Theory Of NP-completeness Research Articles

A Theory of NP-completeness and Ill-conditioning for Approximate Real Computations

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs. NP question in this new theory to the classical P vs. NP problem.

On the Subexponential-Time Complexity of CSP

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice. Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential-time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. An instance of the constraint satisfaction problem with n variables over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). In this paper we study the existence of subexponential-time algorithms, that is, algorithms running in do(n) steps, for various natural restrictions of the constraint satisfaction problem. We consider both the constraint satisfaction problem in which all the constraints are given extensionally as tables, and that in which all the constraints are given intensionally in the form of global constraints. We provide tight characterizations of the subexponential-time complexity of the aforementioned problems with respect to several natural structural parameters, which allows us to draw a detailed landscape of the subexponential-time complexity of the constraint satisfaction problem. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving the constraint satisfaction problem.

Absorbing random walks and the NAE2SAT problem

In this paper, we propose a simple randomized algorithm for the NAE2SAT problem. The analysis of the algorithm uses the theory of symmetric, absorbing random walks. Inasmuch as the NAE2SAT problem is in the complexity class L (Deterministic Logarithmic Space) and L ⊆ P ⊆ RP, the existence of such an algorithm is not surprising. However, the simplicity of our approach and the insights revealed by its analysis make the current study worthwhile. Not-all-equal SAT (NAESAT) is the variant of the satisfiability problem (SAT), in which we are interested in an assignment that satisfies all the clauses, but falsifies at least one literal in each clause. NAESAT is an interesting SAT variant in its own right and has been studied by SAT theoreticians, on account of its connections to the graph colouring problem. It is well-known that the variant of NAESAT with exactly three literals per clause (NAE3SAT) is NP-complete [M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman Company, San Francisco, 1979]. Likewise, there seem to be a number of polynomial time algorithms for NAE2SAT as part of algorithm design folklore [D.S. Johnson, Personal Communication]. Here we show that the NAE2SAT problem admits an extremely simple, literal-flipping algorithm, in precisely the same way that 2SAT does. On a satisfiable instance involving n variables, the probability that our algorithm does not find a satisfying assignment is at most 1/24. The randomized algorithm takes O(1) extra space, in the presence of a verifier and provides an interesting insight into checking whether a graph is bipartite. It must be noted that space-parsimonious randomized algorithms for a problem, such as the one proposed in this paper, invariably lead to error–bounded, online algorithms for the same. As part of our analysis, we argue that a restricted variant of NAE2SAT is L-complete. We note that the bounds derived in this paper are sharper than the ones in Papadimitriou [C.H. Papadimitriou, On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1–4 October 1991, IEEE, ed., IEEE, Washington, DC, 1991, pp. 163–169].

SYSTEMATIC VERSUS LOCAL SEARCH AND GA TECHNIQUES FOR INCREMENTAL SAT

Propositional satisfiability (SAT) problem is fundamental to the theory of NP-completeness. Indeed, using the concept of "polynomial-time reducibility" all NP-complete problems can be polynomially reduced to SAT. Thus, any new technique for satisfiability problems will lead to general approaches for thousands of hard combinatorial problems. In this paper, we introduce the incremental propositional satisfiability problem that consists of maintaining the satisfiability of a propositional formula anytime a conjunction of new clauses is added. More precisely, the goal here is to check whether a solution to a SAT problem continues to be a solution anytime a new set of clauses is added and if not, whether the solution can be modified efficiently to satisfy the old formula and the new clauses. We will study the applicability of systematic and approximation methods for solving incremental SAT problems. The systematic method is based on the branch and bound technique, whereas the approximation methods rely on stochastic local search (SLS) and genetic algorithms (GAs). A comprehensive empirical study, conducted on a wide range of randomly generated consistent SAT instances, demonstrates the efficiency in time of the approximation methods over the branch and bound algorithm. However, these approximation methods do not guarantee the completeness of the solution returned. We show that a method we propose that uses nonsystematic search in a limited form together with branch and bound has the best compromise, in practice, between time and the success ratio (percentage of instances completely solved).

The complexity of fixed-parameter problems

We describe how to found and develop the theory of fixed-parameter complexity and hardness in a manner both fully analgous to the classical theory of NP -completeness and much simpler than the original development. We present some recent advance made using this approach.

Imitation of Complex Strategies

Researchers examining loosely coupled systems, knowledge management, and complementary practices in organizations have proposed, informally, that the complexity of a successful business strategy can deter imitation of the strategy. This paper explores this proposition rigorously. A simple model is developed that parametrizes the two aspects of strategic complexity: the number of elements in a strategy and the interactions among those elements. The model excludes conventional resource-based and game-theoretic barriers to imitation altogether. The model is used to show that complexity makes the search for an optimal strategy intractable in the technical sense of the word provided by the theory of NP-completeness. Consequently, would-be copycats must rely on search heuristics or on learning, not on algorithmic “solutions,” to match the performance of superior firms. However, complexity also undermines heuristics and learning. In the face of complexity, firms that follow simple hill-climbing heuristics are quickly snared on low “local peaks,” and firms that try to learn and mimic a high performer's entire strategy suffer large penalties from small errors. The model helps to explain why some winning strategies remain unmatched even though they are open to public scrutiny; why certain bundles of organizational practices diffuse slowly even though they lead to superior performance; and why some strategies yield superior returns even after many of their critical ingredients are adopted by competitors. The analysis also suggests roles for management science and managerial choice in a world of complex strategies.

Hybrid approach to decision-making for job-shop scheduling

The job-shop scheduling problem is one of the hardest problems (NP-complete problem) {Garey and Johnson 1979, Computer and Intractability: A Guide to the Theory of NP-completeness (San Francisco: Freeman)}. In lots of cases, the combination of goals and resources exponentially increases the search space, and the generation of consistent and satisfying schedules is more and more difficult. This article shows the coupling of three approaches in order to contribute to the solving of the job-shop scheduling problem: genetic algorithms (GAs); constraint logic programming (CLP); and multi-criteria decision-making (MCDM). The GAs are searching algorithms based on the mechanics of natural selection. They employ a probabilistic search for locating the globally optimal solution. They start with an initial population often randomly generated. The difficulty holds in the creation of this initial population which is considered an important parameter of GAs. CLP is a concept based on operation research (OR) and artificial intelligence (AI). It tends to rid itself of their drawbacks and to regroup their advantages. It aims at providing a set of solutions to a problem expressed in term of constraints. A MCDM approach aims at providing an analysis of a set of possible alternatives according to a set of conflicting criteria in order to help the user. This article explains first how to use the CLP to generate a first population. Second it describes the application of GAs to provide a set of job-shop scheduling minimizing the global makespan. Among the set of job-shop scheduling minimizing the global makespan, it is then possible for the decision-maker to select the most satisfying scheduling according to other criteria, e.g. balance of workload or makespan of each manufacturing order. A large-sized example illustrates clearly the advantages of our hybrid method.

Computational complexity of a problem in molecular structure prediction.

The computational task of protein structure prediction is believed to require exponential time, but previous arguments as to its intractability have taken into account only the size of a protein's conformational space. Such arguments do not rule out the possible existence of an algorithm, more selective than exhaustive search, that is efficient and exact. (An efficient algorithm is one that is guaranteed, for all possible inputs, to run in time bounded by a function polynomial in the problem size. An intractable problem is one for which no efficient algorithm exists.) Questions regarding the possible intractability of problems are often best answered using the theory of NP-completeness. In this treatment we show the NP-hardness of two typical mathematical statements of empirical potential energy function minimization of macromolecules. Unless all NP-complete problems can be solved efficiently, these results imply that a function minimization algorithm can be efficient for protein structure prediction only if it exploits protein-specific properties that prohibit the simple geometric constructions that we use in our proofs. Analysis of further mathematical statements of molecular structure prediction could constitute a systematic methodology for identifying sources of complexity in protein folding, and for guiding development of predictive algorithms.

Domino Games and Complexity

Domino games which describe computations of alternating Turing machines in the same way as dominoes (tiling systems) encode computations of deterministic and nondeterministic Turing machines are considered. The domino games are two-person games in the course of which the players build up domino tilings of a square of prescribed size. Acceptance of an alternating Turing machine corresponds to a winning strategy for one player—the number of moves in the game is the number of alternations of the Turing machine. Let $\operatorname{ATIME}(T(n), m)$ denote the class of all sets that are accepted by some alternating Turing machine in time $T(n)$ with at most m alternations. It is shown that any problem in such a complexity class can be reduced to the strategy problem for some domino game. In particular domino games which are complete in the classes $\Sigma_{m}^{p}$, and $\Pi_{m}^{p}$, of the polynomial time hierarchy are found. This corresponds to the approach of Savelsbergh and van Emde Boas and of Lewis and Papadimitriou, who have shown that the theory of NP-completeness may also be founded on a finite domino problem instead of the satisfiability problem for propositional formulae. Similar generalizations are possible for domino connectability problems: Domino thread games where the players build up a thread of dominoes are introduced; the first player wins if he can ultimately connect two given points. It is shown that a certain class of such games captures deterministic time complexity. In particular, games are found whose strategy problems are P-complete. In the last section applications to the complexity of simple prefix classes in logical theories are briefly discussed.

Piecing together complexity

To illustrate the "remarkable extent to which complexity theory operates by means of analogs from computability theory," Richard Karp created this conceptual map or jigsaw puzzle. To lay out the puzzle in the plane, he used a "graph planarity algorithm." The more distantly placed parts might not at first seem related, "but in the end, the theory of NP-completeness does bring them all together," Karp says. The upper right portion of the puzzle shows concepts related to combinatorial explosions and the notion of a "good" or "efficient" algorithm. In turn, "Complexity" connects these concepts to the upper left portion, which represents the concerns of early computability theorists. The traveling salesman problem is closer to the upper right corner because it is probably intractable. It therefore borders on "NP-completeness" and "Combinatorial explosion." To some extent, however, certain divisions blur. "Linear programming," for example, has an anomalous status—the most widely used algorithms for solving such problems in practice are not good in the theoretical sense, and those that are good in the theoretical sense are often not good in practice. One example is the ellipsoid method that was the object of so much attention six years ago. It ran in polynomial time, but the polynomial was of such a high degree that the method proved good in the technical sense, but ineffective in practice. "The reason is that our notion of polynomial-time algorithms doesn't exactly capture the notion of an intuitively efficient algorithm," Karp explains. "When you get up to n 5 or n 6 , then it's hard to justify saying that it is really efficient. So Edmonds's concept of a good algorithm isn't quite a perfect formal counterpart of good in the intuitive sense." Further, the simplex algorithm is good in every practical sense, Karp says, but not good according to the standard paradigm of complexity theory. The most recent addition to linear programming solutions, an algorithm devised by Narendra Karmarkar that some think challenges the simplex algorithm, is good in the technical sense and also appears to be quite effective in practice, says Karp. The good algorithm segment is adjacent to "Heuristics" because a heuristic algorithm may work well, but lack a theoretical pedigree. Some heuristic algorithms are always fast, but sometimes fail to give good solutions. Others always give an optimal solution, but are not guaranteed to be fast. The simplex algorithm is of the latter type. "Undecidability, " "Combinatorial explosion," and "Complexity" are on the same plane because they are analogs of one another; undecidability involves unbounded search, whereas combinatorial explosions are by definition very long but not unbounded searches. Complexity theory bridges the gap because, instead of asking whether a problem can be solved at all, it poses questions about the resources needed to solve a problem. The lower left region contains the segments Karp has been concerned with most recently and that contain open-ended questions. "Randomized algorithm," for example, is situated opposite "Probabilistic analysis" because both are alternatives to worst-case analyses of deterministic algorithms. Randomized algorithms might be able to solve problems in polynomial time that deterministic ones cannot and that could mean an extension of the notion of good algorithms. Perhaps through software designs for non-von Neumann machines, algorithms can be made more efficient in practice through parallelism. Finally, some parts of the puzzle are not yet defined. Says Karp, "They correspond to the unknown territory that remains to be explored in the future."

Computers and Intractability: A Guide to the Theory of NP-Completeness (Michael R. Garey and David S. Johnson)

Computers and Intractability: A Guide to the Theory of NP-Completeness (Michael R. Garey and David S. Johnson)