37 research outputs found
Graph tilings in incompatibility systems
Given two graphs and , an \emph{-tiling} of is a collection of
vertex-disjoint copies of in and an \emph{-factor} is an -tiling
that covers all vertices of . K\"{u}hn and Osthus managed to characterize,
up to an additive constant, the minimum degree threshold which forces an
-factor in a host graph . In this paper we study a similar tiling problem
in a system that is locally bounded. An \emph{incompatibility system}
over is a family with
. We say that two
edges are \emph{incompatible} if for some
, and otherwise \emph{compatible}. A subgraph of is
\emph{compatible} if every pair of edges in are compatible. An
incompatibility system is \emph{-bounded} if for any
vertex and any edge incident with , there are at most
two-subsets in containing . This notion was partly motivated by a
concept of transition system introduced by Kotzig in 1968, and first formulated
by Krivelevich, Lee and Sudakov to study the robustness of Hamiltonicity of
Dirac graphs.
We prove that for any and any graph with vertices, there
exists a constant such that for any sufficiently large with , if is an -vertex graph with
and is a -bounded incompatibility system over , then there exists a compatible
-factor in , where the value is either the chromatic number
or the critical chromatic number and we provide a
dichotomy. Moreover, the error term is inevitable in general case
Approximate Counting of k-Paths: Deterministic and in Polynomial Space
A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^km epsilon^{-2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 +/- epsilon. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k+O(log^3k)}m log n whenever epsilon^{-1}=k^{O(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^km epsilon^{-2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results.
- We present a deterministic 4^{k+O(sqrt{k}(log^2k+log^2 epsilon^{-1}))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices.
- Additionally, we present a randomized 4^{k+O(log k(log k + log epsilon^{-1}))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method.
Thus, the algorithm by Brand et al. runs in time 4^{k+o(k)}m whenever epsilon^{-1}=2^{o(k)}, while our deterministic and randomized algorithms run in time 4^{k+o(k)}m log n whenever epsilon^{-1}=2^{o(k^{1/4})} and epsilon^{-1}=2^{o(k/(log k))}, respectively. Prior to our work, no 2^{O(k)}n^{O(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth
Counting small induced subgraphs satisfying monotone properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result