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Deduction is a recently introduced graph searching process in which searchers clear the vertex set of a graph with one move each, with each searcher's movement determined by which of its neighbors are protected by other searchers. In this paper, we show that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search. We give a structural characterization, new bounds and a spectrum result on the number of searchers required. We consider the complexity of computing this parameter, giving an NP-completeness result for arbitrary graphs, and exhibiting families of graphs for which the parameter can be computed in polynomial time. We also describe properties of the deduction process related to the timing of searcher movement and the success of terminal layouts., Comment: 29 pages, 7 figures

This paper studies $MPC^{5+}_v$, which is to cover as many vertices as possible in a given graph $G=(V,E)$ by vertex-disjoint $5^+$-paths (i.e., paths each with at least five vertices). $MPC^{5+}_v$ is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of $\frac {19}7\approx 2.714$ and runs in $O(|V|^6)$ time. In this paper, we present a new approximation algorithm for $MPC^{5+}_v$ which achieves a ratio of $2.511$ and runs in $O(|V|^{2.5} |E|^2)$ time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion., Comment: Full version of an extended abstract in COCOON 2024

We contribute the first randomized algorithm that is an integration of arbitrarily many deterministic algorithms for the fully online multiprocessor scheduling with testing problem. When there are two machines, we show that with two component algorithms its expected competitive ratio is already strictly smaller than the best proven deterministic competitive ratio lower bound. Such algorithmic results are rarely seen in the literature. Multiprocessor scheduling is one of the first combinatorial optimization problems that have received numerous studies. Recently, several research groups examined its testing variant, in which each job $J_j$ arrives with an upper bound $u_j$ on the processing time and a testing operation of length $t_j$; one can choose to execute $J_j$ for $u_j$ time, or to test $J_j$ for $t_j$ time to obtain the exact processing time $p_j$ followed by immediately executing the job for $p_j$ time. Our target problem is the fully online version, in which the jobs arrive in sequence so that the testing decision needs to be made at the job arrival as well as the designated machine. We propose an expected $(\sqrt{\varphi + 3} + 1) (\approx 3.1490)$-competitive randomized algorithm as a non-uniform probability distribution over arbitrarily many deterministic algorithms, where $\varphi = \frac {\sqrt{5} + 1}2$ is the Golden ratio. When there are two machines, we show that our randomized algorithm based on two deterministic algorithms is already expected $\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)$-competitive. Besides, we use Yao's principle to prove lower bounds of $1.6682$ and $1.6522$ on the expected competitive ratio for any randomized algorithm at the presence of at least three machines and only two machines, respectively, and prove a lower bound of $2.2117$ on the competitive ratio for any deterministic algorithm when there are only two machines., Comment: 21 pages with 1 plot; an extended abstract to be submitted

Sorting a permutation by reversals is a famous problem in genome rearrangements. Since 1997, quite some biological evidence were found that in many genomes the reversed regions are usually flanked by a pair of inverted repeats. This type of reversals are called symmetric reversals, which, unfortunately, were largely ignored until recently. In this paper, we investigate the problem of sorting by symmetric reversals, which requires a series of symmetric reversals to transform one chromosome $A$ into the another chromosome $B$. The decision problem of sorting by symmetric reversals is referred to as {\em SSR} (when the input chromosomes $A$ and $B$ are given, we use {\em SSR(A,B)}) and the corresponding optimization version (i.e., when the answer for {\em SSR(A,B)} is yes, using the minimum number of symmetric reversals to convert $A$ to $B$), is referred to as {\em SMSR(A,B)}. The main results of this paper are summarized as follows, where the input is a pair of chromosomes $A$ and $B$ with $n$ repeats. (1) We present an $O(n^2)$ time algorithm to solve the decision problem {\em SSR(A,B)}, i.e., determine whether a chromosome $A$ can be transformed into $B$ by a series of symmetric reversals. (2) We design an $O(n^2)$ time algorithm for a special 2-balanced case of {\em SMSR(A,B)}, where chromosomes $A$ and $B$ both have duplication number 2 and every repeat appears twice in different orientations in $A$ and $B$. (3) We show that SMSR is NP-hard even if the duplication number of the input chromosomes are at most 2, hence showing that the above positive optimization result is the best possible. As a by-product, we show that the \emph{minimum Steiner tree} problem on \emph{circle graphs} is NP-hard, settling the complexity status of a 38-year old open problem., Comment: 35 pages, 12 figures

Due to hybridization events in evolution, studying two different genes of a set of species may yield two related but different phylogenetic trees for the set of species. In this case, we want to measure the dissimilarity of the two trees. The rooted subtree prune and regraft (rSPR) distance of the two trees has been used for this purpose. The problem of computing the rSPR distance of two given trees has many applications but is unfortunately NP-hard. The previously best approximation algorithm for rSPR distance achieves a ratio of 2.5 and it was open whether a better approximation algorithm for rSPR distance exists. In this paper, we answer this question in the affirmative by presenting a cubic-time approximation algorithm for rSPR distance that achieves a ratio of 2. Our algorithm is based on the new notion of key and a number of new structural lemmas. The algorithm is fairly simple and the proof of its correctness is intuitively understandable albeit complicated.

Given a vertex-weighted connected graph $G = (V, E)$, the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree $T$ of $G$ such that the total weight of the internal vertices in $T$ is maximized. The un-weighted variant, denoted as MIST, is NP-hard and APX-hard, and the currently best approximation algorithm has a proven performance ratio $13/17$. The currently best approximation algorithm for MwIST only has a performance ratio $1/3 - \epsilon$, for any $\epsilon > 0$. In this paper, we present a simple algorithm based on a novel relationship between MwIST and the maximum weight matching, and show that it achieves a better approximation ratio of $1/2$. When restricted to claw-free graphs, a special case been previously studied, we design a $7/12$-approximation algorithm., Comment: Revised complete version, with 27 pages, 27 figures; an extended abstract appears in the Proceedings of COCOON 2017

Motivation: Genome rearrangement plays an important role in evolutionary biology and has profound impacts on phenotype in organisms ranging from microbes to humans. The mechanisms for genome rearrangement events remain unclear. Lots of comparisons have been conducted among different species. To reveal the mechanisms for rearrangement events, comparison of different individuals/strains within the same species or genus (pan-genomes) is more helpful since they are much closer to each other. Results: We study the mechanism for inversion events via core-genome scaffold comparison of different strains within the same species. We focus on two kinds of bacteria, Pseudomonas aeruginosa and Escherichia coli, and investigate the inversion events among different strains of the same specie. We find an interesting phenomenon that long (larger than 10,000 bp) inversion regions are flanked by a pair of Inverted Repeats (IRs) (with lengths ranging from 385 bp to 27476 bp) which are often Insertion Sequences (ISs).This mechanism can also explain why the breakpoint reuses for inversion events happen. We study the prevalence of the phenomenon and find that it is a major mechanism for inversions. The other observation is that for different rearrangement events such as transposition and inverted block interchange, the two ends of the swapped regions are also associated with repeats so that after the rearrangement operations the two ends of the swapped regions remain unchanged. To our knowledge, this is the first time such a phenomenon is reported for transposition event., Comment: 8 pages, 11 figures

Given a graph G, the {\em maximum internal spanning tree problem} (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of 3/4. In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, 13/17) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.