Lower bounds on the complexity ofMSO1model-checking

Abstract

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic secondorder logic with edge-set quantifications (i.e., MSO 2 model-checking) is decidable in linear time on any class of graphs of bounded tree-width. Recently, Kreutzer and Tazari [15] proved a corresponding complexity lower-bound-that MSO 2 model-checking is not even in XP wrt. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is polylogarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).In this paper we present a closely related result. We show that even MSO 1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari; (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO 1 -definable properties on vertex labeled graphs instead of MSO 2 -definable properties on unlabeled graphs.Our result has an interesting consequence in the realm of digraph width measures: Strengthening the recent result [10], we show that no subdigraphmonotone measure can be "algorithmically useful", unless it is within a polylogarithmic factor of undirected tree-width.

Tractable Lineages on Treelike Instances

“…Thus, tractability on bounded-treewidth instances is really the best we can get, on arity-2 signatures. Surprisingly, we show that q h can be taken to be a (non-monotone) FO query; this is in stark contrast with non-probabilistic query evaluation [38,26] where FO queries are fixed-parameter tractable under much milder conditions than bounded treewidth [37]. This provides the lower bound of a dichotomy, the upper bound being our result in [2].…”

“…In Section 5, we explain how this dichotomy result can be adapted to non-probabilistic MSO query evaluation and match counting on subgraph-closed graph families. While the necessity of bounded-treewidth for non-probabilistic query evaluation was studied before [38,26], our use of a recent polynomial bound on grid minors [10] allows us to obtain stronger results in this context, which we review. Our work thus answers the conjecture of [30] (Conjecture 8.3) for MSO, which [38] answered for MSO 2 , under similar complexity-theoretic assumptions.…”

“…Second, another line of work [42,38,26] has shown necessity of bounded treewidth when a class of graphs is closed under some operations: extracting topological minors in [42], extracting subgraphs in [38], and extracting subgraphs and vertex relabeling in [26]. This requires that there are sufficiently many instances of high treewidth, through notions of strong unboundedness [38] and dense unboundedness [26].…”

Parameters Tied to Treewidth

“…As a result of this weakening, it is easier to tie glm(G) to tw(G). This notion has also been applied to prove computational intractability results in monadic second-order logic; see Kreutzer [26], Ganian et al [18], and Kreutzer and Tazari [27,28].…”

Practical algorithms for MSO model-checking on tree-decomposable graphs

“…Roughly speaking, they show that, modulo a certain complexity-theoretical assumption (the Exponential Time Hypothesis [106]), the MSO Model-Checking problem cannot be solved in time O(poly (n)) on C, where poly (n) is a polynomial whose degree depends on the input formula ϕ, and C is a class of graphs that has strongly poly-logarithmically unbounded treewidth and is either closed under coloring [120,122], or under taking subgraphs [121]. A related result for the MSO 1 Model-Checking problem was proven in [82]. We note that there are, indeed, classes of strongly poly-logarithmically unbounded treewidth that admit polynomial time algorithms for the MSO 1 Model-Checking problem, e.g., classes of bounded clique-width or rank-width, but those are not closed under taking subgraphs.…”