Probabilistic analysis of the number partitioning problem

Low-Rank Combinatorial Optimization and Statistical Learning by Spatial Photonic Ising Machine

The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.

Landscape Analysis of a Class of NP-Hard Binary Packing Problems

This article presents an exploratory landscape analysis of three NP-hard combinatorial optimisation problems: the number partitioning problem, the binary knapsack problem, and the quadratic binary knapsack problem. In the article, we examine empirically a number of fitness landscape properties of randomly generated instances of these problems. We believe that the studied properties give insight into the structure of the problem landscape and can be representative of the problem difficulty, in particular with respect to local search algorithms. Our work focuses on studying how these properties vary with different values of problem parameters. We also compare these properties across various landscapes that were induced by different penalty functions and different neighbourhood operators. Unlike existing studies of these problems, we study instances generated at random from various distributions. We found a general trend where some of the landscape features in all of the three problems were found to vary between the different distributions. We captured this variation by a single, easy to calculate parameter and we showed that it has a potentially useful application in guiding the choice of the neighbourhood operator of some local search heuristics.

Efficient exploration of discrete energy landscapes

Many physical and chemical processes, such as folding of biopolymers, are best described as dynamics on large combinatorial energy landscapes. A concise approximate description of the dynamics is obtained by partitioning the microstates of the landscape into macrostates. Since most landscapes of interest are not tractable analytically, the probabilities of transitions between macrostates need to be extracted numerically from the microscopic ones, typically by full enumeration of the state space or approximations using the Arrhenius law. Here, we propose to approximate transition probabilities by a Markov chain Monte Carlo method. For landscapes of the number partitioning problem and an RNA switch molecule, we show that the method allows for accurate probability estimates with significantly reduced computational cost.

Phase transition and landscape statistics of the number partitioning problem

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy-hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the difficulty measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In addition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the p spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model.

Extreme value statistics and traveling fronts: application to computer science

We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem that is then further mapped to a modified directed polymer problem on a Cayley tree. We employ the techniques of traveling fronts to solve the polymer problem and translate back to derive exact asymptotic properties in the original search tree problem. The second mapping allows us not only to again derive the already known results for random binary trees but to obtain exact results for search trees where the entries arrive according to an arbitrary distribution, not necessarily randomly. Besides it allows us to derive the asymptotic shape of the full probability distribution of height and not just its moments. Our results are then generalized to m-ary search trees with arbitrary distribution.

Entropy-based analysis of the number partitioning problem

In this paper we apply the multicanonical method of statistical physics on the number partitioning problem (NPP). This problem is a basic NP-hard problem from computer science, and can be formulated as a spin-glass problem. We compute the spectral degeneracy, which gives us information about the number of solutions for a given cost E and cardinality difference m. We also study an extension of this problem for Q partitions. We show that a fundamental difference on the spectral degeneracy of the generalized (Q>2) NPP exists, which could explain why it is so difficult to find good solutions for this case.

Random costs in combinatorial optimization

The random cost problem is the problem of indentifying the minimum in a list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. On the one hand this explains the bad performance of heuristic approaches to the number partitioning problem, but on the other hand it allows one to calculate the probability distributions of the optimum and suboptimum costs.