We consider the logarithmic space counting classes #L, opt-L, and span-L, which are defined analogously to their polynomial time counterparts. We obtain complete functions for these three classes in terms of graphs and finite automata. We show that #L and opt-L are both contained in NC', but that, surprisingly, span-L seems to be a much harder counting class than #L and opt-L. We deinonst,rate that span-Lfunctions can be computed in polynomial time if and onlyif P = NP = PH = P(#P), i.e. iff the class P(#P) and all the classes of the polyiiomial time hierarchy are contained in P. This result follows from the fact that span-L and #P are very similar: span-L C #P, and any function in #P can be represented as a subtraction of two functions in span-L. Nevertheless, #P C span-L would imply NL = P = NP. We furthermore investigate various restrictions of the classes opt-L and span-L, and show, e.g., that if opt-L coincides with one of its restricted versions. then L = NL follows. Iiit ro duct ioiiDuring the past several years the topic of "counting" appeared in many different settings in complexity theory. In the case of logarithmic space, for example, powerful counting techniques revealed the zntrznszc computational power of various machine models to achieve counting, above all NI,, and LOGCFL (see [la] [26] [3]). On the other hand, counting was used to zncrease the computational power of polyiiomial time machines by defining counting variants of NP, like the function classes opt-P (see [ls]), #P (see I291 [30]) and spaii-P ([17], see also [25] [IS]). These classes have been shown to contain interesting functional counting variants of NP-complete problems. Recently, Toda pointed out tlie enormous power that such classes can have: tlie whole polynomial time hierarchy is contained in P(#P) [27].This raises the question, whether the phenomenon of increasing the computational power by counting is a general one and shows up by logarithmic space classes as well, and if so to what extent, or whether e.g. due to the intrinsic counting power of NL here no more power is gained. Stated alternatively, this question concerns the complexity of functional counting variants of NLcomplete problems. For example, it is well-known that the non-emptiness problem L ( M ) # O? for a given finite automaton M is NL-complete with respect to logarithmic space reductions. This holds irrespectively of whether the automaton Af is deterministic (DFA) [14] or nondeterministic (NFA). Consider the following functional counting variants of this problem: computing the number of words accepted by a DFA or NFA which are smaller than a given word z, the "ranking functions" for DFA or NFA, and the problem of computing the maximal word smaller than or equal to a given word z accepted by a NFA, the "maximal word function" for NFA. How difficult are these functions to compute? Can they, e.g., be computed by a deterministic log space Turing machine with oracle in NL?We show that these functions are, respectively, many-one complete for the three logarithmic spac...

The paper's main contributions are a compendium of problems that are complete for symmetric logarithmic space (SL), a collection of material relating to SL, a list of open problems, and an extension to the number of problems known to be SL-complete. Complete problems are one method of studying SL, a class for which programming is nonintuitive. Our exposition helps make the class SL less mysterious and more accessible to other researchers.

We study universal stability of directed and undirected graphs in the adversarial queuing model for static packet routing. In this setting, packets are injected in some edge and have to traverse a predefined path before leaving the system. Restrictions on the allowed packet trajectory provide a way to analyze stability under different packet trajectories. We consider five packet trajectories, two for directed graphs and three for undirected graphs, and provide polynomial time algorithms for testing universal stability when considering each of them. In each case we obtain a different characterization of the universal stability property in terms of a set of forbidden subgraphs. Thus we show that variations of the allowed packet trajectory lead to nonequivalent characterizations. Using those characterizations we are also able to provide polynomial time algorithms for testing stability under the ntg-lis (Nearest To Go-Longest In System) protocol.

In this paper we study the universal stability of undirected graphs in the adversarial queueing model for packet routing. In this setting, packets must be injected in some edge and have to traverse a path before leaving the system. Restrictions on the allowed types of path that packets must traverse provide different packet models. We consider three natural models, and provide polynomial time algorithms for testing universal stability on them. In the three cases, we obtain a different characterization, in terms of forbidden subgraphs, thus showing that slight variations lead to non-equivalent models.We extend those results to show that universal stability of digraphs, in the case in which packets follow directed paths without repeating vertices, can be decided in polynomial time.All the instability results are obtained for the ntg-lis protocol. Therefore, the property of universal stability is equivalent to ntg-lis-stability, in all the cases.

In this article, we propose several variations of the adversarial queueing model and address stability issues of networks and protocols in those proposed models. The first such variation is the priority model, which is directed at static network topologies and takes into account the case in which packets can have different priorities. Those priorities are assigned by an adversary at injection time. A second variation, the variable priority model, is an extension of the priority model in which the adversary may dynamically change the priority of packets at each time step. Two more variations, namely the failure model and the reliable model, are proposed to cope with dynamic networks. In the failure and reliable models the adversary controls, under different constraints, the failures that the links of the topology might suffer. Concerning stability of networks in the proposed adversarial models, we show that the set of universally stable networks in the adversarial model remains the same in the priority, variable priority, failure, and reliable models. From the point of view of protocols (or queueing policies), we show that several protocols that are universally stable in the adversarial queueing model remain so in the priority, failure, and reliable models. However, we show that the longest-in-system (LIS) protocol, which is universally stable in the adversarial queueing model, is not universally stable in any of the other models we propose. Moreover, we show that no queueing policy is universally stable in the variable priority model. Finally, we analyze the problem of deciding stability of a given network under a fixed protocol. We provide a characterization of the networks that are stable under firstin-first-out (FIFO) and LIS in the failure model (and therefore in the reliable and priority models). This characterization allows us to show that the stability problem under FIFO and LIS in the failure model can be solved in polynomial time.