Fifty years of P vs. NP and the possibility of the impossible

Advances in algorithms, machine learning, and hardware can help tackle many NP-hard problems once thought impossible.

The knapsack problem with special neighbor constraints

The knapsack problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two knapsack problems which contain additional constraints in the form of directed graphs whose vertex set corresponds to the item set. In the one-neighbor knapsack problem, an item can be chosen only if at least one of its neighbors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all its neighbors are chosen. For both problems, we consider uniform and general profits and weights. We prove upper bounds for the time complexity of these problems when restricting the graph constraints to special sets of digraphs. We discuss directed co-graphs, minimal series-parallel digraphs, and directed trees.

A note on no-free-lunch theorem

The No-Free-Lunch theorem is an interesting and important theoretical result in machine learning. Based on philosophy of No-Free-Lunch theorem, we discuss extensively on the limitation of a data-driven approach in solving NP-hard problems.

Quadrature Photonic Spatial Ising Machine

The mining in physics and biology for accelerating the hardcore algorithm to solve non-deterministic polynomial (NP) hard problems has inspired a great amount of special-purpose ma-chine models. Ising machine has become an efficient solver for various combinatorial optimization problems. As a computing accelerator, large-scale photonic spatial Ising machine have great advantages and potentials due to excellent scalability and compact system. However, current fundamental limitation of photonic spatial Ising machine is the configuration flexibility of problem implementation in the accelerator model. Arbitrary spin interaction is highly desired for solving various NP hard problems. Moreover, the absence of external magnetic field in the proposed photonic Ising machine will further narrow the freedom to map the optimization applications. In this paper, we propose a novel quadrature photonic spatial Ising machine to break through the limitation of photonic Ising accelerator by synchronous phase manipulation in two and three sections. Max-cut problem solution with graph order of 100 and density from 0.5 to 1 is experimentally demonstrated after almost 100 iterations. We derive and verify using simulation the solution for Max-cut problem with more than 1600 nodes and the system tolerance for light misalignment. Moreover, vertex cover problem, modeled as an Ising model with external magnetic field, has been successfully implemented to achieve the optimal solution. Our work suggests flexible problem solution by large-scale photonic spatial Ising machine.

Application of genetic algorithm to industrial scheduling and problems of parameters evaluation

Genetic algorithms are widely used in various mathematical and real world problems. They are approximate metaheuristic algorithms, commonly used for solving NP-hard problems in combinatorial optimisation. Industrial scheduling is one of the classical NP-hard problems. We analyze three classical industrial scheduling problems: job-shop, flow-shop and open-shop. Canonical genetic algorithm is applied for those problems varying its parameters. We analyze some aspects of parameters such as selecting optimal parameters of algorithm, influence on algorithm performance. Finally, three strategies of algorithm – combination of parameters and new conceptualmodel of genetic algorithm are proposed.

Single-based and Population-Based Metaheuristics Algorithms Performances in Solving NP-hard Problems

Metaheuristics is one of the most well-known field of researches uses to find optimum solution for Non-deterministic polynomial hard problems (NP-Hard), that are difficult to find an optimal solution in a polynomial time. Over time many algorithms have been developed based on the heuristics to solve difficult real-life problems, this paper will introduce Metaheuristic-based algorithms and its classifications, Non-deterministic polynomial hard problems. It also will compare the performance two metaheuristic-based algorithms (Elephant Herding optimization algorithm and Tabu Search) to solve Traveling Salesman Problem (TSP), which is one of the most known problem belongs to Non-deterministic polynomial hard problem and widely used in the performance evaluations for different metaheuristics-based optimization algorithms. the experimental results of the paper compare the results of EHO and TS for solving 10 different problems from the TSPLIB95.