The list update problem is a classical online problem, with an optimal competitive ratio that is still open, known to be somewhere between 1.5 and 1.6. An algorithm with competitive ratio 1.6, the smallest known to date, is COMB, a randomized combination of BIT and the TIMESTAMP algorithm TS. This and almost all other list update algorithms, like MTF, are projective in the sense that they can be defined by looking only at any pair of list items at a time. Projectivity (also known as “list factoring”) simplifies both the description of the algorithm and its analysis, and so far seems to be the only way to define a good online algorithm for lists of arbitrary length. In this article, we characterize all projective list update algorithms and show that their competitive ratio is never smaller than 1.6 in the partial cost model. Therefore, COMB is a best possible projective algorithm in this model.

We consider the Minimum Linear Arrangement problem and the (Uniform) Sparsest Cut problem. So far, these two notorious NP-hard graph problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme, unless NP-complete problems can be solved in randomized subexponential time. Furthermore, we show that the same techniques can be used for the Maximum Edge Biclique problem, for which we obtain a hardness factor similar to previous results but under a more standard assumption.

We consider the task of network exploration by a mobile agent (robot) with small memory. The agent has to traverse all nodes and edges of a network (represented as an undirected connected graph), and return to the starting node. Nodes of the network are unlabeled and edge ports are locally labeled at each node. The agent has no a priori knowledge of the topology of the network or of its size, and cannot mark nodes in any way. Under such weak assumptions, cycles in the network may prevent feasibility of exploration, hence we restrict attention to trees. We present an algorithm to accomplish tree exploration (with return) using O (log n )-bit memory for all n -node trees. This strengthens the result from Diks et al. [2004], where O (log 2 n )-bit memory was used for tree exploration, and matches the lower bound on memory size proved there. We also extend our O (log n )-bit memory traversal mechanism to a weaker model in which ports at each node are ordered in circular manner, however, the explicit values of port numbers are not available.

We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.

Summary form only given. We introduce the weighted version of the range assignment problem in which the cost a station s pays to transmit to another station depends on the distance between the stations and on the energy cost of station s. Most of the algorithm results for the unweighted range assignment problem can not be applied to the weighted version. We thus provide a set of algorithmic results for this version and discuss some interesting related open questions.