Graph coloring is a notoriously challenging problem, especially when considering the online setting where each arriving vertex must be colored immediately and irreversibly. Even on trees, which are trivially two-colorable, achieving anything better than a logarithmic competitive ratio becomes impossible if the order of arrival is adversarially determined. We investigate tree coloring in a slightly relaxed model where vertices arrive online but in random order, focusing specifically on algorithms with predictions of varying reliability. Furthermore, we extend our analysis to all two-colorable graphs and provide matching lower bounds for both cases.

Abstract We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we identify a broad class of online problems for which the existence of a randomized online algorithm with constant expected competitive ratio r implies the existence of a randomized online algorithm that has a competitive ratio of $$(1+\varepsilon )r$$ ( 1 + ε ) r with high probability, measured with respect to the optimal profit or cost, respectively. The class of problems includes some of the well-studied online problems such as paging, k-server, and metrical task systems on finite metric spaces.

The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization.

In STOC'16, Fraigniaud et al. consider the problem of finding a treasure hidden in one of many boxes that are ordered by importance. That is, if a treasure is in a more important box, then one would like to find it faster. Assuming there are many searchers, the authors suggest that using an algorithm that requires no coordination between searchers can be highly beneficial. Indeed, besides saving the need for a communication and coordination mechanism, such algorithms enjoy inherent robustness. The authors proceed to solve this linear search problem in the case of countably many boxes and an adversary placed treasure, and prove that the best speed-up possible by $k$ non-coordinating searchers is precisely $\frac{k}{4}(1+1/k)^2$. In particular, this means that asymptotically, the speed-up is four times worse compared to the case of full coordination. We suggest an important variant of the problem, where the treasure is placed uniformly at random in one of a finite, large, number of boxes. We devise non-coordinating algorithms that achieve a speed-up of $6/5$ for two searchers, a speed-up of $3/2$ for three searchers, and in general, a speed-up of $k(k+1)/(3k-1)$ for any $k \geq 1$ searchers. Thus, as $k$ grows to infinity, the speed-up approaches three times worse compared to the case of full coordination. Moreover, these bounds are tight in a strong sense as no non-coordinating search algorithm for $k$ searchers can achieve better speed-ups. We also devise non-coordinating algorithms that use only logarithmic memory in the size of the search domain, and yet, asymptotically, achieve the optimal speed-up. Finally, we note that all our algorithms are extremely simple and hence applicable.

We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem.

Abstract Mobile agents operating in networked environments face threats from other agents as well as from the hosts (i.e., network sites) they visit. A black hole is a harmful host that destroys incoming agents without leaving any trace. To determine the location of such a harmful host is a dangerous but crucial task, called black hole search. The most important parameter for a solution strategy is the number of agents it requires (the size); the other parameter of interest is the total number of moves performed by the agents (the cost). It is known that at least two agents are needed; furthermore, with full topological knowledge, Ω(n log n) moves are required in arbitrary networks. The natural question is whether, in specific networks, it is possible to obtain (topology‐dependent but) more cost efficient solutions. It is known that this is not the case for rings. In this article, we show that this negative result does not generalizes. In fact, we present a general strategy that allows two agents to locate the black hole with O(n) moves in common interconnection networks: hypercubes, cube‐connected cycles, star graphs, wrapped butterflies, chordal rings, as well as in multidimensional meshes and tori of restricted diameter. These results hold even if the networks are anonymous. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(2), 61–71 2006

One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplace matrix. Determining this number is of theoretical interest as well as of practical impact. Sparse graphs with small spectra exhibit excellent structural properties and can act as interconnection topologies. In this paper, for any n we present graphs, for which the product of their vertex degree and the number of different eigenvalues is small. It is known that load balancing can be performed on such graphs in a small number of steps.