The two-dimensional knapsack problem with splittable items in stacks

Greedy algorithms for a class of knapsack problems with binary weights

A faster FPTAS for knapsack problem with cardinality constraint

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.

PurposeThe purpose of this paper is to find the best approximation algorithm for solving the more general case of single‐supplier multi‐retailer capacitated economic lot‐sizing (SM‐CELS) problem in deterministic inventory theory, which is the non‐deterministic polynomial (NP)‐hard problem.Design/methodology/approachSince few theoretical results have been published on polynomial time approximation algorithms for SM‐CELS problems, this paper develops a fully polynomial time approximation scheme (FPTAS) for the problem with monotone production and holding‐backlogging cost functions. First the optimal solution of a rounded problem is presented as the approximate solution and its straightforward dynamic‐programming (DP) algorithm. Then the straightforward DP algorithm is converted into an FPTAS by exploiting combinatorial properties of the recursive function.FindingsAn FPTAS is designed for the SM‐CELS problem with monotone cost functions, which is the strongest polynomial time approximation result.Research limitations/implicationsThe main limitation is that the supplier only manufactures without holding any products when the model is applied.Practical implicationsThe paper presents the best result for the SM‐CELS problem in deterministic inventory theory.Originality/valueThe LP‐rounding technique, an effective approach to design approximation algorithms for NP‐hard problems, is successfully applied to the SM‐CELS problem in this paper.

This paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two-dimensional knapsack problems (2KP). The population correlation structure among 2KP coefficients, the level of constraint slackness, and the type of correlation (product moment or rank) are varied in this study. Representative branch-and-bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance. Population correlation structure, and in particular the interconstraint component of the correlation structure, is found to be a significant factor influencing the performance of both the algorithm and the heuristic. In addition, the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance.

Understanding the interactions between different combinatorial optimisation problems in real-world applications is a challenging task. Recently, the traveling thief problem (TTP), as a combination of the classical traveling salesperson problem and the knapsack problem, has been introduced to study these interactions in a systematic way. We investigate the underlying non-linear packing while traveling (PWT) problem of the TTP where items have to be selected along a fixed route. We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS) when maximising the benefit that can be gained over the baseline travel cost. Our experimental investigations show that our new approaches outperform current state-of-the-art approaches on a wide range of benchmark instances.

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of $K$-approximation functions and the calculus of $K$-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed.

A universal approach for continuous or discrete nonlinear programmings with multiple variables and constraints

With the proliferation of wireless cellular technology (e.g., 5G, 4G LTE), managing quality of experience (QoE) for wireless clients streaming different videos over these networks is becoming increasingly important. In this paper, we consider the problem of managing playout stalls experienced by multiple clients streaming different videos from a cellular base station. We build on our prior work [1], where we formulate an epoch based lead aware multiple video transmission (LMVT) problem to minimize the total number of playout stalls experienced by mobile clients. In [1], [2], we show that the LMVT problem is NP-hard and propose a pseudo polynomial dynamic programming solution and a simple greedy heuristic to solve the problem. In this paper, we first show that even a simpler and less restrictive version of the LMVT problem remains hard. We then design a fully polynomial time approximation scheme (FPTAS) for the LVMT problem by demonstrating that it is equivalent to the Santa Claus Problem [3], [4]. The FPTAS provides a bounded performance guarantee, differing by a factor of 1/1+∊B from the optimal solution (where B is the number of time slots in an epoch and ε is a small constant).

Chauhan et al. Oper Res Lett 33 (2005), 249–254 presented a fully polynomial time approximation scheme (FPTAS) for a supply scheduling problem, which is to minimize a total cost associated with the sizes of deliveries from several providers to one manufacturer. The cost functions are assumed non‐decreasing and their arguments are assumed continuously divisible. In this note, we give a motivation for considering the case of non‐monotone cost functions with continuously divisible or discrete arguments. For this more general case, we suggest a modification of the FPTAS in [Chauhan et al., Oper Res Lett 33 (2005), 249–254]. We also suggest a different FPTAS to handle concave cost functions. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

The closest string and substring problems find applications in PCR primer design, genetic probe design, motif finding, and antisense drug design. For their importance, the two problems have been extensively studied recently in computational biology. Unfortunately both problems are NP-complete. Researchers have developed both fixed-parameter algorithms and approximation algorithms for the two problems. In terms of fixed-parameter, when the radius d is the parameter, the best-known fixed-parameter algorithm for closest string has time complexity O(ndd+1), which is still superpolynomial even if d = O(log n). In this paper we provide an O(n|Σ|O(d)) algorithm where Σ is the alphabet. This gives a polynomial time algorithm when d = O(log n) and Σ has constant size. Using the same technique, we additionally provide a more efficient subexponential time algorithm for the closest substring problem. In terms of approximation, both closest string and closest substring problems admit polynomial time approximation schemes (PTAS). The best known time complexity of the PTAS is O(nO(Ɛ-2 log 1/Ɛ)). In this paper we present a PTAS with time complexity O(nO(Ɛ-2)). At last, we prove that a restricted version of the closest substring has the same parameterized complexity as closest substring, answering an open question in the literature.

We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colours) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can accommodate. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the class-constrained bin-packing (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 0–1 class-constrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colours of items depends on the input size. Finally, we introduce the generalized class-constrained packing problem (GCCP), where each item may have more than one colour. We show that GCCP is APX-hard, already for the case of a single knapsack, where all items have the same size and the same value. Our optimization problems have several important applications, including storage management for multimedia systems, production planning, and multiprocessor scheduling. Copyright © 2001 John Wiley & Sons, Ltd.

In the 90’s Clark, Colbourn and Johnson wrote a seminal paper, where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of (unit) d-dimensional balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS ’18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs, however the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for solving maximum clique in a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

Approximation of satisfactory bisection problems

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.