Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

On the exact separation of cover inequalities of maximum-depth

We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly $${\mathcal {N}}{\mathcal {P}}$$ N P -hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

X(3): an exactly separable γ-rigid version of the X(5) critical point symmetry

A γ-rigid version (with γ = 0) of the X(5) critical point symmetry is constructed. The model, to be called X(3) since it is proved to contain three degrees of freedom, utilizes an infinite well potential, is based on exact separation of variables, and leads to parameter free (up to overall scale factors) predictions for spectra and B(E2) transition rates, which are in good agreement with existing experimental data for 172Os and 186Pt. An unexpected similarity of the β1-bands of the X(5) nuclei 150Nd, 152Sm, 154Gd, and 156Dy to the X(3) predictions is observed.

Chirp Signal Generator Based on Direct Digitalsynthesizer (Dds) for a Radar At 300 Ghz

In the Modern day electronic fighting Systems the utilization of range has been broadened with a proficient way which makes the recognizable proof of the signal difficult. Modern radars use recurrence and stage tweaked signs to spread their range to enhance the handling gain. The way toward finding the balance arrangement of a perceived signal, the moderate development between signal distinguishing proof and demodulation, is an imperative task of a wise recipient, with various customary resident and military applications. Obviously, with no learning of the transmitted data and various cloud parameters at the authority, for instance, the signal control, carrier repeat and stage offsets, data, and so on., daze distinguishing proof of the adjustment is a troublesome assignment. This turns out to be considerably additionally difficult in certifiable situations. Wideband direct recurrence tweaked (LFM) signal is generally utilized in exact separation estimating radar framework. As customary techniques for creating LFM signal have a great deal of detriments, for example, shakiness and nonlinearity we propose an alternate answer for wideband LFM signal generator in L-band dependent on DDS and recurrence augmentation. The proposed strategy creates the baseband LFM signal utilizing the DDS, and after that includes the baseband motion into the recurrence duplication framework; last we can accomplish an unadulterated L-band wideband LFM signal. The estimation result demonstrates that the proposed can satisfy every prerequisite of anticipant palatably.

An exact approach for the multicommodity network optimization problem with a step cost function

We investigate the Multicommodity Network Optimization Problem with a Step Cost Function (MNOP-SCF) where the available facilities to be installed on the edges have discrete step-increasing cost and capacity functions. This strategic long-term planning problem requires installing at most one facility capacity on each edge so that all the demands are routed and the total installation cost is minimized. We describe a path-based formulation that we solve exactly using an enhanced constraint generation based procedure combined with columns and new cuts generation algorithms. The main contribution of this work is the development of a new exact separation model that identifies the most violated bipartition inequalities coupled with a knapsack-based problem that derives additional cuts. To assess the performance of the proposed approach, we conducted computational experiments on a large set of randomly generated instances. The results show that it delivers optimal solutions for large instances with up to 100 nodes, 600 edges, and 4950 commodities while in the literature, the best developed approaches are limited to instances with 50 nodes, 100 edges, and 1225 commodities.