The complexity of optimization problems

Petri nets are known to be useful for modeling concurrent systems. Once modeled by a Petri net, the behavior of a concurrent system can be characterized by the set of all executable transition sequences, which in turn can be viewed as a language over an alphabet of symbols corresponding to the transitions of the underlying Petri net. In this paper, we study the language issue of Petri nets from a computational complexity viewpoint. We analyze the complexity of theregularity problem(i.e., the problem of determining whether a given Petri net defines an irregular language or not) for a variety of classes of Petri nets, includingconflict-free,trap-circuit,normal,sinkless,extended trap-circuit,BPP, andgeneralPetri nets. (Extended trap-circuit Petri nets are trap-circuit Petri nets augmented with a specific type ofcircuits.) As it turns out, the complexities for these Petri net classes range from NL (nondeterministic logspace), PTIME (polynomial time), and NP (nondeterministic polynomial time), to EXPSPACE (exponential space). In the process of deriving the complexity results, we develop adecomposition approachwhich, we feel, is interesting in its own right, and might have other applications to the analysis of Petri nets as well. As a by-product, an NP upper bound of the reachability problem for the class of extended trap-circuit Petri nets (which properly contains that of trap-circuit (and hence, conflict-free) and BPP-nets, and is incomparable with that of normal and sinkless Petri nets) is derived.

We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class . Thus, if the polynomial time hierarchy does not collapse, then -complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The first asserts that statistical zero knowledge is contained in [11, 2], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in [19]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability is less than (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specifically, it is in ). In the standard setting of negligible error probability, there exist PSPACE-complete languages which have sub-linear knowledge complexity. However, if we insist, for example, that the error probability is less than , then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless . In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions , the verifier runs in time polynomial in and and fails with probability at most to detect an additive error in the entropy. We believe that this protocol is of independent interest. Subsequent to our work Goldreich and Vadhan [16] established that the problem of comparing the entropies of two samplable distributions if they are noticeably different is a natural complete promise problem for the class of statistical zero knowledge ( ).

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

......... Non-deterministic polynomial time (commonly termed ‘NP-complete’) problems are relevant to many computational tasks of practical interest—such as the ‘travelling salesman problem’—but are difficult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit ‘phase boundaries’, across which dramatic changes occur in the computational difficulty and solution character—the problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K-satisfiability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the efficiency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems. Many computational tasks of practical interest are surprisingly difficult to solve even using the fastest available machines. Such problems, found for example in planning, scheduling, machine learning, hardware design, and computational biology, generally belong to the class of NP-complete problems 1‐3 . NP stands for ‘nondeterministic polynomial time’, which denotes an abstract computational model with a rather technical definition. Intuitively speaking, this class of computational tasks consists of problems for which a potential solution can be checked efficiently for correctness, yet finding such a solution appears to require exponential time in the worst case. A good analogy can be drawn from mathematics: proving open conjectures in mathematics is extremely difficult, but verifying any given proof (or solution) is generally relatively straightforward. The class of NP-complete problems lies at the foundations of the theory of computational complexity in modern computer science. Literally thousands of computational problems have been shown to be NP-complete. The completeness property of NPcomplete problems means that if an efficient algorithm for solving just one of these problems could be found, one would immediately have an efficient algorithm for all NP-complete problems. However,

The problem of partitioning a set of geographically distributed tasks among a team of UAVs that must travel to perform the tasks arises in many contexts, from surveillance to environmental monitoring. When UAVs have limits in their travel distances, it is important to select subsets of tasks for each UAV that achieve highest team values while remaining feasible given limits on UAV resources. This class of problems has been studied previously in computer science and operations research as orienteering problems. This problem is known to be NP-Hard. Previous algorithms for solving orienteering problems as integer programming problems are slow; faster algorithms have been proposed based on heuristic techniques used in traveling salesperson problems. In this paper, we develop a different class of fast algorithms, based on a decomposition of the orienteering problem into a knapsack assignment problem and a subsequent traveling salesperson problem. We combine greedy algorithms for knapsack problems with the use of spanning trees to estimate traveling salesperson tour lengths to obtain new, approximate algorithms for orienteering problems. We conduct experiments with single and multiple UAV algorithms on benchmark problems as well as problems derived from a LANDAT dataset, comparing our new algorithms with other algorithms in the literature. Our results indicate that our new algorithms are consistently fast, and perform better than competing algorithms.

The problem of patient admission scheduling (PAS) is a nondeterministic polynomial time (NP)-hard combinatorial optimization problem with numerous constraints. Researchers have divided the constraints of this problem into hard (i.e., feasible solution) and soft constraints (i.e., quality solution). The majority of research has dealt with PAS using integer linear programming (ILP) and single objective meta-heuristic searching-based approaches. ILP-based approaches carry high computational demand and the risk of non-feasibility for a large dataset. In a single objective optimization, there is a risk of local minima due to the non-convexity of the problem. In this article, we present the first pareto front-based optimization for PAS using set of meta-heuristic approaches. We selected four multi-objective optimization methods. Problem-specific operators were developed for each of them. Next, we compared them with single objective optimization approaches, namely, simulated annealing and particle swarm optimization. In addition, this article also deals with the dynamical aspect of this problem by comparing historical window-based decomposition with day decomposition, as has previously been proposed in the literature. An evaluation of the models proposed in the article and comparison with traditional models reveals the superiority of our proposed multi-objective optimization with window incorporation in terms of optimality.

Cell association is a significant research issue in future mobile communication systems due to the unacceptably large computational time of traditional schemes. This article proposes a polynomial-time cell association scheme which not only completes the association in polynomial time but also fits for a generic optimization objective function. On the one hand, traditional cell association as a non-deterministic polynomial (NP) hard problem with a generic utility function is heuristically transformed into a 2-dimensional assignment optimization and solved by a certain polynomial-time algorithm, which significantly saves computational time. On the other hand, the scheme jointly considers utility maximization and load balancing among multiple base stations (BSs) by maintaining an experience pool storing a set of weighting factor values and their corresponding performances. When an association optimization is required, a suitable weighting factor value is taken from the pool to calculate a long square utility matrix and a certain polynomial-time algorithm will be applied for the association. Comparing with several representative schemes, the proposed scheme achieves large system capacity and high fairness within a relatively short computational time.

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

In this paper, a global function is a function that computes a (local) function in each ordered structure of a specified vocabulary. We design algebras of global functions for a number of complexity classes for which such algebras have not been known, e.g. for the functions computable in nondeterministic logarithmic space, or in nondeterministic polynomial time. In addition, we present a functional analogue of first-order logic and give a new functional characterization of polynomial time.

Batched sparse (BATS) codes are a promising technique for reliable data transmission in multihop wireless networks. A BATS code consists of an outer code and an inner code, the design of which can be separated. In this paper, we investigate the design of the inner code and focus on maximizing the expected batch transfer matrix rank normalized by the total number of packets transmitted by both the source and intermediate nodes. By allowing different recoding decisions at different nodes, the problem is formulated as a nonlinear integer programming (NLIP) problem, which is non-deterministic polynomial (NP)-hard in general. We derive a noniterative formula to simplify the evaluation of the expected batch transfer matrix rank. By considering large finite fields and relaxing the integer constraints, we obtain an upper bound on the optimal value of the original maximization NLIP problem. We also derive a centralized and a decentralized algorithm to find suboptimal solutions of the NLIP problem, with different tradeoffs between complexity and performance. Numerical results show that our proposed inner codes outperform the standard BATS, and achieve very close to the upper bound, which verifies the near optimality of our algorithms.

Statistical physics of hard optimization problemsOptimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

This paper considers two-dimensional (2-D) retiming, which is the problem of retiming circuits that operate on 2-D signals. We begin by discussing two types of parallelism available in 2-D data processing, which we call inter-iteration parallelism and inter-operation parallelism. We then present two novel techniques for 2-D retiming that can be used to extract inter-operation parallelism. These two techniques are designed to minimize the amount of memory required to implement a 2-D data-flow graph while maintaining a desired clock rate for the circuit. The first technique is based on an integer linear programming (ILP) formulation of the problem, and is called ILP 2-D retiming. This technique considers the entire 2-D retiming problem as a whole, but long central processing unit times are required if the circuit is large. The second technique, called orthogonal 2-D retiming, is a linear programming formulation which is derived by partitioning ILP 2-D retiming into two parts called s- and a-retiming. This technique finds a solution in polynomial time and is much faster than the ILP 2-D retiming technique, but the two sub problems (s- and a-retiming) can give results which are not compatible with one another. To solve this incompatibility problem, a variation of orthogonal 2-D retiming called integer orthogonal 2-D retiming is developed. This technique runs in polynomial time and the s-retiming and a-retiming steps are guaranteed to give compatible results. We show that the techniques presented in this paper can result in memory hardware savings of 50% compared to previously published 2-D retiming techniques.

A class of optimization problems over subsets of zero-one vectors of then-dimensional unit cube given by a special linear congruence relation is considered. The general problem is formulated as a zero-one linear program, minimal and complete descriptions of the associated polytopes by linear inequalities are derived and an\(\mathcal{O}(n \log n)\) time algorithm for the optimization problems is given. Since the number of inequalities that completely describe the polytope grows exponentially withn, we also give a separation algorithm that identifies violated inequalities in time\(\mathcal{O}(n^2 )\). A particular variation of the bin packing problem is a special case of our problem and can thus be solved in polynomial time.