Smooth-DOI in contrast only permit low cost swap operations between nearby items. The use of Detour-DOI permits a faster convergence of CG without weakening the linear programming relaxation. We then argue that these DOI can also be conveniently applied to single source capacitated facility location problems. These problems have been shown to be equivalent to a broad class of logistics optimization problems that include, for example telecommunication network design and production planning. The importance of developing vastly more efficient column generation solvers cannot be overstated. Detour-DOI, which permit large numbers of columns to be expressed with a finite set of variables, contributes to this important endeavor.

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arXiv2021

Family Column Generation: A Principled Stabilized Column Generation Approach

We tackle the problem of accelerating column generation (CG) approaches to set cover formulations in operations research. At each iteration of CG we generate a dual solution that approximately solves the LP over all columns consisting of a subset of columns in the nascent set. We refer to this linear program (LP) as the Family Restricted Master Problem (FRMP), which provides a tighter bound on the master problem at each iteration of CG, while preserving efficient inference. For example, in the single source capacitated facility location problem (SSCFLP) the family of a column $l$ associated with facility $f$ and customer set $N_l$ contains the set of columns associated with $f$ and the customer set that lies in the power set of $N_l$. The solution to FRMP optimization is attacked with a coordinate ascent method in the dual. The generation of direction of travel corresponds to solving the restricted master problem over columns corresponding to the reduced lowest cost column in each family given specific dual variables based on the incumbent dual, and is easily generated without resolving complex pricing problems. We apply our algorithm to the SSCFLP and demonstrate improved performance over two relevant baselines.

cs.DS↗

arXiv2021

Multi-Robot Routing with Time Windows: A Column Generation Approach

Robots performing tasks in warehouses provide the first example of wide-spread adoption of autonomous vehicles in transportation and logistics. The efficiency of these operations, which can vary widely in practice, are a key factor in the success of supply chains. In this work we consider the problem of coordinating a fleet of robots performing picking operations in a warehouse so as to maximize the net profit achieved within a time period while respecting problem- and robot-specific constraints. We formulate the problem as a weighted set packing problem where the elements in consideration are items on the warehouse floor that can be picked up and delivered within specified time windows. We enforce the constraint that robots must not collide, that each item is picked up and delivered by at most one robot, and that the number of robots active at any time does not exceed the total number available. Since the set of routes is exponential in the size of the input, we attack optimization of the resulting integer linear program using column generation, where pricing amounts to solving an elementary resource-constrained shortest-path problem. We propose an efficient optimization scheme that avoids consideration of every increment within the time windows. We also propose a heuristic pricing algorithm that can efficiently solve the pricing subproblem. While this itself is an important problem, the insights gained from solving these problems effectively can lead to new advances in other time-widow constrained vehicle routing problems.

cs.RO↗

arXiv2020

Integer Programming for Multi-Robot Planning: A Column Generation Approach

We consider the problem of coordinating a fleet of robots in a warehouse so as to maximize the reward achieved within a time limit while respecting problem and robot specific constraints. We formulate the problem as a weighted set packing problem where elements are defined as being the space-time positions a robot can occupy and the items that can be picked up and delivered. We enforce that robots do not collide, that each item is delivered at most once, and that the number of robots active at any time does not exceed the total number available. Since the set of robot routes is not enumerable, we attack optimization using column generation where pricing is a resource-constrained shortest-path problem.

cs.AI↗

arXiv2020

Relaxed Dual Optimal Inequalities for Relaxed Columns: with Application to Vehicle Routing

We address the problem of accelerating column generation for set cover problems in which we relax the state space of the columns to do efficient pricing. We achieve this by adapting the recently introduced smooth and flexible dual optimal inequalities (DOI) for use with relaxed columns. Smooth DOI exploit the observation that similar items are nearly fungible, and hence should be associated with similarly valued dual variables. Flexible DOI exploit the observation that the change in cost of a column induced by removing an item can be bounded. We adapt these DOI to the problem of capacitated vehicle routing in the context of ng-route relaxations. We demonstrate significant speed ups on a benchmark data set, while provably not weakening the relaxation.

cs.AI↗

arXiv2020

Accelerating Column Generation via Flexible Dual Optimal Inequalities with Application to Entity Resolution

In this paper, we introduce a new optimization approach to Entity Resolution. Traditional approaches tackle entity resolution with hierarchical clustering, which does not benefit from a formal optimization formulation. In contrast, we model entity resolution as correlation-clustering, which we treat as a weighted set-packing problem and write as an integer linear program (ILP). In this case sources in the input data correspond to elements and entities in output data correspond to sets/clusters. We tackle optimization of weighted set packing by relaxing integrality in our ILP formulation. The set of potential sets/clusters can not be explicitly enumerated, thus motivating optimization via column generation. In addition to the novel formulation, we also introduce new dual optimal inequalities (DOI), that we call flexible dual optimal inequalities, which tightly lower-bound dual variables during optimization and accelerate column generation. We apply our formulation to entity resolution (also called de-duplication of records), and achieve state-of-the-art accuracy on two popular benchmark datasets. The project page is available at the following url, https://github.com/lokhande-vishnu/EntityResolution

cs.AI↗

arXiv2020

Smooth and flexible dual optimal inequalities

We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOI). DOI use knowledge of the dual solution space to derive inequalities that might be violated by intermediate solutions to a restricted master problem, and as such are efficient at reducing the number of iterations and the oscillations of the dual variables commonly observed in column generation procedures. We study two novel classes of DOI which are referred to as Flexible DOI (F-DOI) and Smooth-DOI (S-DOI), respectively (and jointly as SF-DOI). F-DOI provide rebates for covering items more than necessary. S-DOI describe the payment of a penalty to permit the under-coverage of items in exchange for the over-inclusion of other items. Unlike other classes of DOI from the literature, the S-DOI and F-DOI rely on very little to no problem-specific knowledge, and as such have the potential to be applied to a vast number of problem domains. In particular, we illustrate the efficiency of the new inequalities by embedding them within a column generation solver for the single source capacitated facility location problem (SSCFLP). A speed-up of a factor of up to 130x can be observed as when compared to a non-stabilized variant of the same CG procedure to achieve the linear relaxation lower bound on problems with dense columns and structured assignments costs.

math.OC↗

arXiv2019

End-to-end Training of CNN-CRF via Differentiable Dual-Decomposition

Modern computer vision (CV) is often based on convolutional neural networks (CNNs) that excel at hierarchical feature extraction. The previous generation of CV approaches was often based on conditional random fields (CRFs) that excel at modeling flexible higher order interactions. As their benefits are complementary they are often combined. However, these approaches generally use mean-field approximations and thus, arguably, did not directly optimize the real problem. Here we revisit dual-decomposition-based approaches to CRF optimization, an alternative to the mean-field approximation. These algorithms can efficiently and exactly solve sub-problems and directly optimize a convex upper bound of the real problem, providing optimality certificates on the way. Our approach uses a novel fixed-point iteration algorithm which enjoys dual-monotonicity, dual-differentiability and high parallelism. The whole system, CRF and CNN can thus be efficiently trained using back-propagation. We demonstrate the effectiveness of our system on semantic image segmentation, showing consistent improvement over baseline models.

cs.CV↗

arXiv2019

Massively Parallel Benders Decomposition for Correlation Clustering

We tackle the problem of graph partitioning for image segmentation using correlation clustering (CC), which we treat as an integer linear program (ILP). We reformulate optimization in the ILP so as to admit efficient optimization via Benders decomposition, a classic technique from operations research. Our Benders decomposition formulation has many subproblems, each associated with a node in the CC instance's graph, which are solved in parallel. Each Benders subproblem enforces the cycle inequalities corresponding to the negative weight edges attached to its corresponding node in the CC instance. We generate Magnanti-Wong Benders rows in addition to standard Benders rows, to accelerate optimization. Our Benders decomposition approach provides a promising new avenue to accelerate optimization for CC, and allows for massive parallelization.

cs.CV↗

arXiv2018

Accelerating Message Passing for MAP with Benders Decomposition

We introduce a novel mechanism to tighten the local polytope relaxation for MAP inference in Markov random fields with low state space variables. We consider a surjection of the variables to a set of hyper-variables and apply the local polytope relaxation over these hyper-variables. The state space of each individual hyper-variable is constructed to be enumerable while the vector product of pairs is not easily enumerable making message passing inference intractable. To circumvent the difficulty of enumerating the vector product of state spaces of hyper-variables we introduce a novel Benders decomposition approach. This produces an upper envelope describing the message constructed from affine functions of the individual variables that compose the hyper-variable receiving the message. The envelope is tight at the minimizers which are shared by the true message. Benders rows are constructed to be Pareto optimal and are generated using an efficient procedure targeted for binary problems.

cs.LG↗

arXiv2017

Efficient Multi-Person Pose Estimation with Provable Guarantees

Multi-person pose estimation (MPPE) in natural images is key to the meaningful use of visual data in many fields including movement science, security, and rehabilitation. In this paper we tackle MPPE with a bottom-up approach, starting with candidate detections of body parts from a convolutional neural network (CNN) and grouping them into people. We formulate the grouping of body part detections into people as a minimum-weight set packing (MWSP) problem where the set of potential people is the power set of body part detections. We model the quality of a hypothesis of a person which is a set in the MWSP by an augmented tree-structured Markov random field where variables correspond to body-parts and their state-spaces correspond to the power set of the detections for that part. We describe a novel algorithm that combines efficiency with provable bounds on this MWSP problem. We employ an implicit column generation strategy where the pricing problem is formulated as a dynamic program. To efficiently solve this dynamic program we exploit the problem structure utilizing a nested Bender's decomposition (NBD) exact inference strategy which we speed up by recycling Bender's rows between calls to the pricing problem. We test our approach on the MPII-Multiperson dataset, showing that our approach obtains comparable results with the state-of-the-art algorithm for joint node labeling and grouping problems, and that NBD achieves considerable speed-ups relative to a naive dynamic programming approach. Typical algorithms that solve joint node labeling and grouping problems use heuristics and thus can not obtain proofs of optimality. Our approach, in contrast, proves that for over 99 percent of problem instances we find the globally optimal solution and otherwise provide upper/lower bounds.

cs.CV↗

arXiv2017

Efficient Column Generation for Cell Detection and Segmentation

We study the problem of instance segmentation in biological images with crowded and compact cells. We formulate this task as an integer program where variables correspond to cells and constraints enforce that cells do not overlap. To solve this integer program, we propose a column generation formulation where the pricing program is solved via exact optimization of very small scale integer programs. Column generation is tightened using odd set inequalities which fit elegantly into pricing problem optimization. Our column generation approach achieves fast stable anytime inference for our instance segmentation problems. We demonstrate on three distinct light microscopy datasets, with several hundred cells each, that our proposed algorithm rapidly achieves or exceeds state of the art accuracy.

cs.CV↗