Capturing polynomial time using Modular Decomposition

Abstract

The question of whether there is a logic that captures polynomial time is one of the main open problems in descriptive complexity theory and database theory. In 2010 Grohe showed that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. We now consider classes of graphs with excluded induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed-point logic with counting captures polynomial time on the class of permutation graphs. Within the proof of the Modular Decomposition Theorem, we show that the modular decomposition of a graph is definable in symmetric transitive closure logic with counting. We obtain that the modular decomposition tree is computable in logarithmic space. It follows that cograph recognition and cograph canonization is computable in logarithmic space. CC Creative Commons24:2 B. Grußien Vol. 15:1 does not capture PTIME. In order to obtain a candidate for a logic capturing PTIME on all structures, Immerman proposed in 1987 to add to FP the ability to count [Imm87a]. Although the resulting logic, fixed-point logic with counting (FP+C), is not strong enough to capture PTIME on all finite structures [CFI92], it does so on many interesting classes of structures: FP+C captures PTIME, for example, on planar graphs [Gro98], all classes of graphs of bounded treewidth [GM99] and on K 5 -minor free graphs [Gro08]. Note that all these classes can be defined by a list of forbidden minors. In fact, Grohe showed in 2010 that FP+C captures PTIME on all graph classes with excluded minors [Gro10b]. This leads to the question whether a similar result can be obtained for graph classes that are characterized by excluded induced subgraphs, i.e., graph classes that are closed under taking induced subgraphs. For FP+C such a general result is not possible: Capturing PTIME on the class of chordal graphs, comparability graphs or co-comparability graphs is as hard as capturing PTIME on the class of all graphs for any "reasonable" logic [Gro10a, Lau11]. Yet, this gives us reason to consider the three mentioned graph classes and their subclasses more closely. So far, there are results showing that FP+C captures PTIME on the class of chordal line graphs [Gro10a] and on the class of interval graphs (chordal co-comparability graphs) [Lau10].We add to these results the following results: FP+C captures PTIME on the class of permutation graphs (comparability co-comparability graphs) (see Section 5) and on the class of chordal comparability graphs (see [Gru17a]). Both results are based on modular decom...

On the Parallel Parameterized Complexity of the Graph Isomorphism Problem

“…Recall that cographs are graphs without any induced P 4 . The colored graph isomorphism for cographs was shown to be in L using logspace algorithm to find the modular decomposition [24]. From this along with Theorem 3 and Lemma 1, we deduce that the graph isomorphism problem for distance to cographs is in Para-AC 1 .…”

“…The proof of (1), follows from Theorem 5 and the logspace algorithm for colored GI for interval graphs (see [26]). The proof of (2), follows from Corollary 4 and the logspace isomorphism algorithm for colored GI for cographs [24]. ⊓ ⊔…”

Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

“…Transductions (also known as syntactical interpretations) define certain structures within other structures. Detailed introductions with a lot of examples can be found in [Gro13,Gru17b]. In the following we briefly introduce transductions, consider compositions of tranductions, and present the new notion of counting transductions.…”

“…6 It is shown in [GGHL12, Remark 4.8], and in more detail in [Gru17b,Section 8.4] that the class of all colored directed trees that have a linear order on the colors admits LREC-definable canonization. This can easily be extended to LO-colored directed trees since an LO-colored directed tree is a special kind of colored directed tree that has a linear order on its colors.…”