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In few years, graph cuts have become a leading method for solving a wide range of problems in computer vision. However, graph cuts involve the construction of huge graphs which sometimes do not fit in memory. Currently, most of the maxflow algorithms are impracticable to solve such large scale problems. In the image segmentation context, some authors have proposed heuristics [1, 2, 3, 4] to get round this problem. In this paper, we introduce a new strategy for reducing graphs. During the creation of the graph, before creating a new node, we test if the node is really useful to the max-flow computation. The nodes of the reduced graph are typically located in a narrow band surrounding the object edges. Empirically, solutions obtained on the reduced graphs are identical to the solutions on the complete graphs. A parameter of the algorithm can be tuned to obtain smaller graphs when an exact solution is not needed. The test is quickly computed and the time required by the test is often compensated by the time that would be needed to create the removed nodes and the additional time required by the computation of the cut on the larger graph. As a consequence, we sometimes even save time on small scale problems.

ABSTRACT. The 0-1 exact k-item quadratic knapsack problem (E − k QK P) consists of maximizing a quadratic function subject to two linear constraints: the first one is the classical linear capacity constraint; the second one is an equality cardinality constraint on the number of items in the knapsack. Most instances of this NP-hard problem with more than forty variables cannot be solved within one hour by a commercial software such as CPLEX 12.1. We propose therefore a fast and efficient heuristic method which produces both good lower and upper bounds on the value of the problem in reasonable time. Specifically, it integrates a primal heuristic and a semidefinite programming reduction phase within a surrogate dual heuristic. A large computational experiments over randomly generated instances with up to 200 variables validates the relevance of the bounds produced by our hybrid dual heuristic, which yields known optima (and prove optimality) in 90% (resp. 76%) within 100 seconds on the average.

T he multicommodity-ring location routing problem (MRLRP) studied in this paper is an NP-hard minimization problem arising in city logistics. The aim is to locate a set of urban distribution centers (UDCs) and to connect them via a ring in which massive flows of goods will circulate. Goods are transported from gates located outside the city to a UDC, and either join a second UDC through the ring before being delivered in electric vans to the final customers or are delivered directly to the customers from the first UDC. The reverse trip with pickup and transportation to the gates is also possible. A delivery service path starts at a particular UDC, then visits a subset of customers and ends at the same UDC, another UDC, or a self-service parking lot (SPL). A pickup route can start from an SPL or a UDC and ends at a UDC. The objective is to minimize the sum of the installation costs of the ring, flow transportation costs, and routing costs. The MRLRP belongs to the class of location-routing problems (LRP). We model it with a set-partitioning-like representation of delivery and pickup trips and arc-flow elements to describe goods transportation in the ring and between the ring and the gates. We present three approaches to solving the MRLRP: an exact method for small-size instances, a matheuristic for instances of a larger size, and a hybrid approach that applies the exact method to the columns output by the matheuristic. Numerical results are provided for an exhaustive set of instances, obtained by extending benchmark instances of the capacitated LRP with additional MRLRP features.

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