The complexity of approximating a nonlinear program

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.     

Guaranteed performance heuristics for the bottleneck travelling salesman problem

We consider constant-performance, polynomial-time, nonexact algorithms for the minimax or bottleneck version of the Travelling Salesman Problem. It is first shown that no such algorithm can exist for problems with arbitrary costs unless P = NP. However, when costs are positive and satisfy the triangle inequality, we use results pertaining to the squares of biconnected graphs to produce a polynomial-time algorithm with worst-case bound 2 and show further that, unless P = NP, no polynomial alternative can improve on this value.     

Approximation Results for Kinetic Variants of TSP

   Abstract. We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results: • If the points all move with the same velocity, then there is a polynomial time approximation scheme for the Kinetic TSP. • The Kinetic TSP cannot be approximated better than by a factor of 2 by a polynomial time algorithm unless P = NP, even if there are only two moving points in the instance. • The Kinetic TSP cannot be approximated better than by a factor of

On the complexity of finding a local minimizer of a quadratic function over a polytope

Mathematical Programming - We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $$c^n$$ (for any constant $$c \ge 0$$ ) of a local...     

NP-hardness of deciding convexity of quartic polynomials and related problems

We show that unless P = NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.     

Inapproximability of finding maximum hidden sets on polygons and terrains

How many people can hide in a given terrain, without any two of them seeing each other? We are interested in finding the precise number and an optimal placement of people to be hidden, given a terrain with n vertices. In this paper, we show that this is not at all easy: The problem of placing a maximum number of hiding people is almost as hard to approximate as the problem, i.e., it cannot be approximated by any polynomial-time algorithm with an approximation ratio of n for some >0, unless P=NP. This is already true for a simple polygon with holes (instead of a terrain). If we do not allow holes in the polygon, we show that there is a constant >0 such that the problem cannot be approximated with an approximation ratio of 1+.     

Rescheduling with new orders and general maximum allowable time disruptions

We study the rescheduling with new orders on a single machine under the general maximum allowable time disruptions. Under the restriction of general maximum allowable time disruptions, each original job has an upper bound for its time disruption (regarded as the maximum allowable time disruption of the job), or equivalently, in every feasible schedule, the difference of the completion time of each original job compared to that in the pre-schedule does not exceed its maximum allowable time disruption. We also consider a stronger restriction which additionally requires that, in a feasible schedule, the starting time of each original job is not allowed to be scheduled smaller than that in the pre-schedule. Scheduling objectives to be minimized are the maximum lateness and the total completion time, respectively, and the pre-schedules of original jobs are given by EDD-schedule and SPT-schedule, respectively. Then we have four problems for consideration. For the two problems for minimizing the maximum lateness, we present strong NP-hardness proof, provide a simple 2-approximation polynomial-time algorithm, and show that, unless \(\text {P}= \text {NP}\), the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 2. For the two problems for minimizing the total completion time, we present strong NP-hardness proof, provide a simple heuristic algorithm, and show that, unless \(\text {P}= \text {NP}\), the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 4/3. Moreover, by relaxing the maximum allowable time disruptions of the original jobs, we present a super-optimal dual-approximation polynomial-time algorithm. As a consequence, if the maximum allowable time disruption of each original job is at least its processing time, then the two problems for minimizing the total completion time are solvable in polynomial time. Finally, we show that, under the agreeability assumption (i.e., the nondecreasing order of the maximum allowable time disruptions of the original jobs coincides with their scheduling order in the pre-schedule), the four problems in consideration are solvable in polynomial time.     

Knapsack with variable weights satisfying linear constraints

We introduce a variant of the knapsack problem, in which the weights of items are also variables but must satisfy a system of linear constraints, and the capacity of knapsack is given and known. We discuss two models: (1) the value of each item is given; (2) the value-to-weight ratio of each item is given. The goal is to determine the weight of each item, and to find a subset of items such that the total weight is no more than the capacity and the total value is maximized. We provide several practical application scenarios that motivate our study, and then investigate the computational complexity and corresponding algorithms. In particular, we show that if the number of constraints is a fixed constant, then both problems can be solved in polynomial time. If the number of constraints is an input, then we show that the first problem is NP-Hard and cannot be approximated within any constant factor unless \(\mathrm{P}= \mathrm{NP}\), while the second problem is NP-Hard but admits a polynomial time approximation scheme. We further propose approximation algorithms for both problems, and extend the results to multiple knapsack cases with a fixed number of knapsacks and identical capacities.     

Generating All Vertices of a Polyhedron Is Hard

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.     

Two results in negation-free logic

Negation-free propositional logic (or first-order logic) is clearly less expressive than the corresponding full system with negation. However, we present two complexity results for logic without negation that are no different from those for the original system. First, the problem of determining logical implication between sentences composed solely of conjunctions and disjunctions is shown to be as difficult as that between arbitrary sentences. Second, we show that the problem of determining a minimum satisfying assignment for a propositional formula in negation-free conjunctive normal form, even with no more than two disjuncts per clause, is NP-complete. We also show that unless P = NP, no polynomial time approximation scheme can exist for this problem.     

单台机以总完工时间为目标的批排序问题

研究单台机,工件加工时间相等,大小不同的批排序问题,给出了一个最坏情况界为9+3~(1/2)/6≈1.7817的多项式时间近似算法,并证明了即使工件总大小不超过2,该问题也不存在FPTAS,除非P=NP.     

Horn函数可满足性的复杂性(英文)

本文证明了 Ｈｏｒｎ函数的极大可满足性即使是限制在如下两种情况中的任何一种也是 ＭＡＸ ＳＮＰ困难的,第一种情况是每个公式都是二次的,第二种是公式中每一个非单位子句有且只有一个补元,这意味着在这档两种情况下没有多项式的近似算法,除非Ｐ＝ＮＰ．     

Guarantees for the success frequency of an algorithm for finding Dodgson-election winners

In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the Θ ₂ ^p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates—although the number of voters may still be polynomial in the number of candidates—a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.     

An FPTAS for optimizing a class of low-rank functions over a polytope

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.     

Weighted matching with pair restrictions

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.     

Vertex partitions of r-edge-colored graphs

Let G be an edge-colored graph.The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G.In the authors' previous work,it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P=NP.In this paper the authors show that for any fixed integer r≥5,if the edges of a graph G are colored by r colors,called an r-edge-colored graph,the problem remains NP-complete.Similar result holds for the monochromatic path(cycle)partition problem.Therefore,to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given.     

A class of polynomially solvable 0-1 programming problems and an application

It is well known that general 0-1 programming problems are NP-Complete and their optimal solutions cannot be found with polynomial-time algorithms unless P=NP. In this paper, we identify a specific class of 0-1 programming problems that is polynomially solvable, and propose two polynomial-time algorithms to find its optimal solutions. This class of 0-1 programming problems commits to a wide range of real-world industrial applications. We provide an instance of representative in the field of supply chain man...     

An FPTAS for a vector subset search problem

Under study is a strongly NP-hard problem of finding a subset of a given size of a finite set of vectors in Euclidean space which minimizes the sum of squared distances from the elements of this subset to its center. The center of the subset is defined as the average vector calculated with all subset elements. It is proved that, unless P=NP, in the general case of the problem there is no fully polynomial time approximation scheme (FPTAS). Such a scheme is provided in the case when the dimension of the space is fixed.     

A note on the approximability of the toughness of graphs

We show that, if NP≠ZPP, for any >0, the toughness of a graph with n vertices is not approximable in polynomial time within a factor of . We give a 4-approximation for graphs with toughness bounded by and we show that this result cannot be generalized to graphs with a bounded toughness. More exactly we prove that there is no constant approximation for graphs with bounded toughness, unless P=NP.     

Computing the minimal covering set

We present the first polynomial-time algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set–the minimal upward covering set and the minimal downward covering set–unless P equals NP. Finally, we observe a strong relationship between von Neumann–Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.