Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem

The strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given sizes (cardinalities) minimizing the sum (over both clusters) of the intracluster sums of squared distances from the elements of the clusters to their centers is considered. It is assumed that the center of one of the sought clusters is specified at the desired (arbitrary) point of space (without loss of generality, at the origin), while the center of the other one is unknown and determined as the mean value over all elements of this cluster. It is shown that unless P = NP, there is no fully polynomial-time approximation scheme for this problem, and such a scheme is substantiated in the case of a fixed space dimension.     

A polynomial-time approximation scheme for the Euclidean problem on a cycle cover of a graph

We study the minimum-weight k-size cycle cover problem (Min-k-SCCP) of finding a partition of a complete weighted digraph into k vertex-disjoint cycles of minimum total weight. This problem is a natural generalization of the known traveling salesman problem (TSP) and has a number of applications in operations research and data analysis. We show that the problem is strongly NP-hard in the general case and preserves intractability even in the geometric statement. For the metric subclass of the problem, a 2-approximation algorithm is proposed. For the Euclidean Min-2-SCCP, a polynomial-time approximation scheme based on Arora’s approach is built.     

A fully polynomial approximation scheme for the total tardiness problem

A fully polynomial approximation scheme is presented for the problem of sequencing jobs for processing by a single machine so as to minimize total tardiness. This result is obtained by modifying the author's pseudopolynomial algorithm for the same problem.     

A fully polynomial time approximation scheme for the Replenishment Storage problem

The Replenishment Storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system with specified individual reorder cycle lengths. The reorders can only take place at integer time units. This problem was shown to be weakly NP-hard for constant joint cycle length (the least common multiple of all individual cycle lengths). We devise here the first known FPTAS for the RSP with different individual cycle lengths and constant joint cycle length.     

A polynomial-time approximation scheme for the two machine flow shop problem with several availability constraints

In this paper, we develop a polynomial-time approximation scheme for the two-machine flow shop scheduling problem with several availability constraints on the first machine, under the resumable scenario.     

A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem

We propose a fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. First, an exact pseudo-polynomial algorithm is developed based on a two-variable extension of the well-known matrix-tree theorem. The scaling and approximate binary search techniques are then utilized to yield a fully polynomial approximation scheme.     

A new fully polynomial time approximation scheme for the interval subset sum problem

The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals \(\left\{ [a_{i,1},a_{i,2}]\right\} _{i=1}^n\) and a target integer T, the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0–1 knapsack problem. We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme for solving the general ISSP problem. The time and space complexities of the proposed scheme are \({{\mathcal {O}}}\left( n \max \left\{ 1 / \epsilon ,\log n\right\} \right) \) and \(\mathcal{O}\left( n+1/\epsilon \right) ,\) respectively, where \(\epsilon \) is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with \(n=100{,}000\) and \(\epsilon =0.1\%\) within 1 s.     

A polynomial-time algorithm for the change-making problem

Optimally making change—representing a given value with the fewest coins from a set of denominations—is in general NP-hard. In most real money systems however, the greedy algorithm is optimal. We give a polynomial-time algorithm to determine, for a given coin system, whether the greedy algorithm is optimal.     

An approximating polynomial algorithm for a sequence partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.     

Strongly polynomial-time approximation for a class of bicriteria problems

Consider the following problem: given a ground set and two minimization objectives of the same type find a subset from a given subset-class that minimizes the first objective subject to a budget constraint on the second objective. Using Megiddo's parametric method we improve an earlier weakly polynomial time algorithm.     

A polynomial-time simplex method for the maximumk-flow problem

A generalization of the maximum-flow problem is considered in which every unit of flow sent from the source to the sink yields a payoff of $k. In addition, the capacity of any arce can be increased at a per-unit cost of $c _e . The problem is to determine how much arc capacity to purchase for each arc and how much flow to send so as to maximize the net profit. This problem can be modeled as a circulation problem. The main result of this paper is that this circulation problem can be solved by the network simplex method in at mostkmn pivots. Whenc _e = 1 for each arce, this yields a strongly polynomial-time simplex method. This result uses and extends a result of Goldfarb and Hao which states that the standard maximum-flow problem can be solved by the network simplex method in at mostmn pivots.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

An approximation scheme for a problem of search for a vector subset

We consider the following clustering problem: Given a vector set, find a subset of cardinality k and minimum square deviation from its mean. The distance between the vectors is defined by the Euclideanmetric. We present an approximation scheme (PTAS) that allows us to solve this problem with an arbitrary relative error ? in time O(n ^2/?+1(9/?)^3/? d), where n is the number of vectors of the input set and d denotes the dimension of the space.     

A fully polynomial approximation algorithm for the 0–1 knapsack problem

A fully polynomial ?-approximation algorithm is developed for the 0–1 knapsack problem. The algorithm uses results of Lawler and Ibarra and Kim. A pseudo-polynomial dynamic programming algorithm is first suggested which solves the problem in O(nb log n) time and O(b) space.     

Approximation algorithm for the problem of partitioning a sequence into clusters

We consider the problem of partitioning a finite sequence of Euclidean points into a given number of clusters (subsequences) using the criterion of the minimal sum (over all clusters) of intercluster sums of squared distances from the elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is at the origin, while the center of each of the other clusters is unknown and determined as the mean value over all elements in this cluster. Additionally, the partition obeys two structural constraints on the indices of sequence elements contained in the clusters with unknown centers: (1) the concatenation of the indices of elements in these clusters is an increasing sequence, and (2) the difference between an index and the preceding one is bounded above and below by prescribed constants. It is shown that this problem is strongly NP-hard. A 2-approximation algorithm is constructed that is polynomial-time for a fixed number of clusters.     

A polynomial-time algorithm for the paired-domination problem on permutation graphs

A set S of vertices in a graph H=(V,E) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O(mn) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges.     

Worst-case analysis of an approximation scheme for the subset-sum problem

Given a set of n integers, the Subset-Sum Problem is to find that subset whose sum is closent to, without exceeding, a given integer W. In this paper we analyse Martello and Toth's polynomial approximation scheme for the problem, showing that its worst-case performance is better than the lower bound they proved, although not so good as conjectured.