Dynamic survey on rank-width and related width parameters of graphs
Sang-il Oum
Aug 19, 2013
This is an incomplete on-going survey on rank-width and its related parameters. I intend to expand it slowly. By no means, this will be complete. Please feel free to leave comments or give me suggestions.1 Definitions
Rank-width was introduced by Oum and Seymour [35]. Clique-width is defined and investigated first by Courcelle and Olariu [5] published in 2000, but the operations for clique-width has been introduced by Courcelle, Engelfriet, and Rozenberg [4] in 1993. NLC-width was introduced by Wanke [40] in 1994. Boolean-width was introduced by Bui-Xuan, Telle, and Vatshelle [2].2 Well-quasi-ordering
Oum [31] proved that graphs of bounded rank-width are well-quasi-ordered under taking pivot-minors. This result has been generalized to- skew-symmetric or symmetric matrices over a fixed finite field by Oum [34],
- σ-symmetric matrices over a fixed finite field by Kante [20].
3 Forbidden vertex-minors
Oum [29] showed that if a graph has rank-width k, then so is its vertex-minor. This, together with the well-quasi-ordering theorem [31] implies that for each k, there exists a list of finitely many graphs such that a graph has rank-width at most k if and only if none of its vertex-minors is isomorphic to a graph in the list. Oum [29] showed that for each k, the forbidden vertex-minor for rank-width at most k has at most (6^{k+1}−1)/5 vertices. So in theory, we can enumerate all of them by an algorithm for fixed k. However, it is not clear how to find the list of forbidden vertex-minors for graphs of linear rank-width at most k, as there are no analogous upper bound on the size of forbidden vertex-minors. Jeong, Kwon, and Oum [16] showed that there are at least 3^{Ω(3k)} forbidden vertex-minors for linear rank-width at most k.4 Hardness
Computing rank-width is NP-hard. It can be easily deduced by combining the following known facts. This reduction is mentioned in [30].- Seymour and Thomas [37] proved that computing branch-width is NP-hard. Kloks, Kratochvíl, and Müller [21] even showed that it is NP-hard to compute branch-width of interval graphs.
- Hicks and McMurray Jr. [13] and independently Mazoit and Thomassé [26] showed that if a graph G is not a forest, then the branch-width of the cycle matroid M(G) is equal to the branch-width of G.
- Oum [29] showed that the branch-width of a binary matroid is equal to the rank-width of its fundamental graph. Every graphic matroid is binary.
5 Finding an approximate rank-decomposition for a fixed k
Oum and Seymour [35] provided an algorithm to find a rank-decomposition of width at most 3k+1 or confirm that the input graph has rank-width bigger than k in time O(8^{k} n^{9} logn). (Here, n is the number of vertices of the input graph.) The running time was later improved by Oum [30] to O(8^{k} n^{4}). This allows us to find a rank-decomposition of width at most c′ times rank-width if the rank-width is at most clogn in polynomial time. This is used in the quantum information theory (see Van den Nest, Dür, Vidal, and Briegel [38]) for their study on measurement-based quantum computation. If we do not mind having bigger function in k, then in the same paper [30], it is possible to find a rank-decomposition of width 3k−1 or confirm that the rank-width is bigger than k in time O(f(k)n^{3}). We do not know whether there is an algorithm to find an approximate rank-decomposition of width O(k) in time O(c_{1}^{k} n^{c}) for c ≤ 3 when an input graph has rank-width at most k. These algorithms can be used as a tool to construct an expression for clique-width decomposition, which are essential in many algorithmic applications.6 Deciding rank-width at most k for fixed k
There is a general algorithm of Oum and Seymour [36] which can construct a branch-decomposition of any symmetric submodular function in time O(n^{8k+c}), and if we apply it to rank-width, we get an algorithm of running time O(n^{8k+12}logn). Note that a simple general algorithm for path-width of any symmetric submodular function was developed by Nagamochi [28], which is applicable to linear rank-width in time O(n^{2k+4}). Courcelle and Oum [6] first constructed an algorithm to decide, for fixed k, whether rank-width is at most k in time O(f(k)n^{3}). But their algorithm uses the monadic second-order logic formula and does not provide an explicit rank-decomposition even if it exists. This problem was solved later. Hlinený and Oum [14] constructed an algorithm to decide whether rank-width is at most k and find a rank-decomposition of width at most k, if it exists, in time O(f(k)n^{3}). Here, f(k) is growing very fast with k, because the algorithm uses the monadic second-order logic as well as the list of forbidden minors for branch-width at most k for matroids representable over a fixed finite field. Geelen, Gerards, Robertson, and Whittle [11] proved that the forbidden minors for matroids of branch-width at most k have at most (6^{k}−1)/5 elements. This implies that we can construct an explicit algorithm for testing rank-width at most k and constructing a rank-decomposition of width at most k if it exists. (If there was no upper bound, then it may be impossible to enumerate all forbidden minors.) One can also decide whether the linear rank-width is at most k in time O(n^{3}) by using the well-quasi-ordering theorem [31] and monadic second-order logic formula to test vertex-minors [6]. But it is not known how to find the list of forbidden vertex-minors. Wahlström [39] showed that deciding whether clique-width is at most k and finding a k-expression can be done in time O^{*}((2k+1)^{n}). Espelage, Gurski, and Wanke [7] constructed an algorithm to decide whether a graph has clique-width at most k for graphs of bounded tree-width.7 Relation to Tree-width
Kante [19] showed that rank-width is at most 4k+2 if the tree-width is k. Later, Oum [32] showed that rank-width is at most k+1 if tree-width is k. In fact, it is shown thatCorneil and Rotics [3] showed that the clique-width is at most 3·2^{k−1} if the tree-width is k. Moreover, they proved that for each k, there is a graph G having tree-width k and clique-width at least 2^{k/2−1}. This also implies that there are graphs having rank-width at most k+1 and clique-width at least 2^{k/2−1} for each k. Kwon and Oum [22] proved that every graph of rank-width k is a pivot-minor of a graph of tree-width at most 2k. They also proved that every graph of linear rank-width k is a pivot-minor of a graph of path-width k+1. In other words, a set I of graphs has bounded rank-width if and only if it is a set of pivot-minors of graphs of bounded tree-width. Fomin, Oum, and Thilikos [10] showed that when graphs are planar, or H-minor-free, then having bounded tree-width is equivalent to having bounded rank-width. For instance, if a graph G is planar and has rank-width k, then tree-width is at most 72k−2. If G of rank-width k has no K_{r} minor with r > 2, then tree-width is at most 2^{O(rloglogr)}k. This is already proven in Courcelle and Olariu [5] without explicit bounds because they use logical tools.if G has branch-width k, then the incidence graph of G has rank-width k or k−1, unless k=0.
8 Exact Algorithms
There are small number of papers dealing with computing exact value of rank-width or related width parameters.Width Parameter | Running time | Paper |
Rank-width | O^{*}(2^{n}) | Oum [33] |
Linear rank-width | O^{*}(2^{n}) | forklore (trivial) |
9 Random graphs
Lee, Lee, and Oum [23] showed that asymptotically almost surely the Erdös-Rényi random graph G(n,p) has rank-width n/3−O(1) if p is a constant between 0 and 1. Furthermore, [1/(n)] << p ≤ [1/2], then the rank-width is [(n)/3]−o(n), and if p = c/n and c > 1, then rank-width is at least r n for some r = r(c). Since the clique-width is at least rank-width, this also gives a lower bound for clique-width. Adler, Bui-Xuan, Rabinovich, Renault, Telle, and Vatshelle [1] claimed that boolean-width of G(n,p) for fixed 0 < p < 1 is Θ(log^{2} n) asymptotically almost surely. Remark.
Johansson [17] (also in his Ph.D. thesis
[18]) claimed in a conference paper that NLC-width
and clique-width of a random graph G(n,p) for a fixed 0 < p < 1 is
Ω(n) almost surely. But we believe that the proof is incorrect.
In 2009, Marecek [25] studied the rank-width
of G(n,1/2), though I believe that the first version of his paper on
arxiv^{1} is
incorrect; Later versions have different proofs.
10 Explicit graphs
Jelínek [15] proved that an n×n grid has rank-width n−1.11 Software
Philipp Klau Krause implemented a simple dynamic programming algorithm to compute the rank-width of a graph^{2}. This software is now included in the open source mathematics software package called SAGE; see the manual^{3}. Apparently Friedmanský wrote Master's thesis on "implementation of optimization of a graph algorithm for computing rank-width"^{4} under the supervision of Robert Ganian.Acknowledgment
I would like to thank careful readers who kindly emailed me suggestions for this survey.- August 2013: Jisu Jeong found a typo.
- August 2013: Chiheon Kim pointed out a mistake on the statement on the consequence of a theorem by Corneil and Rotics.
- July 2013: J. Marecek explained his paper on arxiv.
- July 2013: F. Gurski provided a journal version of his paper with E. Wanke cited in the survey.
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