A Dynamically Programmable Quantum Photonic Microprocessor for Graph Computation

This paper presents an on-chip Gaussian Boson Sampling microprocessor, powered by photonic technology, proficiently solving graph combinatorial problems like max cut and vertex cover. It demonstrates photonic quantum computing's potential to accelerate traditionally insurmountable computations.

The objective of this chapter is to illustrate the method of DNA computing and based on this understanding, to explain the application to group controlling of elevators.Many non-deterministic polynomial time (NP) complete problems are solved using heuristic and approximate methods instead of providing a complete solution or mathematical optimal approaches. The central reason comes from the huge computation time needed to solve such combinatorial problems by means of a conventional silicon computer based on von Neumann's architecture. To solve such NP complete problems, this chapter provides an alternative and innovative method based on nano size of DNA to obtain infinitively in a life such as silicon. This chapter is an introductory explanation.The group management is treated as an example of its application. Inefficient situations have been encountered such that all elevators are moving in the same direction or that all elevators arrive at the same floor even during rush hours in the morning. To resolve such situations all elevators should be controlled to assign the best elevator to passengers according to time-to-time change of passengers. The group control system is employed in selection of driving patterns according to the change of traffic volumes or driving management in accidents. Such a group control realises comfortable, safe, and economical management of elevators. The objective of this chapter is to apply DNA computing to calculate complex and huge combinatorial problems of a group of elevators and huge number of floors. The optimal solution will be presented to the group control of elevators on the basis of the DNA computing.

Combinatorial optimization (CO) has a broad range of applications in various fields, including operations research, computer science, and artificial intelligence. However, many of these problems are classified as nondeterministic polynomial-time (NP)-complete or NP-hard problems, which are known for their computational complexity and cannot be solved in polynomial time on traditional digital computers. To address this challenge, continuous-time Ising machine solvers have been developed, utilizing different physical principles to map CO problems to ground state finding. However, most Ising machine prototypes operate at speeds comparable to digital hardware and rely on binarizing node states, resulting in increased system complexity and further limiting operating speed. To tackle these issues, a novel device-algorithm co-design method is proposed for fast sub-optimal solution finding with low hardware complexity. On the device side, a piezoelectric lithium niobate (LiNbO3) microelectromechanical system (MEMS) oscillator network-based Ising machine without second-harmonic injection locking (SHIL) is devised to solve Max-cut and graph coloring problems. The LiNbO3 oscillator operates at speeds greater than 9 GHz, making it one of the fastest oscillatory Ising machines. System-wise, an innovative grouping method is used that achieves a performance guarantee of 0.878 for Max-cut and 0.658 for graph coloring problems, which is comparable to Ising machines that utilize binarization.

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

A continuous approach using NCP function for approximating the solution of the max-cut problem is proposed. The max-cut problem is relaxed into an equivalent nonlinearly constrained continuous optimization problem and a feasible direction method without line searches is presented for generating an optimal solution of the relaxed continuous optimization problem. The convergence of the algorithm is proved. Numerical experiments and comparisons on some max-cut test problems show that we can get the satisfactory solution of max-cut problems with less computation time. Furthermore, this is the first time that the feasible direction method is combined with NCP function for solving max-cut problem, and similar idea can be generalized to other combinatorial optimization problems.

Maximal independent sets in caterpillar graphs

This paper is devoted to the study of complexity of finding a special covering for a set, as well as to obtaining some important applications of special decomposition. We formulate the problem of existence of a special covering as a decision problem. To determine the complexity class in which this problem is located, we study the relationship between this problem and the Boolean satisfiability problem, treating them as formal languages. We prove that these problems are polynomially equivalent, which means that the problem of existence of a special covering for a set is an -complete problem. In this article we also introduce a new concept ‘Replaceable Subsets’. The properties of such subsets are used to fill in the missing elements needed to obtain a special set covering. It is proved that when searching for missing elements to fill a special covering, the order in which these elements are considered does not matter. This result is of great importance in the search for satisfiability of Boolean functions.

The solutions of certain natural decision problems such as the halting problem and the boolean satisfiability problem contain large amounts of useful information about computation that is highly organized and readily available to efficient computational processes. Such problems are computationally useful. This dissertation investigates the complexity and distribution of these computationally useful problems. The main results of this dissertation are of the following three general types. (1) Useful problems contain highly organized information. (2) Very useful problems are so highly organized that they are unusually simple and hence rare. (3) Useful problems are, as a whole, not rare and thus are not necessarily simple. A result of type (1) is proven in Chapter 3. Bennett recently extended algorithmic information theory to include a notion of computational depth that appears to quantify the level of organization in binary strings and sequences. The main result of Chapter 3 states that every weakly useful sequence is strongly deep. (A sequence x is weakly useful if a non-negligible set of recursive problems are decidable within a fixed recursive time bound when given access to x.) Results of type (2) are presented in Chapters 4 and 5. These results say that the $\le\sbsp{\rm m}{\rm P}$-complete problems for E = DTIME(2$\sp{\rm linear}$) and the $\le\sbsp{\rm m}{\rm p/poly}$-complete problems for ESPACE = DSPACE(2$\sp{\rm linear}$) are unusually simple and hence rare. Complete problems are very useful because every problem in E or ESPACE is efficiently decidable when given access to one of these problems. Chapter 6 develops a result of type (3). This result says that the weakly $\le\sbsp{\rm m}{\rm P}$-complete problems for E and ESPACE are not rare and hence are not necessarily simple. Weakly complete problems are useful because every problem in a non-negligible subset of E or ESPACE is efficiently decidable when given access to one of these problems. The above results (and others along the way) are obtained through a systematic investigation of the measure-theoretic structure of complexity classes.

The minimum sensor cover is an important optimization problem in wireless sensor networks, which can be seen as a special case of the minimum set cover problem. The minimum set cover problem is a well-known problem in combinatorial optimization, which has no polynomial-time (ρlnδ)-approximation for 0<ρ<1 unless NP⊆DTIME(n O(loglogn)) where δ is the maximum cardinality of a subset in the input collection. However, the minimum sensor cover problem has polynomial-time constant-approximations because this special case has a geometric structure. The design technique, called partition, is employed to take the advantage of this geometric structure. Especially, the double partition plays an important role. In this article, we would like to give an exploratory essay for the double partition technique together with research progress on approximations for the minimum sensor cover problem.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

Some APX-completeness results for cubic graphs

Here we study linear programming applied to the random K-SAT problem, a fundamental problem in computational complexity. The K-SAT problem is to decide whether a Boolean formula with N variables and structured as a conjunction of M clauses, each being a disjunction of K variables or their negations is satisfiable or not. The ensemble of random K-SAT attracted considerable interest from physicists because for a specific ratio αs = M/N it undergoes in the limit of large N a sharp phase transition from a satisfiable to an unsatisfiable phase. In this study we will concentrate on finding for linear programming algorithms “easy-hard” transitions between phases of different typical hardness of the problems on either side. Linear programming is widely applied to solve practical optimization problems, but has been only rarely considered in the physics community. This is a deficit, because those typically studied types of algorithms work in the space of feasible {0, 1}N configurations while linear programming operates outside the space of valid configurations hence gives a very different perspective on the typical-case hardness of a problem. Here, we demonstrate that the technique leads to one simple-to-understand transition for the well known 2-SAT problem. On the other hand we detect multiple transitions in 3-SAT and 4-SAT. We demonstrate that, in contrast to the previous work on vertex cover and therefore somewhat surprisingly, the hardness transitions are not driven by changes of any of various standard percolation or solution space properties of the problem instances. Thus, here a more complex yet undetected property must be related to the easy-hard transition.

We develop and assess centralized and decentralized signal control systems with short-term origin-destination (OD) demands as inputs. Considering each intersection turning movement as a virtual link, we assign traffic demand to paths based on minimal instantaneous travel time. Then, the optimal control is formulated using a G/G/n/FIFO open queueing network model (QNM). We also solve the issue of optimal control using a three-step naïve method for the centralized system with the new inputs. Because the optimization of large-scale network traffic signals can involve sizeable numbers of decision variables and nonlinear constraints, making it a nondeterministic polynomial time (NP) complete problem, we further decompose the centralized system into a decentralized system where the network is divided into subnetworks. Each subnetwork has a dedicated agent that optimizes signals within it. Furthermore, traffic demand for the entire network is decomposed into demands for subnetworks via path decomposition index (PDI). The proposed control systems are applied to test scenarios constructed using different demand profiles in grid networks. We also investigate the impact of network decomposition strategy on signal control system performance. Results show that network decomposition with smaller subnetworks results in less computational time (CT) but increased average travel time (ATT) and total travel delay (TTD).

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems (COPs). Many social production problems can be effectively represented as instances of TSPs. However, solving large-scale TSPs remains a significant challenge for conventional Von Neumann computers. Many studies have proposed annealing processors to address large-scale COPs, but most of them focus on unconstrained problems, such as the Maxcut problem. In this paper, a scalable decoupled-clustering annealng processor (DCAP) for efficiently handling large-scale TSPs is presented. A decoupled hierarchical clustering algorithm is proposed for higher convergence speed and improved scalability. Several techniques have been developed in hardware to minimize area overhead and processing time, including a modified spin connection topology for the Ising model, an area-efficient random threshold generator, a one-step spin update scheme and a dynamic prediction method. The DCAP prototype is implemented on FPGA with an operating frequency of 125MHz. We tested our design on various TSP instances from the TSPLIB. Results show that our design outperforms the CPU- and GPU-based Neuro-Ising scheme by achieving maximum speedups of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$780\times $ </tex-math></inline-formula> and a 42% improvement in accuracy. With multi-chip interconnection, DCAP is able to handle problems of scale up to 85900 cities.