Online Bin Covering: Expectations vs. Guarantees

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all "reasonable" algorithms have a competitive ratio of 1/2. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures. Two classic algorithms for online bin packing that have natural dual versions are Harmonic and Next-Fit. Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonic is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice.Comment: IMADA-preprint-c

Adding Isolated Vertices Makes some Online Algorithms Optimal

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm. It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This was fixe

The Frequent Items Problem in Online Streaming under Various Performance Measures

In this paper, we strengthen the competitive analysis results obtained for a fundamental online streaming problem, the Frequent Items Problem. Additionally, we contribute with a more detailed analysis of this problem, using alternative performance measures, supplementing the insight gained from competitive analysis. The results also contribute to the general study of performance measures for online algorithms. It has long been known that competitive analysis suffers from drawbacks in certain situations, and many alternative measures have been proposed. However, more systematic comparative studies of performance measures have been initiated recently, and we continue this work, using competitive analysis, relative interval analysis, and relative worst order analysis on the Frequent Items Problem.Comment: IMADA-preprint-c

Keyword-aware Optimal Route Search

Identifying a preferable route is an important problem that finds applications in map services. When a user plans a trip within a city, the user may want to find "a most popular route such that it passes by shopping mall, restaurant, and pub, and the travel time to and from his hotel is within 4 hours." However, none of the algorithms in the existing work on route planning can be used to answer such queries. Motivated by this, we define the problem of keyword-aware optimal route query, denoted by KOR, which is to find an optimal route such that it covers a set of user-specified keywords, a specified budget constraint is satisfied, and an objective score of the route is optimal. The problem of answering KOR queries is NP-hard. We devise an approximation algorithm OSScaling with provable approximation bounds. Based on this algorithm, another more efficient approximation algorithm BucketBound is proposed. We also design a greedy approximation algorithm. Results of empirical studies show that all the proposed algorithms are capable of answering KOR queries efficiently, while the BucketBound and Greedy algorithms run faster. The empirical studies also offer insight into the accuracy of the proposed algorithms.Comment: VLDB201

Fully Dynamic Single-Source Reachability in Practice: An Experimental Study

Given a directed graph and a source vertex, the fully dynamic single-source reachability problem is to maintain the set of vertices that are reachable from the given vertex, subject to edge deletions and insertions. It is one of the most fundamental problems on graphs and appears directly or indirectly in many and varied applications. While there has been theoretical work on this problem, showing both linear conditional lower bounds for the fully dynamic problem and insertions-only and deletions-only upper bounds beating these conditional lower bounds, there has been no experimental study that compares the performance of fully dynamic reachability algorithms in practice. Previous experimental studies in this area concentrated only on the more general all-pairs reachability or transitive closure problem and did not use real-world dynamic graphs. In this paper, we bridge this gap by empirically studying an extensive set of algorithms for the single-source reachability problem in the fully dynamic setting. In particular, we design several fully dynamic variants of well-known approaches to obtain and maintain reachability information with respect to a distinguished source. Moreover, we extend the existing insertions-only or deletions-only upper bounds into fully dynamic algorithms. Even though the worst-case time per operation of all the fully dynamic algorithms we evaluate is at least linear in the number of edges in the graph (as is to be expected given the conditional lower bounds) we show in our extensive experimental evaluation that their performance differs greatly, both on generated as well as on real-world instances