Comparator Circuits over Finite Bounded Posets
The comparator circuit model was originally introduced by Mayr et al. (1992) (and further studied by Cook et al. (2014)) to capture problems that are not known to be P-complete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems many-one logspace reducible to the Comparator Circuit Value Problem and we know that NLOG⊆CC⊆P. Cook et al. (2014) showed that CC is also the class of languages decided by polynomial size comparator circuit families. We study generalizations of the comparator circuit model that work over fixed finite bounded posets. We observe that there are universal comparator circuits even over arbitrary fixed finite bounded posets. Building on this, we show the following: • Comparator circuits of polynomial size over fixed finite distributive lattices characterize the class CC. When the circuit is restricted to be skew, they characterize LOG. Noting that (uniform) polynomial sized Boolean circuits (resp. skew) characterize P (resp. NLOG), this indicates a comparison between P vs CC and NLOG vs LOG problems.• Complementing this, we show that comparator circuits of polynomial size over arbitrary fixed finite lattices characterize the class P even when the comparator circuit is skew.• In addition, we show a characterization of the class NP by a family of polynomial sized comparator circuits over fixed finite bounded posets. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira's theorem (Spira, 1971) can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture the class NC1.These results generalize results in Cook et al. (2014) regarding the power of comparator circuits. Our techniques involve design of comparator circuits and finite posets. We then use known results from lattice theory to show that the posets that we obtain can be embedded into appropriate lattices. Our results give new methods to establish CC upper bounds for problems and also indicate potential new approaches towards the problems P vs CC and NLOG vs LOG using lattice theoretic methods.
Space complexity of reachability testing in labelled graphs
Fix an algebraic structure (A,⁎). Given a graph G=(V,E) and the labelling function ϕ (ϕ:E→A) for the edges, two nodes s, t∈V, and a subset F⊆A, the A-REACH problem asks if there is a path p (need not be simple) from s to t whose yield (result of the operation in the ordered set of the labels of the edges constituting the path) is in F. On the complexity frontier of this problem, we show the following results. • When A is a group whose size is polynomially bounded in the size of the graph (hence equivalently presented as a multiplication table at the input), and the graph is undirected, the A-REACH problem is in L. Building on this, using a decomposition in [4], we show that, when A is a fixed quasi-group, and the graph is undirected, the A-REACH problem is in L. In a contrast, we show NL-hardness of the problem over bidirected graphs, when A is a matrix group over Q, or an aperiodic monoid. • As our main theorem, we prove a dichotomy for graphs labelled with fixed aperiodic monoids by showing that for every fixed aperiodic monoid A, A-REACH problem is either in L or is NL-complete. • We show that there exists a monoid M, such that the reachability problem in general DAGs can be reduced to A-REACH problem for planar non-bipartite DAGs labelled with M. In contrast, we show that if the planar DAGs that we obtain above are bipartite, the problem can be further reduced to reachability testing in planar DAGs and hence is in UL.
On the complexity of l-reachability
We initiate a complexity theoretic study of the language based graph reachability problem (L-REACH) : Fix a language L. Given a graph whose edges are labelled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following: • Restricting the language using formal language theory we show that the complexity of L-REACH increases with the power of the formal language class. We show that there is a regular language for which the L-REACH is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-REACH does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DAGREACH is LogCFL-complete. • We use L-REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC0 for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH-1(P)1 question. This leads to the intriguing possibility that by proving DAGREACH-1(P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.
Space complexity of reachability testing in labelled graphs
Fix an algebraic structure (A, ∗). Given a graph G = (V, E) and the labelling function φ (φ: E → A) for the edges, two nodes s, t ∈ V, and a subset F ⊆ A, the A-Reach problem asks if there is a path p (need not be simple) from s to t whose yield (result of the operation in the ordered set of the labels of the edges constituting the path) is in F. On the complexity frontier of this problem, we show the following results. – When A is a group whose size is polynomially bounded in the size of the graph (hence equivalently presented as a multiplication table at the input), and the graph is undirected, the A-Reach problem is in L. Building on this, using a decomposition in [4], we show that, when A is a fixed quasi-group, and the graph is undirected, the A-Reach problem is in L. In contrast, we show NL-hardness of the problem over bidirected graphs, when A is a matrix group over ℚ. When A is a fixed aperiodic monoid, we show that the problem is NL-complete. – As our main theorem, we prove a dichotomy for graphs labelled with fixed aperiodic monoids by showing that for every fixed aperiodic monoid A, A-Reach problem is either in L or is NL-complete. – We show that there exists a monoid M, such that the reachability problem in general DAGs can be reduced to A-Reach problem for planar non-bipartite DAGs labelled with M. In contrast, we show that if the planar DAGs that we obtain above are bipartite, the problem can be further reduced to reachability testing in planar DAGs and hence is in UL.
Comparator circuits over finite bounded posets
Comparator circuit model was originally introduced in [4] (and further studied in [2]) to capture problems which are not known to be P-complete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems many-one logspace reducible to the Comparator Circuit Value Problem and we know that NLOG ⊆ CC ⊆ P. Cook et al [2] showed that CC is also the class of languages decided by polynomial size comparator circuits. We study generalizations of the comparator circuit model that work over fixed finite bounded posets. We observe that there are universal comparator circuits even over arbitrary fixed finite bounded posets. Building on this, we show that general (resp. skew) comparator circuits of polynomial size over fixed finite distributive lattices characterizes CC (resp. LOG). Complementing this, we show that general comparator circuits of polynomial size over arbitrary fixed finite lattices exactly characterizes P and that when the comparator circuit is skew they characterize NLOG. In addition, we show a characterization of the class NP by a family of polynomial sized comparator circuits over fixed finite bounded posets. These results generalize the results in [2] regarding the power of comparator circuits. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira’s theorem[5] can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture exactly NC1. Our techniques involve design of comparator circuits and finite posets. We then use known results from lattice theory to show that the posets that we obtain can be embedded into appropriate lattices. Our results gives new methods to establish CC upper bound for problems also indicate potential new approaches towards the problems P vs CC and NLOG vs LOG using lattice theoretic methods.