Non-deterministic Graph Research Articles

Non-deterministic graph searching in trees

Non-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q≥0, the q-limited search number, sq(G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s0(G) corresponds to the pathwidth of a graph G, and s∞(G) to its treewidth. Determining sq(G) is NP-complete for any fixed q≥0 in general graphs and s0(T) can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any q>0.We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by rsq and is shown to be equal to the non-deterministic graph searching parameter sq for q=0,1, and at most twice sq for any q≥2 (for any graph G).Our main result is a polynomial time algorithm that computes rsq(T) for any tree T and any q≥0. This provides a 2-approximation of sq(T) for any tree T, and shows that the decision problem associated to s1 is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest.

Deterministic vs non-deterministic graph property testing

A graph property \(\mathcal{P}\) is said to be testable if one can check whether a graph is close or far from satisfying \(\mathcal{P}\) using few random local inspections. Property \(\mathcal{P}\) is said to be non-deterministically testable if one can supply a “certificate” to the fact that a graph satisfies \(\mathcal{P}\) so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovasz and Vesztergombi [9], who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemeredi’s regularity lemma.

Non-Deterministic Graph Property Testing

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

Monotonicity of non-deterministic graph searching

In graph searching, a team of searchers are aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers permanently know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack. Fomin et al. [F.V. Fomin, P. Fraigniaud, N. Nisse, Nondeterministic graph searching: From pathwidth to treewidth, in: Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, MFCS’05, 2005, pp. 364–375] introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that permanently knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching was however left open. In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-deterministic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.

Nondeterministic Graph Searching: From Pathwidth to Treewidth

We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this game-theoretic approach and graph decompositions called q -branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched treewidth for all q≥0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers.

Lazy Context Cloning for Non-Deterministic Graph Rewriting

We define a rewrite strategy for a class of non-confluent constructor-based term graph rewriting systems and prove its correctness. Our strategy and its extension to narrowing are intended for the implementation of non-strict non-deterministic functional logic programming languages. Our strategy is based on a graph transformation, called bubbling, that avoids the construction of large contexts of redexes with distinct replacements, an expensive and frequently wasteful operation executed by competitive complete techniques.

Rank-non-increasing transformations on transition graphs

This paper is concerned with cycle rank (or, as sometimes called, cycle complexity) of finite transition graphs, a notion which was introduced by Eggan in connection with the notion of star height of regular events. Certain basic types of transformations on transition graphs are introduced and their effect on the cycle rank of the graphs is investigated. It is then shown that any transition graph G can be converted, by a finite series of such transformations, into an equivalent non-deterministic state graph (i.e., transition graph without any empty-word transitions) which has no more nodes than G and no higher cycle rank. A stronger version of Eggan's Star Height Theorem follows.