An equivalent version of the Caccetta-Häggkvist conjecture in an online load balancing problem

Abstract. We study the competitive ratio of certain online algorithms for a well-studied class of load balancing problems. These algorithms are obtained and analyzed according to a method by Crescenzi et al (2004). We show that an exact analysis of their competitive ratio on certain “uniform ” instances would resolve a fundamental conjecture by Caccetta and Häggkvist (1978). The conjecture is that any digraph on n nodes and minimum outdegree d must contain a directed cycle involving at most ⌈n/d ⌉ nodes. Our results are the first relating this conjecture to the competitive analysis of certain algorithms, thus suggesting a new approach to the conjecture itself. We also prove that, on “uniform ” instances, the analysis by Crescenzi et al (2004) gives only trivial upper bounds, unless we find a counterexample to the conjecture. This is in contrast with other (notable) examples where the same analysis yields optimal (non-trivial) bounds. Key words: Caccetta-Häggkvist conjecture, online load balancing, competitive analysis