Dealing with Hardness

This chapter demonstrates several approaches for dealing with hard problems. These approaches include brute force, heuristics, and approximation. Typically, no single technique will suffice to handle the difficult NP problems one needs to solve. For moderate-sized problems one can search over all possible solutions with the very fast computers available today. One can use algorithms that might not work for every problem but do work for many of the problems one cares about. Other algorithms may not find the best possible solution but still a solution that's good enough. Other times one just cannot get a solution for an NP-complete problem. One has to try to solve a different problem or just give up.

The Prehistory of P versus NP

This chapter explores two separate paths that led to the P versus NP question. In the end it was Steve Cook in the West and Leonid Levin in the East who would first ask whether P = NP. Science does not happen in a vacuum, and both sides have a long history leading to the work of Cook and Levin. The chapter covers just a small part of those research agendas, the struggle in the West to understand efficient computation and the struggle in the East to understand the necessity of perebor. Both would lead to P versus NP. Today, with most academic work available over the Internet and with generally open travel around the world, there is now one large research community instead of two separate ones.

The Golden Ticket

This introductory chapter provides an overview of the P versus NP problem. The P versus NP problem asks, among other things, whether one can quickly find the shortest route for a traveling salesman. P and NP are named after their technical definitions, but it is best not to think of them as mathematical objects but as concepts. “NP” is the collection of problems that have a solution that one wants to find. “P” consists of the problems to which one can find a solution quickly. “P = NP” means one can always quickly compute these solutions, like finding the shortest route for a traveling salesman. “P ≠ NP” means one cannot. Ultimately, the P versus NP problem has achieved the status of one of the great open problems in all of mathematics.

Quantum

This chapter examines the power of quantum computing, as well as the related concepts of quantum cryptography and teleportation. In 1982, the Nobel prize-winning physicist Richard Feynman noticed there was no simple way of simulating quantum physical systems using digital computers. He turned this problem into an opportunity—perhaps a computational device based on quantum mechanics could solve problems more efficiently than more traditional computers. In the decades that followed, computer scientists and physicists, often working together, showed in theory that quantum computers can solve certain problems, such as factoring numbers, much faster. Whether one can actually build large or even medium-scale working quantum computers and determine exactly what these computers can or cannot do still remain significant challenges.

P and NP

This chapter discusses P and NP through Frenemy, an imaginary world where every pair of people comprises either friends or enemies. Frenemy has about 20,000 inhabitants. Every individual seems normal, but put two in close vicinity and a strange thing happens. Either the two take an instant liking to each other, immediately becoming the best of friends, or they take one look at each other and immediately become the worst of enemies. By examining social networking data such as Facebook and Twitter, computer scientists at the Frenemy Institute of Technology have put together a nearly complete database showing which pairs of people are friends and which pairs of people are enemies. The chapter then shows what these researchers can and cannot do with this data. It also presents other NP problems from a few other scientific fields where there is no known efficient algorithms.

The Beautiful World

This chapter examines an efficient algorithm that solves NP problems, the Urbana algorithm. With the Urbana algorithm one can solve all the NP problems quickly, finding the simplest program that classifies data becomes an easy programming exercise. All one needs do is feed in lots of data and the algorithm does the rest. And that lets one learn just about everything. If it turns out that P = NP and the world has efficient algorithms for all NP problems, it will change in ways that will make the Internet seem like a footnote in history. Not only would it be impossible to describe all these changes but the biggest implications of the new technologies would be impossible to predict.

The Future

This chapter explores some of today's great challenges of computing. These challenges include parallel computation, dealing with big data, and the networking of everything. The chapter then argues that P versus NP goes well beyond a simple mathematical puzzle. The P versus NP problem is a way of thinking, a way to classify computational problems by their inherent difficulty. P versus NP also brings communities together. There are NP-complete problems in physics, biology, economics, and many other fields. Physicists and economists work on very different problems, but they share a commonality that can give great benefits from sharing tools and techniques. Tools developed to find the ground state of a physical system can help find equilibrium behavior in a complex economic environment. Ultimately, the inherent difficulty of NP problems leads to new technologies.

Proving P ≠ NP

This chapter focuses on a few of the ideas that people have tried to solve the P versus NP problem. These have not panned out to anything close to a solution to the problem. To prove P ≠ NP one needs to show that no algorithm, even those that have not been discovered yet, can solve some NP problem. It is simply very difficult to show that something cannot be done. However, it is not a logically impossible task. The only known serious approach to the P versus NP problem today is due to Ketan Mulmuley from the University of Chicago. He has shown how solving some difficult problems in a mathematical field called algebraic geometry may lead to a proof that P ≠ NP. However, resolving these algebraic geometry problems will require mathematical techniques far beyond what is available today.

Secrets

This chapter analyzes how, in 1976, Whitfield Diffie and Martin Hellman suggested that one could use NP to hide one's own secrets. The field of cryptography, the study of secret messages, changed forever. Diffie and Hellman, building on earlier work of Roger Merkle, proposed a method to get around the problem of network security using what they called “public-key” cryptography. A computer would generate two keys, a public key and a private key. The computer would store the private key, never putting that key on the network. The public key would be sent over the network broadcast to everyone. Diffie and Hellman's idea was to develop a cryptosystem that used the public key for encrypting messages, turning the real message into a coded one. The public key would not be able to decrypt the message. Only the private key could decrypt the message.

The Hardest Problems in NP

This chapter looks at some of the hardest problems in NP. Most of the NP problems that people considered in the mid-1970s either turned out to be NP-complete or people found efficient algorithms putting them in P. However, some NP problems refused to be so nicely and quickly characterized. Some would be settled years later, and others are still not known. These NP problems include the graph isomorphism, one of the few problems whose difficulty seems somewhat harder than P but not as hard as NP-complete problems like Hamiltonian paths and max-cut. Other NP problems include prime numbers and factoring, and linear programming. The linear programming problem has good algorithms in theory and practice—they just happen to be two very different algorithms.