744 research outputs found
The Graph Motif problem parameterized by the structure of the input graph
The Graph Motif problem was introduced in 2006 in the context of biological
networks. It consists of deciding whether or not a multiset of colors occurs in
a connected subgraph of a vertex-colored graph. Graph Motif has been mostly
analyzed from the standpoint of parameterized complexity. The main parameters
which came into consideration were the size of the multiset and the number of
colors. Though, in the many applications of Graph Motif, the input graph
originates from real-life and has structure. Motivated by this prosaic
observation, we systematically study its complexity relatively to graph
structural parameters. For a wide range of parameters, we give new or improved
FPT algorithms, or show that the problem remains intractable. For the FPT
cases, we also give some kernelization lower bounds as well as some ETH-based
lower bounds on the worst case running time. Interestingly, we establish that
Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which
is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
On the automorphism groups of strongly regular graphs II
We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp((logn)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ăkos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem
Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy
Given two structures G and H distinguishable in FO^k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a FO^k formula distinguishing G from H. Let A^k(n) be the maximum value of A^k(G,H) over n-element structures. We prove the strictness of the quantifier alternation hierarchy of FO^2 in a strong quantitative form, namely A^2(n) >= n/8-2, which is tight up to a constant factor. For each k >= 2, it holds that A^k(n) > log_(k+1) n-2 even over colored trees, which is also tight up to a constant factor if k >= 3. For k >= 3 the last lower bound holds also over uncolored trees, while the alternation hierarchy of FO^2 collapses even over all uncolored graphs.
We also show examples of colored graphs G and H on n vertices that can be distinguished in FO^2 much more succinctly if the alternation number is increased just by one: while in Sigma_i it is possible to distinguish G from H with bounded quantifier depth, in Pi_i this requires quantifier depth Omega(n2). The quadratic lower bound is best possible here because, if G and H can be distinguished in FO^k with i quantifier alternations, this can be done with quantifier depth n^(2k-2)
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