PDF《1 Introduction》

arXiv:1802.

02819v1 [cs.CC]

8 Feb

2018 A Complexity Theory for Labeling Schemes Maurice Chandoo? Abstract. In a labeling scheme the vertices of a given graph from a particular class are assigned short labels such that adjacency can be algorithmically determined from these labels. A representation of a graph from that class is given by the set of its vertex labels. Due to the shortness constraint on the labels such schemes provide space-e?cient representations for various graph classes, such as planar or interval graphs. We consider what graph classes cannot be represented by labeling schemes when the algorithm which determines adjacency is subjected to computational constraints. Keywords: adjacency labeling schemes, descriptive complexity of graph properties, graph class reductions, pointer numbers, lower bounds, weak implicit graph conjecture

1 Introduction Suppose you have a database and in one of its tables you need to store large interval graphs in such a way that adjacency can be determined quickly. This means generic data compression algorithms are not an option. A graph is an interval graph if each of its vertices can be mapped to a closed interval on the real line such that two vertices are adjacent i? their corresponding intervals intersect. A naive approach would be to store the adjacency matrices of the interval graphs. This requires roughly n2 bits. However, there are only 2O(n log n) interval graphs on n vertices which means that a space-wise optimal representation of interval graphs would only use O(n log n) bits. Adjacency lists perform even worse space-wise if the graphs are dense. A simple and asymptotically optimal solution for this problem is to use the interval representation. More speci?cally, given an interval graph G with n vertices write down its interval model (the set of intervals that correspond to its vertices), enumerate the endpoints of the intervals from left to right and label each vertex with the two endpoints of its interval, see Figure 1. The set of vertex labels is a representation of the graph and, moreover, adjacency of two vertices can be determined quickly by comparing their four endpoints. Each endpoint is a number in the range 1,2n and therefore one vertex label requires

2 log 2n bits. Thus the representation of the whole graph requires O(n log n) bits which is asymptotically optimal. The idea behind this representation can be generalized to other graph classes as follows. Let C be a graph class with similarly many graphs on n vertices as interval graphs. We say C has a labeling scheme (or implicit representation) if the vertices of every graph in C can be assigned binary labels of length O(log n) such that adjacency can be decided by an (e?cient) algorithm A which gets two labels as input. The algorithm A must only depend on C. We remark that labeling schemes can also be constructed for graph classes which have asymptotically more graphs than interval graphs by adjusting the label length. However, many important graph classes do indeed only have 2O(n log n) graphs on n vertices and therefore we shall restrict our attention to them;

we call such graph classes small. ? Leibniz Universit¨ at Hannover, Institut f¨ ur Theoretische Informatik, Appelstr. 4,

30167 Hannover, Germany;

E- Mail: [email protected]

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7 Figure 1: Interval model and the resulting labeling of the interval graph The central question of this paper is what small graph classes do not admit an implicit represen- tation when imposing computational constraints on the label decoding algorithm A. We propose a formal framework inspired by classical computational complexity in order to investigate this open- ended question in a structured way. A complexity class in this setting is a set of graph classes, usually de?ned in terms of labeling schemes. The introduced complexity classes can be seen as a complexity measure for the adjacency structure of a graph class. The concept of labeling schemes was introduced by Muller [Mul88] and by Kannan, Naor and Rudich [KNR92]. Among the ?rst and most basic questions was whether every small and hereditary (= closed under vertex deletion) graph class has a labeling scheme. If one omits the hereditary condition then this is not true due to a simple counting argument [Spi03, Thm. 7]. Being hereditary is a uniformity condition that is commonly required of graph classes in order for them to have some meaningful structure. This restriction can be slightly weakened without a?ecting the previous question by considering all graph classes which are a subset of a small and hereditary graph class, the justi?cation being that if a graph class C has an implicit representation then obviously every subset of C has an implicit representation as well. For instance, this weaker restriction also includes trees which are not hereditary but their hereditary closure, namely forests, are small and hereditary. This question remains unsolved and has been named the implicit graph conjecture (IGC) by Spinrad: every small and hereditary graph class has an implicit representation. He also gives an overview of graph classes known to have a labeling scheme and those not known to have one, which still remains quite accurate more than a decade later [Spi03]. The following is a brief account on results related to the IGC. De?nitions of the mentioned graph classes and other properties are given in the next section. Let us call a small and hereditary graph class which is not known to have a labeling scheme a candidate for the IGC. Identifying such graph classes is important when trying to study the limitations of labeling schemes. The challenge is to determine whether a given hereditary graph class is small. If a graph class has a labeling scheme then it must be small since every graph on n vertices in that class can be described using O(n log n) bits. Clearly, this argument cannot be used to ?nd candidates for the IGC. Another method to establish that a hereditary graph class is small is to apply Warren'