New Results on Online Resource Minimization

We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known strong lower bound of n for the general problem. For the preemptive setting, we show that the natural algorithm LLF achieves a constant ratio for agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)). We also give an O(log n)-competitive algorithm for the general preemptive problem, which improves upon the known O(p_max/p_min)-competitive algorithm. Our algorithm maintains a dynamic partition of the job set into loose and tight jobs and schedules each (temporal) subset individually on separate sets of machines. The key is a characterization of how the decrease in the relative laxity of jobs influences the optimum number of machines. To achieve this we derive a compact expression of the optimum value, which might be of independent interest. We complement the general algorithmic result by showing lower bounds that rule out that other known algorithms may yield a similar performance guarantee

Improved Bounds for Open Online Dial-a-Ride on the Line

We consider the open, non-preemptive online Dial-a-Ride problem on the real line, where transportation requests appear over time and need to be served by a single server. We give a lower bound of 2.0585 on the competitive ratio, which is the first bound that strictly separates online Dial-a-Ride on the line from online TSP on the line in terms of competitive analysis, and is the best currently known lower bound even for general metric spaces. On the other hand, we present an algorithm that improves the best known upper bound from 2.9377 to 2.6662. The analysis of our algorithm is tight

Online-Bearbeitung kritischer Aufgaben : Deadline-Scheduling und Convex-Body-Chasing

In this thesis, we study two fundamental online optimization problems in which tasks arrive over time and require to be processed either immediately or until a certain deadline. The goal is to finish all tasks and minimize some cost function. In the real world, such an optimization problem has to be solved for instance by an on-board computer of a car that needs to handle safety-relevant and thus time-critical tasks.In der vorliegenden Arbeit betrachten wir zwei fundamentale Probleme aus der Online-Optimierung, in denen Jobs, die nach und nach ankommen, entweder sofort oder bis zu einer bestimmten Deadline bearbeitet werden müssen. Das Ziel ist es, alle Jobs rechtzeitig zu erledigen und dabei eine bestimmte Kostenfunktion zu minimieren. In der Praxis müssen solche Probleme zum Beispiel von dem Bordcomputer eines Autos gelöst werden, der sicherheitsrelevante und daher zeitkritische Jobs bearbeiten muss.DFG, GRK 1408, Methods for Discrete StructuresDFG, ME 3825-1, Scheduling under Uncertainty: On Performance-Adaptivity Tradeoff

SUPERSET: A (Super)Natural Variant of the Card Game SET

We consider Superset, a lesser-known yet interesting variant of the famous card game Set. Here, players look for Supersets instead of Sets, that is, the symmetric difference of two Sets that intersect in exactly one card. In this paper, we pose questions that have been previously posed for Set and provide answers to them; we also show relations between Set and Superset. For the regular Set deck, which can be identified with F^3_4, we give a proof for the fact that the maximum number of cards that can be on the table without having a Superset is 9. This solves an open question posed by McMahon et al. in 2016. For the deck corresponding to F^3_d, we show that this number is Omega(1.442^d) and O(1.733^d). We also compute probabilities of the presence of a superset in a collection of cards drawn uniformly at random. Finally, we consider the computational complexity of deciding whether a multi-value version of Set or Superset is contained in a given set of cards, and show an FPT-reduction from the problem for Set to that for Superset, implying W[1]-hardness of the problem for Superset

Routing Games with Progressive Filling

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived

Online Multistage Subset Maximization Problems

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one

Simple Algorithms for Stochastic Score Classification with Small Approximation Ratios

We revisit the Stochastic Score Classification (SSC) problem introduced by Gkenosis et al. (ESA 2018): We are given $n$ tests. Each test $j$ can be conducted at cost $c_j$, and it succeeds independently with probability $p_j$. Further, a partition of the (integer) interval $\{0,\dots,n\}$ into $B$ smaller intervals is known. The goal is to conduct tests so as to determine that interval from the partition in which the number of successful tests lies while minimizing the expected cost. Ghuge et al. (IPCO 2022) recently showed that a polynomial-time constant-factor approximation algorithm exists. We show that interweaving the two strategies that order tests increasingly by their $c_j/p_j$ and $c_j/(1-p_j)$ ratios, respectively, -- as already proposed by Gkensosis et al. for a special case -- yields a small approximation ratio. We also show that the approximation ratio can be slightly decreased from $6$ to $3+2\sqrt{2}\approx 5.828$ by adding in a third strategy that simply orders tests increasingly by their costs. The similar analyses for both algorithms are nontrivial but arguably clean. Finally, we complement the implied upper bound of $3+2\sqrt{2}$ on the adaptivity gap with a lower bound of $3/2$. Since the lower-bound instance is a so-called unit-cost $k$-of-$n$ instance, we settle the adaptivity gap in this case