Randomization can be as helpful as a glimpse of the future in online computation

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $\Omega(n)$ bits of advice to achieve a competitive ratio better than $H_k=\Omega(\log k)$, where $k$ is the cache size. Previously, it was only known that $\Omega(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $\Omega(n\log n)$ bits of advice. Previously, it was only known that $\Omega(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Ros\'en [ICALP'09]

Serving Online Requests with Mobile Servers

We study an online problem in which a set of mobile servers have to be moved in order to efficiently serve a set of requests that arrive in an online fashion. More formally, there is a set of $n$ nodes and a set of $k$ mobile servers that are placed at some of the nodes. Each node can potentially host several servers and the servers can be moved between the nodes. There are requests $1,2,\ldots$ that are adversarially issued at nodes one at a time. An issued request at time $t$ needs to be served at all times $t' \geq t$. The cost for serving the requests is a function of the number of servers and requests at the different nodes. The requirements on how to serve the requests are governed by two parameters $\alpha\geq 1$ and $\beta\geq 0$. An algorithm needs to guarantee at all times that the total service cost remains within a multiplicative factor of $\alpha$ and an additive term $\beta$ of the current optimal service cost. We consider online algorithms for two different minimization objectives. We first consider the natural problem of minimizing the total number of server movements. We show that in this case for every $k$, the competitive ratio of every deterministic online algorithm needs to be at least $\Omega(n)$. Given this negative result, we then extend the minimization objective to also include the current service cost. We give almost tight bounds on the competitive ratio of the online problem where one needs to minimize the sum of the total number of movements and the current service cost. In particular, we show that at the cost of an additional additive term which is roughly linear in $k$, it is possible to achieve a multiplicative competitive ratio of $1+\varepsilon$ for every constant $\varepsilon>0$.Comment: 25 page

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure

Iterative Beam Search for Simple Assembly Line Balancing with a Fixed Number of Work Stations

The simple assembly line balancing problem (SALBP) concerns the assignment of tasks with pre-defined processing times to work stations that are arranged in a line. Hereby, precedence constraints between the tasks must be respected. The optimization goal of the SALBP-2 version of the problem concerns the minimization of the so-called cycle time, that is, the time in which the tasks of each work station must be completed. In this work we propose to tackle this problem with an iterative search method based on beam search. The proposed algorithm is able to obtain optimal, respectively best-known, solutions in 283 out of 302 test cases. Moreover, for 9 further test cases the algorithm is able to produce new best-known solutions. These numbers indicate that the proposed iterative beam search algorithm is currently a state-of-the-art method for the SALBP-2

A Multi-Objective Particle Swarm Optimization for Mixed-Model Assembly Line Balancing with Different Skilled Workers

This paper presents a multi-objective Particle Swarm Optimization (PSO) algorithm for worker assignment and mixed-model assembly line balancing problem when task times depend on the worker’s skill level. The objectives of this model are minimization of the number of stations (equivalent to the maximization of the weighted line efficiency), minimization of the weighted smoothness index and minimization of the total human cost for a given cycle time. In addition, the performance of proposed algorithm is evaluated against a set of test problems with different sizes. Also, its efficiency is compared with a Simulated Annealing algorithm (SA) in terms of the quality of objective functions. Results show the proposed algorithm performs well, and it can be used as an efficient algorithm. This paper presents a multi-objective Particle Swarm Optimization (PSO) algorithm for worker assignment and mixed-model assembly line balancing problem when task times depend on the worker’s skill level. The objectives of this model are minimization of the number of stations (equivalent to the maximization of the weighted line efficiency), minimization of the weighted smoothness index and minimization of the total human cost for a given cycle time. In addition, the performance of proposed algorithm is evaluated against a set of test problems with different sizes. Also, its efficiency is compared with a Simulated Annealing algorithm (SA) in terms of the quality of objective functions. Results show the proposed algorithm performs well, and it can be used as an efficient algorith