A compendium of problems complete for symmetric logarithmic space

Abstract

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.

Undirected ST-connectivity in log-space

“…Since λ(H) ≤ 1/2, we have by Corollary 9 that λ(Gi−1 z H) ≤ 1 − 3/8(1 − λ) < 1 − 1/3(1 − λ). By the definition of Gi and by Proposition 7 we have that λ(Gi) < [1 − 1/3(1 − λ)] 8 . We now consider two cases.…”

“…[47], and putting some very interesting problems into SL. To give just one example, the planarity of bounded-degree undirected graphs was placed in SL as a corollary (we refer again to [8] for a list of SL-complete problems).…”

“…This fundamental combinatorial problem has received a lot of attention in the last few decades and was studied in a large variety of computational models. It is a basic building block for more complex graph algorithms and is complete for the class SL of problems solvable by symmetric, non-deterministic, log-space computations [24] (see [8] for a recent study of SL and quite a few of its complete problems).…”

Cyclic Extensions of Order Varieties

“…Proof -By Lemma 4.9, EntropicExt reduces to the problem of telling, given the undirected graph ω with vertex set ω, whether no vertex (a, b) ∈ ω is connected to (b, a). This is an undirected non-reachability problem, hence in coSL, since undirected reachability is SL-complete [3]. Since SL=L [17], we have coSL=L, and EntropicExt is therefore in L. By Lemma 3.4, so is OV-Cyc.…”

“…We then prove that the linear entropic extensibility of ω is equivalent to a simple condition on the forcing classes of ω, and also equivalent to the condition that the cocomparability graph of ω be a comparability graph, or equivalently that the dimension of ω be at most 2. The latter problem amouts to testing whether an undirected graph is a comparability graph, a problem known to be SL-complete [3], SL being the class of langages decidable by a symmetric Turing machine [11] within logarithmic space. By using the recent result that SL = L [17], we may conclude that the problem of linear entropic extensibility is in L. This problem is then trivially L-complete (or SL-complete) for log-space reductions, and it would be interesting to decide whether it is complete for even weaker reductions, e.g.…”