Ruben van der Zwaan

Reducing a target interval to a few exact queries

Many combinatorial problems involving weights can be formulated as a so-called ranged problem. That is, their input consists of a universe u, a (succinctly-represented) set family \mathcal{f} \subseteq 2^{u}f?2 u \mathcal{f} \subseteq 2^{u}, a weight function ?:u?{1,…,n}, and integers 0?=?l?=?u?=?8. Then the problem is to decide whether there is an x \in \mathcal{f}x?fx \in \mathcal{f} such that l?=? e?x ?(e)?=?u. Well-known examples of such problems include knapsack, subset sum, maximum matching, and traveling salesman. In this paper, we develop a generic method to transform a ranged problem into an exact problem (i.e. A ranged problem for which l?=?u). We show that our method has several intriguing applications in exact exponential algorithms and parameterized complexity, namely: , in exact exponential algorithms, we present new insight into whether subset sum and knapsack have efficient algorithms in both time and space. In particular, we show that the time and space complexity of subset sum and knapsack are equivalent up to a small polynomial factor in the input size. We also give an algorithm that solves sparse instances of knapsack efficiently in terms of space and time. In parameterized complexity, we present the first kernelization results on weighted variants of several well-known problems. In particular, we show that weighted variants of vertex cover and dominating set, traveling salesman, and knapsack all admit polynomial randomized turing kernels when parameterized by |u|. Curiously, our method relies on a technique more commonly found in approximation algorithms.

Minimum latency submodular cover

We study the submodular ranking problem in the presence of metric costs. The input to the minimum latency submodular cover (MLSC) problem consists of a metric (V,d) with source r∈V and m monotone submodular functions f1, f2, ..., fm: 2V→[0,1]. The goal is to find a path originating at r that minimizes the total cover time of all functions; the cover time of function fi is the smallest value t such that fi has value one for the vertices visited within distance t along the path. This generalizes many previously studied problems, such as submodular ranking [1] when the metric is uniform, and group Steiner tree [14] when m=1 and f1 is a coverage function. We give a polynomial time

A branch and price procedure for the container premarshalling problem

O(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |V|)

Minimum Latency Submodular Cover

-approximation algorithm for MLSC, where e>0 is the smallest non-zero marginal increase of any

Dynamic pricing problems with elastic demand

\{f_i\}_{i=1}^m

Complexity and approximability of the k-way vertex cut†

and δ>0 is any constant. This result is enabled by a simpler analysis of the submodular ranking algorithm from [1]. We also consider the stochastic submodular ranking problem where elements V have random instantiations, and obtain an adaptive algorithm with an O(log1/ e) approximation ratio, which is best possible. This result also generalizes several previously studied stochastic problems, eg. adaptive set cover [15] and shared filter evaluation [24,23].

Internet routing between autonomous systems

During the loading phase of a vessel, only the containers that are on top of their stack are directly accessible. If the container that needs to be loaded next is not the top container, extra moves have to be performed, resulting in an increased loading time. One way to resolve this issue is via a procedure called premarshalling. The goal of premarshalling is to reshuffle the containers into a desired lay-out prior to the arrival of the vessel, in the minimum number of moves possible. This paper presents an exact algorithm based on branch and bound, that is evaluated on a large set of instances. The complexity of the premarshalling problem is also considered, and this paper shows that the problem at hand is NP-hard, even in the natural case of stacks with fixed height.

Approximating Real-Time Scheduling on Identical Machines

We study the Minimum Latency Submodular Cover (MLSC) problem, which consists of a metric (V, d) with source r ∈ V and m monotone submodular functions f1, f2, …, fm: 2V → [0, 1]. The goal is to find a path originating at r that minimizes the total “cover time” of all functions. This generalizes well-studied problems, such as Submodular Ranking [Azar and Gamzu 2011] and the Group Steiner Tree [Garg et al. 2000]. We give a polynomial time O(log 1/ε ċ log2+δ|V|)-approximation algorithm for MLSC, where ε > 0 is the smallest non-zero marginal increase of any {fi}mi = 1 and δ > 0 is any constant. We also consider the Latency Covering Steiner Tree (LCST) problem, which is the special case of MLSC where the fis are multi-coverage functions. This is a common generalization of the Latency Group Steiner Tree [Gupta et al. 2010; Chakrabarty and Swamy 2011] and Generalized Min-sum Set Cover [Azar et al. 2009; Bansal et al. 2010] problems. We obtain an O(log 2|V|)-approximation algorithm for LCST. Finally, we study a natural stochastic extension of the Submodular Ranking problem and obtain an adaptive algorithm with an O(log 1/ε)-approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [Goemans and Vondrák 2006] and Shared Filter Evaluation [Munagala et al. 2007; Liu et al. 2008].

How to cut a graph into many pieces

We consider a dynamic pricing problem for a company that sells a single product to a group of price-sensitive customers over a finite time horizon. The objective is to set the prices over time so as to maximize revenue. Two price-sensitivity models are studied: multiplicative and additive demand change. We develop a polynomial-time algorithm for the multiplicative model. In contrast, we prove that the problem under additive demand change is NP-hard and admits an FPTAS.

Vertex Ranking with Capacity

In this article, we consider k-way vertex cut: the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k-way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed–parameter tractable on planar graphs with the size of the separator as the parameter.