Discrete Optimization最新文献

In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by rounding a solution to the standard LP with a slight $(1+ϵ)$ violation in the capacities.

Our result shows that the standard LP is not too bad.

Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.

The result is also extended to give the first approximation for uniform hard capacitated $k$-facility location problem violating the capacities by a factor of $(1+ϵ)$ and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of $\frac{2}{1+ϵ}$.

An L-sequence of a graph $G$ is a sequence of distinct vertices $S=(v_1,\dots ,v_k)$ such that $N\left[v_i\right]\setminus {\cup }_{j=1}^{i-1}N\left(v_j\right)\ne 0̸$. The length of a longest L-sequence is called the L-Grundy domination number, denoted ${\gamma }_{gr}^L\left(G\right)$. In this paper, we prove ${\gamma }_{gr}^L\left(G\right)\le n\left(G\right)-\delta \left(G\right)+1$, which was conjectured by Brešar, Gologranc, Henning, and Kos. We also prove some initial results about characteristics of $n$-vertex graphs satisfying ${\gamma }_{gr}^L\left(G\right)=n$.

Large scale set covering problems have often been approached by constructive greedy heuristics, and much research has been devoted to the design and evaluation of various greedy criteria for such heuristics. A criterion proposed by Caprara et al. (1999) is based on reduced costs with respect to the yet unfulfilled constraints, and the resulting greedy heuristic is reported to be superior to those based on original costs or ordinary reduced costs.

We give a theoretical justification of the greedy criterion proposed by Caprara et al. by deriving it from a global optimality condition for general non-convex optimisation problems. It is shown that this criterion is in fact greedy with respect to incremental contributions to a quantity which at termination coincides with the deviation between a Lagrangian dual bound and the objective value of the feasible solution found.

The Capacitated Vehicle Routing Problem (CVRP) consists of finding the cheapest way to serve a set of customers with a fleet of vehicles of a given capacity. While serving a particular customer, each vehicle picks up its demand and carries its weight throughout the rest of its route. While costs in the classical CVRP are measured in terms of a given arc distance, the Cumulative Vehicle Routing Problem (CmVRP) is a variant of the problem that aims to minimize total energy consumption. Each arc’s energy consumption is defined as the product of the arc distance by the weight accumulated since the beginning of the route.

The purpose of this work is to propose several different formulations for the CmVRP and to study their Linear Programming (LP) relaxations. In particular, the goal is to study formulations based on combining an arc-item concept (that keeps track of whether a given customer has already been visited when traversing a specific arc) with another formulation from the recent literature, the Arc-Load formulation (that determines how much load goes through an arc).

Both formulations have been studied independently before – the Arc-Item is very similar to a multi-commodity-flow formulation in Letchford and Salazar-González (2015) and the Arc-Load formulation has been studied in Fukasawa et al. (2016) – and their LP relaxations are incomparable. Nonetheless, we show that a formulation combining the two (called Arc-Item-Load) may lead to a significantly stronger LP relaxation, thereby indicating that the two formulations capture complementary aspects of the problem. In addition, we study how set partitioning based formulations can be combined with these formulations. We present computational experiments on several well-known benchmark instances that highlight the advantages and drawbacks of the LP relaxation of each formulation and point to potential avenues of future research.

The Quadratic Knapsack Problem (QKP) is a well-known $\mathrm{NP}$-hard combinatorial optimisation problem, with many practical applications. We present a ‘cut-and-branch’ algorithm for the QKP, in which a cutting-plane phase is followed by a branch-and-bound phase. The cutting-plane phase is more sophisticated than the existing ones in the literature, incorporating several classes of cutting planes, two primal heuristics, and several rules for eliminating variables and constraints. Computational results show that the algorithm is competitive.

Quadratic Unconstrained Binary Optimization (QUBO) modeling has become a unifying framework for solving a wide variety of both unconstrained as well as constrained optimization problems. More recently, QUBO (or equivalent $-1/+1$ Ising Spin) models are a requirement for quantum annealing computers. Noisy Intermediate-Scale Quantum (NISQ) computing refers to classical computing preparing or compiling problem instances for compatibility with quantum hardware architectures. The process of converting a constrained problem to a QUBO compatible quantum annealing problem is an important part of the quantum compiler architecture and specifically when converting constrained models to unconstrained the choice of penalty magnitude is not trivial because using a large penalty to enforce constraints can overwhelm the solution landscape, while having too small a penalty allows infeasible optimal solutions. In this paper we present NISQ approaches to bound the magnitude of the penalty scalar $M$ and demonstrate efficacy on a benchmark set of problems having a single equality constraint and present a QUBO partitioning approach validated by experimentation.

We address the problem of minimizing a quadratic function subject to linear constraints over binary variables. We introduce the exact solution method called EXPEDIS where the constrained problem is transformed into a max-cut instance, and then the whole machinery available for max-cut can be used to solve the transformed problem. We derive the theory in order to find a transformation in the spirit of an exact penalty method; however, we are only interested in exactness over the set of binary variables. In order to compute the maximum cut we use the solver BiqMac. Numerical results show that this algorithm can be successfully applied on various classes of problems.

In this paper, we study the maximum diversity problem (MDP) which is equivalent to the quadratic unconstrained binary optimization (QUBO) problem with cardinality constraint. The MDP aims to select a subset of elements with given cardinality such that the sum of pairwise distances between any two elements in the selected subset is maximized. For solving this computationally challenging problem, we propose a two-phase tabu search based evolutionary algorithm (TPTS/EA), which integrates several distinguishing features to ensure the diversity and the quality of the evolution, such as a two-phase tabu search algorithm which consists of a dynamic candidate list (DCL) strategy-based traditional tabu search in the first phase and a solution-based tabu search procedure to refine the search in the second phase, and two path-relinking based recombination operators to generate new offspring solutions. Tested on three sets of totally 140 public instances in the literature, the study demonstrates the efficacy of the proposed TPTS/EA algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPTS/EA algorithm is able to improve the previous best known results for 2 instances, while matching the previous best-known solutions for 130 instances. We also provide experimental evidences to highlight the beneficial effect of several important components in our TPTS/EA algorithm.