Priority Algorithms with Advice for Disjoint Path Allocation Problems

TL;DR

This paper studies priority algorithms with advice for disjoint path allocation problems, providing bounds on advice complexity and approximation ratios on paths and trees, extending online algorithm analysis to a more powerful model.

Contribution

It introduces and analyzes priority algorithms with advice for DPA, establishing bounds on advice complexity and approximation ratios on paths and trees, extending prior online algorithm results.

Findings

Matching bounds on advice for optimality in LWDPA.

Generalization of DPA optimality results from paths to degree-3 trees.

Advice bounds for achieving optimal solutions on higher-degree trees.

Abstract

We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. While online algorithms for DPA have been thoroughly studied in the past, we extend the analysis of this optimization problem by considering the more powerful class of priority algorithms. Like an online algorithm, a priority algorithm receives its input only sequentially and must output irrevocable decisions for individual input items before having seen the input in its entirety. However, in contrast to the online setting, a priority algorithm may choose an order on the set of all possible input items and the actual input is then presented according to this order. A priority algorithm is a natural model for the intuitively well-understood concept of a greedy algorithm. Apart from analyzing the classical priority setting, we also consider priority algorithms with advice. Originally conceived to study online algorithms from an information-theoretic point of view, the concept of advice has recently been extended to the priority framework. In this paper, we analyze the classical variant of the DPA problem on the graph class of paths, the related problem of Length-Weighted DPA, and finally, DPA on the graph class of trees. We show asymptotically matching upper and lower bounds on the advice necessary for optimality in LWDPA and generalize the known optimality result for DPA on paths to trees with maximal degree at most 3. On trees with maximal degree greater than 3, we prove matching upper and lower bounds on the approximation ratio in the advice-free priority setting. Finally, we present upper and lower bounds on the advice necessary to achieve optimality on such trees.