Fast algorithms for some dominating induced matching problems

We describe O(n)$O(n)$ time algorithms for finding the minimum weighted dominating induced matching of chordal, dually chordal, biconvex, and claw-free graphs. For the first three classes, we prove tight O(n)$O(n)$ bounds on the maximum number of edges that a graph having a dominating induced matching may contain. By applying these bounds, and employing existing O(n+m)$O(n+m)$ time algorithms we show that they can be reduced to O(n)$O(n)$ time. For claw-free graphs, we describe a variation of the existing algorithm for solving the unweighted version of the problem, which decreases its complexity from O(n²)$O(n^2)$ to O(n)$O(n)$, while additionally solving the weighted version. The same algorithm can be easily modified to count the number of DIM's of the given graph.     

Computing phylogenetic roots with bounded degrees and errors is NP-complete

In this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E)$G=(V,E)$, we wish to find a tree T such that (1) the degree of each internal node of T   is at least 3 and at most Δ$\Delta$, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E   and {{u,v}:u,v$\{\{u,v\}:u,v$ are leaves of T   and _dT(u,v)≤k}$d_T(u,v)\le k\}$, is as small as possible, where _dT(u,v)$d_T(u,v)$ denotes the distance between u$u$ and v$v$ in tree T  . We show that this problem is NP-hard for all fixed constants Δ,k≥3$\Delta ,k\ge 3$.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

Compatibility of unrooted phylogenetic trees is FPT

A collection of _T1,_T2,…,_Tk$T_1,T_2,\dots ,T_k$ of unrooted, leaf labelled (phylogenetic) trees, all with different leaf sets, is said to be compatible   if there exists a tree T$T$ such that each tree _Ti$T_i$ can be obtained from T$T$ by deleting leaves and contracting edges. Determining compatibility is NP-hard, and the fastest algorithm to date has worst case complexity of around Ω(n^k)$\mathrm{\Omega }(n^k)$ time, n$n$ being the number of leaves. Here, we present an O(nf(k))$\mathrm{O}(\mathit{nf}(k))$ algorithm, proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable   (FPT) with respect to the number k$k$ of trees.     

Rate-adaptive weighted fair queueing for energy-aware scheduling

In this paper we study energy-aware scheduling that trades energy consumption against a traditional performance measure of delay. We use the power-rate function f(x)=c+x^α$f(x)=c+x^{\alpha }$ for x>0$x>0$ and f(0)=0$f(0)=0$ to model the power consumption, where c>0$c>0$ represents the base power. We give a definition of a rate-adaptive version of the Weighted Fair Queueing scheduling algorithm, and prove its energy consumption is within a bounded factor of the best possible when the algorithm guarantees the classic end-to-end delay for every connection.