Verbesserte Approximationsalgorithmen für Packungs- und Ablaufplanungsprobleme

This thesis presents approximation algorithms for geometric packing and scheduling problems. First, improved AFPTAS for the Bin Packing Problem (BP) and its generalization, the Variable-sized Bin Packing Problem (VBP), are explained. Our algorithms have to solve the unbounded variant of the Knapsack Problem (KP) and of the Knapsack Problem with Inversely Proportional Profits (KPIP) as subproblems. In the normal 0-1 variant of KP, an item can be chosen only once. In the bounded variant, an individual bounded number of copies can be taken of every item. The unbounded variant (UKP) allows for an infinite number of copies of every item. KPIP is a generalization of KP in which we have not only one, but several knapsack sizes. The profit of an item is inversely proportional to the size of the knapsack into which it has been packed. This makes it non-trivial to choose the knapsack size that maximizes the profit over all knapsack sizes. Similar to KP, there are the 0-1, the bounded, and the unbounded variant of KPIP. We first present FPTAS for every of the three variants of KPIP. They are faster than the natural approach to separately solve for every knapsack size the corresponding Knapsack Problem. Second, we present an FPTAS for UKP that is faster and needs less storage space than previously known algorithms. Finally, we combine the approaches of the KPIP and of the UKP FPTAS to get an FPTAS for the Unbounded KPIP that has again a better time and space complexity. All these results improve the running time for our BP and VBP algorithms. As a corollary, we also improve the running time for a Strip Packing AFPTAS. Finally, we consider Scheduling on Unrelated Machines of which we study the special case with a constant number K of machine types: one job has the same processing time on every machine of the same type. We present a PTAS for this special case. The algorithm has a better running time than the previously known algorithm for general (but constant) K.Diese Dissertation stellt Approximationsalgorithmen für geometrische Packungs- und Ablaufplanungsprobleme (Packing and Scheduling Problems) vor. Zuerst werden verbesserte AFPTAS für das Behälterproblem (Bin Packing, BP) und seine Verallgemeinerung, das Behälterproblem mit verschiedenen Behältergrößen (Variable-sized Bin Packing, VBP), erklärt. Unsere Algorithmen müssen die unbeschränkte (unbounded) Variante des Rucksackproblems (Knapsack Problem, KP) und des Rucksackproblems mit invers proportionalen Profiten (Knapsack Problem with Inversely Proportional Profits, KPIP) als Unterprobleme lösen. Bei KP gibt es die Varianten 0-1, beschränkt und unbeschränkt. KPIP ist eine Verallgemeinerung des Rucksackproblems mit mehreren Rucksackgrößen, die in dieser Dissertation eingeführt wird. Wie bei KP gibt es bei KPIP ebenfalls die Varianten 0-1, beschränkt und unbeschränkt. Wir stellen zuerst FPTAS für alle drei Varianten von KPIP vor. Sie sind schneller als der natürliche Ansatz, für jede Rucksackgröße das entsprechende Rucksackproblem einzeln zu lösen. Danach stellen wir ein FPTAS für UKP vor, das schneller ist und weniger Speicherplatz benötigt als zuvor bekannte Algorithmen. Schließlich kombinieren wir den Ansatz für KPIP und für das unbeschränkte KPIP, um ein FPTAS für die unbeschränkte Variante von KPIP zu erhalten, das wiederum eine kleinere Zeit- und Speicherkomplexität besitzt. All diese Resultate verbessern die Laufzeit unserer BP- und VBP-Algorithmen. Als Korollar verbessern wir außerdem die Laufzeit eines AFPTAS für das geometrische Zuschnittproblem (Strip Packing). Schließlich betrachten wir das Ablaufplanungsproblem auf heterogenen Maschinen (Scheduling on Unrelated Machines), bei dem wir den Spezialfall mit einer konstanten Anzahl K an Maschinentypen untersuchen: Eine Aufgabe (Job) hat auf jeder Maschine desselben Typs die gleiche Ausführungszeit. Wir stellen für diesen Spezialfall ein PTAS vor. Der Algorithmus ist schneller als das zuvor bekannte Verfahren für allgemeines (aber konstantes) K

An FPTAS for the Δ\Delta-modular multidimensional knapsack problem

It is known that there is no EPTAS for the $m$-dimensional knapsack problem unless $W[1] = FPT$. It is true already for the case, when $m = 2$. But, an FPTAS still can exist for some other particular cases of the problem. In this note, we show that the $m$-dimensional knapsack problem with a $\Delta$-modular constraints matrix admits an FPTAS, whose complexity bound depends on $\Delta$ linearly. More precisely, the proposed algorithm complexity is $$O(T_{LP} \cdot (1/\varepsilon)^{m+3} \cdot (2m)^{2m + 6} \cdot \Delta),$$ where $T_{LP}$ is the linear programming complexity bound. In particular, for fixed $m$ the arithmetical complexity bound becomes $$O(n \cdot (1/\varepsilon)^{m+3} \cdot \Delta).$$ Our algorithm is actually a generalisation of the classical FPTAS for the $1$-dimensional case. Strictly speaking, the considered problem can be solved by an exact polynomial-time algorithm, when $m$ is fixed and $\Delta$ grows as a polynomial on $n$. This fact can be observed combining previously known results. In this paper, we give a slightly more accurate analysis to present an exact algorithm with the complexity bound $$O(n \cdot \Delta^{m + 1}), \quad \text{ for $m$ being fixed}.$$ Note that the last bound is non-linear by $\Delta$ with respect to the given FPTAS

An FPTAS for Stochastic Unbounded Min-Knapsack Problem

In this paper, we study the stochastic unbounded min-knapsack problem ($\textbf{Min-SUKP}$). The ordinary unbounded min-knapsack problem states that: There are $n$ types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least $W$ and the total cost is minimized. The \prob~generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In \prob, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for $\textbf{Min-SUKP}$, i.e., the approximate value our algorithm computes is at most $(1+\epsilon)$ times the optimum, and our algorithm runs in $poly(1/\epsilon,n,\log W)$ time.Comment: 24 page

An Improved FPTAS for 0-1 Knapsack

The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1+epsilon runs in O~(n + (1/epsilon)^{12/5}) time, where O~ hides polylogarithmic factors. In this paper we present an improved algorithm in O~(n+(1/epsilon)^{9/4}) time, with only a (1/epsilon)^{1/4} gap from the quadratic conditional lower bound based on (min,+)-convolution. Our improvement comes from a multi-level extension of Chan\u27s number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items

On Problems Equivalent to (min,+)-Convolution

In the recent years, significant progress has been made in explaining apparent hardness of improving over naive solutions for many fundamental polynomially solvable problems. This came in the form of conditional lower bounds -- reductions from a problem assumed to be hard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others. In the (min,+)-convolution problem, the goal is to compute a sequence c, where c[k] = min_i a[i]+b[k-i], given sequences a and b. This can easily be done in O(n^2) time, but no O(n^{2-eps}) algorithm is known for eps > 0. In this paper we undertake a systematic study of the (min,+)-convolution problem as a hardness assumption. As the first step, we establish equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min,+)-convolution has been used as a building block in algorithms for many problems, notably problems in stringology. It has also already appeared as an ad hoc hardness assumption. We investigate some of these connections and provide new reductions and other results

Approximation Schemes for 0-1 Knapsack

We revisit the standard 0-1 knapsack problem. The latest polynomial-time approximation scheme by Rhee (2015) with approximation factor 1+eps has running time near O(n+(1/eps)^{5/2}) (ignoring polylogarithmic factors), and is randomized. We present a simpler algorithm which achieves the same result and is deterministic. With more effort, our ideas can actually lead to an improved time bound near O(n + (1/eps)^{12/5}), and still further improvements for small n