Jens Svalgaard Kohrt

A constant approximation algorithm for sorting buffers

We consider an algorithmic problem that arises in manufacturing applications. The input is a sequence of objects of various types. The scheduler is fed the objects in the sequence one by one, and is equipped with a finite buffer. The goal of the scheduler/sorter is to maximally reduce the number of type transitions. We give the first polynomial-time constant approximation algorithm for this problem. We prove several lemmas about the combinatorial structure of optimal solutions that may be useful in future research, and we show that the unified algorithm based on the local ratio lemma performs well for a slightly larger class of problems than was apparently previously known.

Comparing online algorithms for bin packing problems

The relative worst-order ratio is a measure of the quality of online algorithms. In contrast to the competitive ratio, this measure compares two online algorithms directly instead of using an intermediate comparison with an optimal offline algorithm.In this paper, we apply the relative worst-order ratio to online algorithms for several common variants of the bin packing problem. We mainly consider pairs of algorithms that are not distinguished by the competitive ratio and show that the relative worst-order ratio prefers the intuitively better algorithm of each pair.

Separating online scheduling algorithms with the relative worst order ratio

The relative worst order ratio is a measure for the quality of online algorithms. Unlike the competitive ratio, it compares algorithms directly without involving an optimal offline algorithm. The measure has been successfully applied to problems like paging and bin packing. In this paper, we apply it to machine scheduling. We show that for preemptive scheduling, the measure separates multiple pairs of algorithms which have the same competitive ratios; with the relative worst order ratio, the algorithm which is “intuitively better” is also provably better. Moreover, we show one such example for non-preemptive scheduling.

List Factoring and Relative Worst Order Analysis

Relative worst order analysis is a supplement or alternative to competitive analysis which has been shown to give results more in accordance with observed behavior of online algorithms for a range of different online problems. The contribution of this paper is twofold. As the first contribution, it adds the static list accessing problem to the collection of online problems where relative worst order analysis gives better results. List accessing is a classic data structuring problem of maintaining optimal ordering in a linked list. It is also one of the classic problems in online algorithms, in that it is used as a model problem, along with paging and a few other problems, when trying out new techniques and quality measures. As the second contribution, this paper adds the non-trivial supplementary proof technique of list factoring to the theoretical toolbox for relative worst order analysis. List factoring is perhaps the most successful technique for analyzing list accessing algorithms, reducing the complexity of the analysis of algorithms on full-length lists to lists of length two.

A theoretical comparison of LRU and LRU-K

The paging algorithm Least Recently Used Second Last Request (LRU-2) was proposed for use in database disk buffering and shown experimentally to perform better than Least Recently Used (LRU). We compare LRU-2 and LRU theoretically, using both the standard competitive analysis and the newer relative worst order analysis. The competitive ratio for LRU-2 is shown to be 2k for cache size k, which is worse than LRU’s competitive ratio of k. However, using relative worst order analysis, we show that LRU-2 and LRU are comparable in LRU-2’s favor, giving a theoretical justification for the experimental results. Many of our results for LRU-2 also apply to its generalization, Least Recently Used Kth Last Request.

The maximum resource bin packing problem

Usually, for bin packing problems, we try to minimize the number of bins used or in the case of the dual bin packing problem, maximize the number or total size of accepted items. This paper presents results for the opposite problems, where we would like to maximize the number of bins used or minimize the number or total size of accepted items. We consider off-line and on-line variants of the problems. For the off-line variant, we require that there be an ordering of the bins, so that no item in a later bin fits in an earlier bin. We find the approximation ratios of two natural approximation algorithms, First-Fit-Increasing and First-Fit-Decreasing for the maximum resource variant of classical bin packing. For the on-line variant, we define maximum resource variants of classical and dual bin packing. For dual bin packing, no on-line algorithm is competitive. For classical bin packing, we find the competitive ratio of various natural algorithms. We study the general versions of the problems as well as the parameterized versions where there is an upper bound of

List factoring and relative worst order analysis

\frac{1}{k}

Comparing First-Fit and Next-Fit for online edge coloring

on the item sizes, for some integer k.

Comparing First-Fit and Next-Fit for Online Edge Coloring

Relative worst order analysis is a supplement or alternative to competitive analysis which has been shown to give results more in accordance with observed behavior of online algorithms for a range of different online problems. The contribution of this paper is twofold. First, it adds the static list accessing problem to the collection of online problems where relative worst order analysis gives better results. Second, and maybe more interesting, it adds the non-trivial supplementary proof technique of list factoring to the theoretical toolbox for relative worst order analysis.

ON-LINE SEAT RESERVATIONS VIA OFF-LINE SEATING ARRANGEMENTS

We study the performance of the algorithms First -Fit and Next -Fit for two online edge coloring problems. In the min-coloring problem, all edges must be colored using as few colors as possible. In the max-coloring problem, a fixed number of colors is given, and as many edges as possible should be colored. Previous analysis using the competitive ratio has not separated the performance of First -Fit and Next -Fit, but intuition suggests that First -Fit should be better than Next -Fit. We compare First -Fit and Next -Fit using the relative worst-order ratio, and show that First -Fit is better than Next -Fit for the min-coloring problem. For the max-coloring problem, we show that First -Fit and Next -Fit are not strictly comparable, i.e., there are graphs for which First -Fit is significantly better than Next -Fit and graphs where Next -Fit is slightly better than First -Fit.