arXiv subjects
Publications and source records attributed to Yixin Cao.
We study the minimum \emph{interval deletion} problem, which asks for the removal of a set of at most $k$ vertices to make a graph of $n$ vertices into an interval graph. We present a parameterized algorithm of runtime $10^k \cdot n^{O(1)}$ for this problem, that is, we show the problem is fixed-parameter tractable.
A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used it to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden induced subgraph characterization and a direct recognition algorithm for normal Helly circular-arc graphs, both of which are resolved by the current paper. Moreover, when the input is not a normal Helly circular-arc graph, our recognition algorithm finds in linear time a minimal forbidden induced subgraph as certificate.
We present a dynamic synchrotron X-ray imaging study of the effective temperature $T_{eff}$ in a vibrated granular medium. By tracking the directed motion and the fluctuation dynamics of the tracers inside, we obtained $T_{eff}$ of the system using Einstein relation. We found that as the system unjams with increasing vibration intensities $Γ$, the structural relaxation time $τ$ increases substantially which can be fitted by an Arrhenius law using $T_{eff}$. And the characteristic energy scale of structural relaxation yielded by the Arrhenius fitting is $E = 0.21 \pm 0.02$ $pd^3$, where $p$ is the pressure and $d$ is the background particle diameter, which is consistent with those from hard sphere simulations in which the structural relaxation happens via the opening up of free volume against pressure.
Given a graph $G = (V,E)$ and a terminal $s\in V$, a cut $X$ for $s$ is a vertex set that contains $s$. We look for a cut that is small in two senses, i.e., there are no more than $k$ vertices in $X$ and no more than $t$ edges leaving $X$. Answering a question asked by Fomin et al. (arXiv:1304.6189), we show the problem is fixed-parameter tractable parameterized by either $k$ or $t$.
We study the \emph{interval completion} problem, which asks for the insertion of a set of at most $k$ edges to make a graph of $n$ vertices into an interval graph. We focus on chordal graphs with no small obstructions, where every remaining obstruction is known to have a shallow property. From such a shallow obstruction we single out a subset 6 or 7 vertices, called the frame, and 5 missed edges in the subgraph induced by the frame. We show that if none of these edges is inserted, then the frame cannot be altered at all, and the whole obstruction is also fixed, by and large, in the sense that their related positions in an interval representation of the objective interval graph have a specific pattern. We propose a simple bounded search process, which effectively transforms a given graph to a graph with the structural property that all obstructions are shallow and have fixed frames. Then we fill in polynomial time all obstructions that have been previously left in indecision. These efforts together deliver a simple parameterized algorithm of time $6^k\cdot n^{O(1)}$ for the problem, significantly improving the only known parameterized algorithm of time $k^{2k}\cdot n^{O(1)}$.
In this letter, we present the first experimental study of bridge structures in three-dimensional dry granular packings. When bridges are small, they are predominantly 'linear', and have an exponential size distribution. Larger, predominantly 'complex' bridges, are confirmed to follow a power-law size distribution. Our experiments, which use X-ray tomography, are in good agreement with the simulations presented here, for the distribution of sizes, end-to-end lengths, base extensions and orientations of predominantly linear bridges. Quantitative differences between the present experiment and earlier simulations suggest that packing fraction is an important determinant of bridge structure.
Kernelization algorithms, usually a preprocessing step before other more traditional algorithms, are very special in the sense that they return (reduced) instances, instead of final results. This characteristic excludes the freedom of applying a kernelization algorithm for the weighted version of a problem to its unweighted instances. Thus with only very few special cases, kernelization algorithms have to be studied separately for weigthed and unweighted versions of a single problem. {\sc feedback arc set on tournament} is currently a very popular problem in recent research of parameterized, as well as approximation computation, and its wide applications in many areas make it appear in all top conferences. The theory of graph modular decompositions is a general approach in the study of graph structures, which only had its surfaces touched in previous work on kernelization algorithms of {\sc feedback arc set on tournament}. In this paper, we study further properties of graph modular decompositions and apply them to obtain the first linear kernel for the unweighted {\sc feedback arc set on tournament} problem, which only admits linear kernel in its weighted version, while quadratic kernel for the unweighted.
Kernelization algorithms for the {\sc cluster editing} problem have been a popular topic in the recent research in parameterized computation. Thus far most kernelization algorithms for this problem are based on the concept of {\it critical cliques}. In this paper, we present new observations and new techniques for the study of kernelization algorithms for the {\sc cluster editing} problem. Our techniques are based on the study of the relationship between {\sc cluster editing} and graph edge-cuts. As an application, we present an ${\cal O}(n^2)$-time algorithm that constructs a $2k$ kernel for the {\it weighted} version of the {\sc cluster editing} problem. Our result meets the best kernel size for the unweighted version for the {\sc cluster editing} problem, and significantly improves the previous best kernel of quadratic size for the weighted version of the problem.