arXiv subjects

Guohui Lin

Publications and source records attributed to Guohui Lin.

At least 19 recordsLinked to original sources

arXiv2026

Covering vertices by sequential stars

We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval $[k, \ell]$, where $1 \le k < \ell$ and $\ell$ can be infinity. This is referred to as sequential {\sc $[k, \ell]$-Star Packing} problem. It is solvable in polynomial time when $k = 1$, but becomes strongly NP-hard when $k \ge 2$. In this paper, we propose either the first or an improved approximation algorithm for the following four sequential settings: 1) a $\frac {k+1}2$-approximation algorithm when $k \ge 3$ and $\ell = \infty$, improving the previous best ratio of $\frac {(k+1)^2}{2k+1}$; 2) a $\frac 43$-approximation algorithm when $k = 2$ and $\ell = \infty$, improving the previous best ratio of $\frac 32$; 3) the first $(1 + \frac \ell{\ell+1})$-approximation algorithm when $2 = k < \ell$; and 4) the first $(1 + \max\left\{\frac {k-1}2, \frac {(k+1) \ell}{3 (\ell+1)}\right\})$-approximation algorithm when $3 \le k < \ell$. Besides the main algorithmic techniques being local search coupled with amortized analysis, we observe augmenting configurations to bridge two distant neighborhoods for a local improvement operation. Additionally, the problem has been shown APX-hard when $k \ge 3$; we prove its APX-hardness for the last remaining case where $k = 2$.

cs.DS↗

arXiv2025

Maximizing social welfare among EF1 allocations at the presence of two types of agents

We study the fair allocation of indivisible items to $n$ agents to maximize the utilitarian social welfare, where the fairness criterion is envy-free up to one item and there are only two different utility functions shared by the agents. We present a $2$-approximation algorithm when the two utility functions are normalized, improving the previous best ratio of $16 \sqrt{n}$ shown for general normalized utility functions; thus this constant ratio approximation algorithm confirms the APX-completeness in this special case previously shown APX-hard. When there are only three agents, i.e., $n = 3$, the previous best ratio is $3$ shown for general utility functions, and we present an improved and tight $\frac 53$-approximation algorithm when the two utility functions are normalized, and a best possible and tight $2$-approximation algorithm when the two utility functions are unnormalized.

cs.GT↗

arXiv2025

An improved local search based algorithm for $k^-$-star partition

We study the $k^-$-star partition problem that aims to find a minimum collection of vertex-disjoint stars, each having at most $k$ vertices to cover all vertices in a simple undirected graph $G = (V, E)$. Our main contribution is an improved $O(|V|^3)$-time $(\frac k2 - \frac {k-2}{8k-14})$-approximation algorithm. Our algorithm starts with a $k^-$-star partition with the least $1$-stars and a key idea is to distinguish critical vertices, each of which is either in a $2$-star or is the center of a $3$-star in the current solution. Our algorithm iteratively updates the solution by three local search operations so that the vertices in each star in the final solution produced cannot be adjacent to too many critical vertices. We present an amortization scheme to prove the approximation ratio in which the critical vertices are allowed to receive more tokens from the optimal solution.

cs.DS↗

arXiv2024

Pre-trained protein language model for codon optimization

Motivation: Codon optimization of Open Reading Frame (ORF) sequences is essential for enhancing mRNA stability and expression in applications like mRNA vaccines, where codon choice can significantly impact protein yield which directly impacts immune strength. In this work, we investigate the use of a pre-trained protein language model (PPLM) for getting a rich representation of amino acids which could be utilized for codon optimization. This leaves us with a simpler fine-tuning task over PPLM in optimizing ORF sequences. Results: The ORFs generated by our proposed models outperformed their natural counterparts encoding the same proteins on computational metrics for stability and expression. They also demonstrated enhanced performance against the benchmark ORFs used in mRNA vaccines for the SARS-CoV-2 viral spike protein and the varicella-zoster virus (VZV). These results highlight the potential of adapting PPLM for designing ORFs tailored to encode target antigens in mRNA vaccines.

q-bio.QM↗

arXiv2024

Approximation algorithms for non-sequential star packing problems

For a positive integer $k \ge 1$, a $k$-star ($k^+$-star, $k^-$-star, respectively) is a connected graph containing a degree-$\ell$ vertex and $\ell$ degree-$1$ vertices, where $\ell = k$ ($\ell \ge k$, $1 \le \ell \le k$, respectively). The $k^+$-star packing problem is to cover as many vertices of an input graph $G$ as possible using vertex-disjoint $k^+$-stars in $G$; and given $k > t \ge 1$, the $k^-/t$-star packing problem is to cover as many vertices of $G$ as possible using vertex-disjoint $k^-$-stars but no $t$-stars in $G$. Both problems are NP-hard for any fixed $k \ge 2$. We present a $(1 + \frac {k^2}{2k+1})$- and a $\frac 32$-approximation algorithms for the $k^+$-star packing problem when $k \ge 3$ and $k = 2$, respectively, and a $(1 + \frac 1{t + 1 + 1/k})$-approximation algorithm for the $k^-/t$-star packing problem when $k > t \ge 2$. They are all local search algorithms and they improve the best known approximation algorithms for the problems, respectively.

cs.DS↗

arXiv2024

Approximately covering vertices by order-$5$ or longer paths

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.

cs.DS↗

arXiv2023

Planar graphs are acyclically edge $(\Delta + 5)$-colorable

An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) conjectured that every simple graph with maximum degree $\Delta$ is acyclically edge $(\Delta + 2)$-colorable -- the well-known acyclic edge coloring conjecture (AECC). Despite many major breakthroughs and minor improvements, the conjecture remains open even for planar graphs. In this paper, we prove that planar graphs are acyclically edge $(\Delta + 5)$-colorable. Our proof has two main steps: Using discharging methods, we first show that every non-trivial planar graph must have one of the eight groups of well characterized local structures; and then acyclically edge color the graph using no more than $\Delta + 5$ colors by an induction on the number of edges.

cs.DM↗

arXiv2023

Randomized algorithms for fully online multiprocessor scheduling with testing

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.

cs.DS↗

arXiv2023

An Approximation Algorithm for Covering Vertices by 4^+-Paths

This paper deals with the problem of finding a collection of vertex-disjoint paths in a given graph G=(V,E) such that each path has at least four vertices and the total number of vertices in these paths is maximized. The problem is NP-hard and admits an approximation algorithm which achieves a ratio of 2 and runs in O(|V|^8) time. The known algorithm is based on time-consuming local search, and its authors ask whether one can design a better approximation algorithm by a completely different approach. In this paper, we answer their question in the affirmative by presenting a new approximation algorithm for the problem. Our algorithm achieves a ratio of 1.874 and runs in O(min{|E|^2|V|^2, |V|^5}) time. Unlike the previously best algorithm, ours starts with a maximum matching M of G and then tries to transform M into a solution by utilizing a maximum-weight path-cycle cover in a suitably constructed graph.

cs.DS↗

arXiv2022

Approximation algorithms for covering vertices by long paths

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the problem is polynomial time solvable; when $k$ is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed $k \ge 4$, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a $k$-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when $k = 4$, the problem admits a $4$-approximation algorithm which was presented recently. We propose the first $(0.4394 k + O(1))$-approximation algorithm for the general problem and an improved $2$-approximation algorithm when $k = 4$. Both algorithms are based on local improvement, and their theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies.

cs.DS↗

arXiv2022

An efficient polynomial-time approximation scheme for parallel multi-stage open shops

Various new scheduling problems have been arising from practical production processes and spawning new research areas in the scheduling field. We study the parallel multi-stage open shops problem, which generalizes the classic open shop scheduling and parallel machine scheduling problems. Given m identical k-stage open shops and a set of n jobs, we aim to process all jobs on these open shops with the minimum makespan, i.e., the completion time of the last job, under the constraint that job preemption is not allowed. We present an efficient polynomial-time approximation scheme (EPTAS) for the case when both m and k are constant. The main idea for our EPTAS is the combination of several categorization, scaling, and linear programming rounding techniques. Jobs and/or operations are first scaled and then categorized carefully into multiple types so that different types of jobs and/or operations are scheduled appropriately without increasing the makespan too much.

cs.DS↗

arXiv2021

Approximation algorithms for the directed path partition problems

Given a directed graph $G = (V, E)$, the $k$-path partition problem is to find a minimum collection of vertex-disjoint directed paths each of order at most $k$ to cover all the vertices of $V$. The problem has various applications in facility location, network monitoring, transportation and others. Its special case on undirected graphs has received much attention recently, but the general directed version is seemingly untouched in the literature. We present the first $k/2$-approximation algorithm, for any $k \ge 3$, based on a novel concept of augmenting path to minimize the number of singletons in the partition. When $k \ge 7$, we present an improved $(k+2)/3$-approximation algorithm based on the maximum path-cycle cover followed by a careful $2$-cycle elimination process. When $k = 3$, we define the second novel kind of augmenting paths and propose an improved $13/9$-approximation algorithm.

cs.DS↗

arXiv2020

Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree $\Delta$ is acyclically edge $(\Delta + 2)$-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

cs.DM↗

arXiv2019

Approximation algorithms for maximally balanced connected graph partition

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc $3$-BGP} admit a $5/4$-approximation and a $3/2$-approximation, respectively; but no approximability result exists for {\sc $k$-BGP} when $k \ge 4$, except a trivial $k$-approximation. In this paper, we present another $3/2$-approximation for our cardinality {\sc $3$-BGP} and then extend it to become a $k/2$-approximation for {\sc $k$-BGP}, for any constant $k \ge 3$. Furthermore, for {\sc $4$-BGP}, we propose an improved $24/13$-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.

cs.DS↗

arXiv2019

On Computing a Center Persistence Diagram

Throughout this paper, a persistence diagram ${\cal P}$ is composed of a set $P$ of planar points (each corresponding to a topological feature) above the line $Y=X$, as well as the line $Y=X$ itself, i.e., ${\cal P}=P\cup\{(x,y)|y=x\}$. Given a set of persistence diagrams ${\cal P}_1,...,{\cal P}_m$, for the data reduction purpose, one way to summarize their topological features is to compute the {\em center} ${\cal C}$ of them first under the bottleneck distance. We consider two discrete versions and one continuous version. For technical reasons, we first focus on the case when $|P_i|$'s are all the same (i.e., all have the same size $n$), and the problem is to compute a center point set $C$ under the bottleneck matching distance. We show, by a non-trivial reduction from the Planar 3D-Matching problem, that this problem is NP-hard even when $m=3$ diagrams are given. This implies that the general center problem for persistence diagrams under the bottleneck distance, when $P_i$'s possibly have different sizes, is also NP-hard when $m\geq 3$. On the positive side, we show that this problem is polynomially solvable when $m=2$ and admits a factor-2 approximation for $m\geq 3$. These positive results hold for any $L_p$ metric when $P_i$'s are point sets of the same size, and also hold for the case when $P_i$'s have different sizes in the $L_\infty$ metric (i.e., for the Center Persistence Diagram problem). This is the best possible in polynomial time for the Center Persistence Diagram under the bottleneck distance unless P = NP. All these results hold for both of the discrete versions as well as the continuous version; in fact, the NP-hardness and approximation results also hold under the Wasserstein distance for the continuous version.

cs.CG↗

arXiv2018

A local search $4/3$-approximation algorithm for the minimum $3$-path partition problem

Given a graph $G = (V, E)$, the $3$-path partition problem is to find a minimum collection of vertex-disjoint paths each of order at most $3$ to cover all the vertices of $V$. It is different from but closely related to the well-known $3$-set cover problem. The best known approximation algorithm for the $3$-path partition problem was proposed recently and has a ratio $13/9$. Here we present a local search algorithm and show, by an amortized analysis, that it is a $4/3$-approximation. This ratio matches up to the best approximation ratio for the $3$-set cover problem.

cs.DS↗

arXiv2018

Improved approximation algorithms for path vertex covers in regular graphs

Given a simple graph $G = (V, E)$ and a constant integer $k \ge 2$, the $k$-path vertex cover problem ({\sc P$k$VC}) asks for a minimum subset $F \subseteq V$ of vertices such that the induced subgraph $G[V - F]$ does not contain any path of order $k$. When $k = 2$, this turns out to be the classic vertex cover ({\sc VC}) problem, which admits a $\left(2 - {\rm \Theta}\left(\frac 1{\log|V|}\right)\right)$-approximation. The general {\sc P$k$VC} admits a trivial $k$-approximation; when $k = 3$ and $k = 4$, the best known approximation results for {\sc P$3$VC} and {\sc P$4$VC} are a $2$-approximation and a $3$-approximation, respectively. On $d$-regular graphs, the approximation ratios can be reduced to $\min\left\{2 - \frac 5{d+3} + \epsilon, 2 - \frac {(2 - o(1))\log\log d}{\log d}\right\}$ for {\sc VC} ({\it i.e.}, {\sc P$2$VC}), $2 - \frac 1d + \frac {4d - 2}{3d |V|}$ for {\sc P$3$VC}, $\frac {\lfloor d/2\rfloor (2d - 2)}{(\lfloor d/2\rfloor + 1) (d - 2)}$ for {\sc P$4$VC}, and $\frac {2d - k + 2}{d - k + 2}$ for {\sc P$k$VC} when $1 \le k-2 < d \le 2(k-2)$. By utilizing an existing algorithm for graph defective coloring, we first present a $\frac {\lfloor d/2\rfloor (2d - k + 2)}{(\lfloor d/2\rfloor + 1) (d - k + 2)}$-approximation for {\sc P$k$VC} on $d$-regular graphs when $1 \le k - 2 < d$. This beats all the best known approximation results for {\sc P$k$VC} on $d$-regular graphs for $k \ge 3$, except for {\sc P$4$VC} it ties with the best prior work and in particular they tie at $2$ on cubic graphs and $4$-regular graphs. We then propose a $1.875$-approximation and a $1.852$-approximation for {\sc P$4$VC} on cubic graphs and $4$-regular graphs, respectively. We also present a better approximation algorithm for {\sc P$4$VC} on $d$-regular bipartite graphs.

cs.DS↗

arXiv2018

Approximation algorithms for the three-machine proportionate mixed shop scheduling

A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flow-shop jobs and a set of open-shop jobs. Mixed shops are in general much more complex to schedule than flow-shops and open-shops, and have been studied since the 1980's. We consider the three machine proportionate mixed shop problem denoted as $M3 \mid prpt \mid C_{\max}$, in which each job has equal processing times on all three machines. Koulamas and Kyparisis [{\it European Journal of Operational Research}, 243:70--74,2015] showed that the problem is solvable in polynomial time in some very special cases; for the non-solvable case, they proposed a $5/3$-approximation algorithm. In this paper, we present an improved $4/3$-approximation algorithm and show that this ratio of $4/3$ is asymptotically tight; when the largest job is a flow-shop job, we present a fully polynomial-time approximation scheme (FPTAS). On the negative side, while the $F3 \mid prpt \mid C_{\max}$ problem is polynomial-time solvable, we show an interesting hardness result that adding one open-shop job to the job set makes the problem NP-hard if this open-shop job is larger than any flow-shop job. We are able to design an FPTAS for this special case too.

cs.DS↗