How to start a heuristic? Utilizing lower bounds for solving the quadratic assignment problem

The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. The QAP is an NP-hard optimization problem which attracts the use of heuristic or metaheuristic algorithms that can find quality solutions in an acceptable computation time. On the other hand, there is quite a broad spectrum of mathematical programming techniques that were developed for finding the lower bounds for the QAP. This paper presents a fusion of the two approaches whereby the solutions from the computations of the lower bounds are used as the starting points for a metaheuristic, called HC12, which is implemented on a GPU CUDA platform. We perform extensive computational experiments that demonstrate that the use of these lower bounding techniques for the construction of the starting points has a significant impact on the quality of the resulting solutions.

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

Framing Effects and Fuzzy Traces: ‘Some’ Observations

Framing effects occur when people respond differently to the same information, just because it is conveyed in different words. For example, in the classic ‘Disease Problem’ introduced by Amos Tversky and Daniel Kahneman, people’s choices between alternative interventions depend on whether these are described positively, in terms of the number of people who will be saved, or negatively in terms of the corresponding number who will die. In this paper, I discuss an account of framing effects based on ‘fuzzy-trace theory’. The central claim of this account is that people represent the numbers in framing problems in a ‘gist-like’ way, as ‘some’; and that this creates a categorical contrast between ‘some’ people being saved (or dying) and ‘no’ people being saved (or dying). I argue that fuzzy-trace theory’s gist-like representation, ‘some’, must have the semantics of ‘some and possibly all’, not ‘some but not all’. I show how this commits fuzzy-trace theory to a modest version of a rival ‘lower bounding hypothesis’, according to which lower-bounded interpretations of quantities contribute to framing effects by rendering the alternative descriptions extensionally inequivalent. As a result, fuzzy-trace theory is incoherent as it stands. Making sense of it requires dropping, or refining, the claim that decision-makers perceive alternatively framed options as extensionally equivalent; and the related claim that framing effects are irrational. I end by suggesting that, whereas the modest lower bounding hypothesis is well supported, there is currently less evidence for the core element of the fuzzy trace account.

On the achievable rate of bandlimited continuous-time AWGN channels with 1-bit output quantization

We consider a real continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. We derive an approximate lower bound on the capacity by lower-bounding the mutual information rate for input signals with exponentially distributed zero-crossing distances, sine-shaped transition waveform, and an average power constraint. The focus is on the behavior in the mid-to-high signal-to-noise ratio (SNR) regime above 10 dB. For hard bandlimited channels, the lower bound on the mutual information rate saturates with the SNR growing to infinity. For a given SNR the loss with respect to the unquantized additive white Gaussian noise channel solely depends on the ratio of channel bandwidth and the rate parameter of the exponential distribution. We complement those findings with an approximate upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the SNR domain of approximately 10–20 dB.

Laboratory experiments on Internal Solitary Waves in ice covered waters

<p>Oceanic internal waves (IWs) propagate along density interfaces and are ubiquitous in stratified water. Their properties are influenced strongly by the nature and form of the upper and lower bounding surfaces of the containing basin(s) in which they propagate.<span>  </span>As the Arctic Ocean evolves to a seasonally more ice-free state, the IW field will be affected by the change. The relationship between IW dynamics and ice is important in understanding (i) the general circulation and thermodynamics in the Arctic Ocean and (ii) local mixing processes that supply heat and nutrients from depth into upper layers, especially the photic zone. This, in turn, has important ramifications for sea ice formation processes and the state of local and regional ecosystems.<span>  </span>Despite this, the effect of diminishing sea ice cover on the IW field (and vice versa) is not well established. A better understanding of IW dynamics in the Arctic Ocean and, in particular, how the IW field is affected by changes in both ice cover and stratification, is central in understanding how the rapidly changing Arctic will adapt to climate change.</p><p> </p><p>An experimental study of internal solitary waves (ISWs) propagating in a stably stratified two-layer fluid in which the upper boundary condition changes from open water to ice are studied for grease, level, and nilas ice. The experiments show that the internal wave-induced flow at the surface is capable of transporting sea-ice in the horizontal direction. In the level ice case, the transport speed of, relatively long ice floes, nondimensionalized by the wave speed is linearly dependent on the length of the ice floe nondimensionalized by the wave length. It will also be shown that bottom roughness associated with different ice types can cause varying degrees of vorticity and small-scale turbulence in the wave-induced boundary layer beneath the ice. Measures of turbulent kinetic energy dissipation under the ice are shown to be comparable to those at the wave density interface. Moreover, in cases where the ice floe protrudes into the pycnocline, interaction with the ice edge can cause the ISW to break or even be destroyed by the process. The results suggest that interaction between ISWs and sea ice may be an important mechanism for dissipation of ISW energy in the Arctic Ocean.</p><p> </p><p><strong>Acknowledgements</strong></p><p>This work was funded through the EU Horizon 2020 Research and Innovation Programme Hydralab+.</p>

Optimizing bipartite matching in real-world applications by incremental cost computation

The Kuhn-Munkres (KM) algorithm is a classical combinatorial optimization algorithm that is widely used for minimum cost bipartite matching in many real-world applications, such as transportation. For example, a ride-hailing service may use it to find the optimal assignment of drivers to passengers to minimize the overall wait time. Typically, given two bipartite sets, this process involves computing the edge costs between all bipartite pairs and finding an optimal matching. However, existing works overlook the impact of edge cost computation on the overall running time. In reality, edge computation often significantly outweighs the computation of the optimal assignment itself, as in the case of assigning drivers to passengers which involves computation of expensive graph shortest paths. Following on from this observation, we observe common real-world settings exhibit a useful property that allows us to incrementally compute edge costs only as required using an inexpensive lower-bound heuristic. This technique significantly reduces the overall cost of assignment compared to the original KM algorithm, as we demonstrate experimentally on multiple real-world data sets, workloads, and problems. Moreover, our algorithm is not limited to this domain and is potentially applicable in other settings where lower-bounding heuristics are available.

Linearization of McCormick relaxations and hybridization with the auxiliary variable method

The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley’s algorithm; computation of all vertices of an n-simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages.

Lower Bounding the AND-OR Tree via Symmetrization

We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜ deg(AND m ˆ OR n ) = ˜ Ω(√ mn ). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].

Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem

The two-dimensional bin packing problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated, and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management, and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Benders decomposition that is based on a simple model in which the two-dimensional items and bins are just represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but we enrich it with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The resulting algorithm behaved very well on the benchmark sets of instances, improving on average on previous algorithms from the literature and solving for the first time a number of open instances. Summary of Contribution: We address the two-dimensional bin packing problem (2D-BPP), which calls for packing a set of rectangular items into a minimal set of larger rectangular bins. The 2D-BPP is a very difficult generalization of the standard one-dimensional bin packing problem, and it has been widely studied in the past because it models many real-world applications, including production, warehouse management, and transportation. We solve the 2D-BPP with a combinatorial Benders decomposition that is based on a model in which the two-dimensional items and bins are represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but it is enriched with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The algorithm we developed has been extensively tested on the most well-known benchmark set from the literature, which contains 500 instances. It behaved very well, improving on average upon previous algorithms, and solving for the first time a number of open instances. We analyzed in detail several configurations before obtaining the best one and discussed several insights from this analysis in the manuscript.