arXiv subjects

Mingyu Xiao

Publications and source records attributed to Mingyu Xiao.

At least 73 records · Page 4Linked to original sources

arXiv2022

Practical Algorithms with Guaranteed Approximation Ratio for TTP with Maximum Tour Length Two

The Traveling Tournament Problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, minimizing the total distance traveled by all $n$ teams ($n$ is even). In this paper, we consider TTP-2, i.e., TTP under the constraint that at most two consecutive home games or away games are allowed for each team. We propose practical algorithms for TTP-2 with improved approximation ratios. Due to the different structural properties of the problem, all known algorithms for TTP-2 are different for $n/2$ being odd and even, and our algorithms are also different for these two cases. For even $n/2$, our approximation ratio is $1+3/n$, improving the previous result of $1+4/n$. For odd $n/2$, our approximation ratio is $1+5/n$, improving the previous result of $3/2+6/n$. In practice, our algorithms are easy to implement. Experiments on well-known benchmark sets show that our algorithms beat previously known solutions for all instances with an average improvement of $5.66\%$.

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arXiv2022

Further Improvements for SAT in Terms of Formula Length

In this paper, we prove that the general CNF satisfiability problem can be solved in $O^*(1.0638^L)$ time, where $L$ is the length of the input CNF-formula (i.e., the total number of literals in the formula), which improves the previous result of $O^*(1.0652^L)$ obtained in 2009. Our algorithm was analyzed by using the measure-and-conquer method. Our improvements are mainly attributed to the following two points: we carefully design branching rules to deal with degree-5 and degree-4 variables to avoid previous bottlenecks; we show that some worst cases will not always happen, and then we can use an amortized technique to get further improvements. In our analyses, we provide some general frameworks for analysis and several lower bounds on the decreasing of the measure to simplify the arguments. These techniques may be used to analyze more algorithms based on the measure-and-conquer method.

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arXiv2022

Listing Maximal k-Plexes in Large Real-World Graphs

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(γ^D)$ time for each constant $k$, where $γ$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

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arXiv2021

A Further Improvement on Approximating TTP-2

The Traveling Tournament Problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, minimizing the total distance traveled by all $n$ teams ($n$ is even). In this paper, we consider TTP-2, i.e., TTP with one more constraint that each team can have at most two consecutive home games or away games. Due to the different structural properties, known algorithms for TTP-2 are different for $n/2$ being odd and even. For odd $n/2$, the best known approximation ratio is about $(1+12/n)$, and for even $n/2$, the best known approximation ratio is about $(1+4/n)$. In this paper, we further improve the approximation ratio from $(1+4/n)$ to $(1+3/n)$ for $n/2$ being even. Experimental results on benchmark sets show that our algorithm can improve previous results on all instances with even $n/2$ by $1\%$ to $4\%$.

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arXiv2021

Exact algorithms for maximum weighted independent set on sparse graphs

The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted version, each vertex in the graph is associated with a weight and we are going to find an independent set of maximum total vertex weight. In this paper, we design several reduction rules and a fast exact algorithm for the maximum weighted independent set problem, and use the measure-and-conquer technique to analyze the running time bound of the algorithm. Our algorithm works on general weighted graphs and it has a good running time bound on sparse graphs. If the graph has an average degree at most 3, our algorithm runs in $O^*(1.1443^n)$ time and polynomial space, improving previous running time bounds for the problem in cubic graphs using polynomial space.

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arXiv2020

Bounded-Degree Cut is Fixed-Parameter Tractable

In the bounded-degree cut problem, we are given a multigraph $G=(V,E)$, two disjoint vertex subsets $A,B\subseteq V$, two functions $\mathrm{u}_A, \mathrm{u}_B:V\to \{0,1,\ldots,|E|\}$ on $V$, and an integer $k\geq 0$. The task is to determine whether there is a minimal $(A,B)$-cut $(V_A,V_B)$ of size at most $k$ such that the degree of each vertex $v\in V_A$ in the induced subgraph $G[V_A]$ is at most $\mathrm{u}_A(v)$ and the degree of each vertex $v\in V_B$ in the induced subgraph $G[V_B]$ is at most $\mathrm{u}_B(v)$. In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a $2^{18k}|G|^{O(1)}$-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems.

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arXiv2020

Enhancing Balanced Graph Edge Partition with Effective Local Search

Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in power-law graphs than vertex partition and thereby has been more widely adopted as the default partition strategy by existing graph systems. The graph edge partition problem, which is to split the edge set into multiple balanced parts to minimize the total number of copied vertices, has been widely studied from the view of optimization and algorithms. In this paper, we study local search algorithms for this problem to further improve the partition results from existing methods. More specifically, we propose two novel concepts, namely adjustable edges and blocks. Based on these, we develop a greedy heuristic as well as an improved search algorithm utilizing the property of the max-flow model. To evaluate the performance of our algorithms, we first provide adequate theoretical analysis in terms of the approximation quality. We significantly improve the previously known approximation ratio for this problem. Then we conduct extensive experiments on a large number of benchmark datasets and state-of-the-art edge partition strategies. The results show that our proposed local search framework can further improve the quality of graph partition by a wide margin.

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arXiv2020

The Complexity of the Partition Coloring Problem

Given a simple undirected graph $G=(V,E)$ and a partition of the vertex set $V$ into $p$ parts, the \textsc{Partition Coloring Problem} asks if we can select one vertex from each part of the partition such that the chromatic number of the subgraph induced on the $p$ selected vertices is bounded by $k$. PCP is a generalized problem of the classical \textsc{Vertex Coloring Problem} and has applications in many areas, such as scheduling and encoding etc. In this paper, we show the complexity status of the \textsc{Partition Coloring Problem} with three parameters: the number of colors, the number of parts of the partition, and the maximum size of each part of the partition. Furthermore, we give a new exact algorithm for this problem.

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arXiv2020

Object Reachability via Swaps under Strict and Weak Preferences

The \textsc{Housing Market} problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomially solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard even when the network is a path. In addition, we consider the computational complexity of finding different optimal assignments for the problem under the network being a path or a star.

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arXiv2020

An Improved Upper Bound for SAT

We show that the CNF satisfiability problem can be solved $O^*(1.2226^m)$ time, where $m$ is the number of clauses in the formula, improving the known upper bounds $O^*(1.234^m)$ given by Yamamoto 15 years ago and $O^*(1.239^m)$ given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement.

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arXiv2019

Algorithms for Manipulating Sequential Allocation

Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most preferred item among all items having not been allocated yet. This problem is well-known to be not strategyproof, i.e., an agent may get more utility by reporting an untruthful preference ranking of items. It arises the problem: how to find the best response of an agent? It is known that this problem is polynomially solvable for only two agents and NP-complete for arbitrary number of agents. The computational complexity of this problem with three agents was left as an open problem. In this paper, we give a novel algorithm that solves the problem in polynomial time for each fixed number of agents. We also show that an agent can always get at least half of its optimal utility by simply using its truthful preference as the response.

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arXiv2019

Upper and Lower Bounds on Approximating Weighted Mixed Domination

A mixed dominating set of a graph $G = (V, E)$ is a mixed set $D$ of vertices and edges, such that for every edge or vertex, if it is not in $D$, then it is adjacent or incident to at least one vertex or edge in $D$. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is $NP$-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight $w_v$ and all edges are assigned the same nonnegative weight $w_e$, and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include: [1.] for $w_e\geq w_v$, a 2-approximation algorithm; [2.] for $w_e\geq 2w_v$, inapproximability within ratio 1.3606 unless $P=NP$ and within ratio 2 under UGC; [3.] for $2w_v > w_e\geq w_v$, inapproximability within ratio 1.1803 unless $P=NP$ and within ratio 1.5 under UGC; [4.] for $w_e< w_v$, inapproximability within ratio $(1-ε)\ln |V|$ unless $P=NP$ for any $ε>0$.

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arXiv2018

Some Reduction Operations to Pairwise Compatibility Graphs

A graph $G=(V,E)$ with a vertex set $V$ and an edge set $E$ is called a pairwise compatibility graph (PCG, for short) if there are a tree $T$ whose leaf set is $V$, a non-negative edge weight $w$ in $T$, and two non-negative reals $d_{\min}\leq d_{\max}$ such that $G$ has an edge $uv\in E$ if and only if the distance between $u$ and $v$ in the weighted tree $(T,w)$ is in the interval $[d_{\min}, d_{\max}]$. PCG is a new graph class motivated from bioinformatics. In this paper, we give some necessary and sufficient conditions for PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete $k$-partite graph, cactus, and some other graph classes are subsets of PCG.

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arXiv2018

Characterizing Star-PCGs

A graph $G$ is called a pairwise compatibility graph (PCG, for short) if it admits a tuple $(T,w, d_{\min},d_{\max})$ of a tree $T$ whose leaf set is equal to the vertex set of $G$, a non-negative edge weight $w$, and two non-negative reals $d_{\min}\leq d_{\max}$ such that $G$ has an edge between two vertices $u,v\in V$ if and only if the distance between the two leaves $u$ and $v$ in the weighted tree $(T,w)$ is in the interval $[d_{\min}, d_{\max}]$. The tree $T$ is also called a witness tree of the PCG $G$. The problem of testing if a given graph is a PCG is not known to be NP-hard yet. To obtain a complete characterization of PCGs is a wide open problem in computational biology and graph theory. In literature, most witness trees admitted by known PCGs are stars and caterpillars. In this paper, we give a complete characterization for a graph to be a star-PCG (a PCG that admits a star as its witness tree), which provides us the first polynomial-time algorithm for recognizing star-PCGs.

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arXiv2016

A Linear-time Algorithm for Integral Multiterminal Flows in Trees

In this paper, we study the problem of finding an integral multiflow which maximizes the sum of flow values between every two terminals in an undirected tree with a nonnegative integer edge capacity and a set of terminals. In general, it is known that the flow value of an integral multiflow is bounded by the cut value of a cut-system which consists of disjoint subsets each of which contains exactly one terminal or has an odd cut value, and there exists a pair of an integral multiflow and a cut-system whose flow value and cut value are equal; i.e., a pair of a maximum integral multiflow and a minimum cut. In this paper, we propose an $O(n)$-time algorithm that finds such a pair of an integral multiflow and a cut-system in a given tree instance with $n$ vertices. This improves the best previous results by a factor of $Ω(n)$. Regarding a given tree in an instance as a rooted tree, we define $O(n)$ rooted tree instances taking each vertex as a root, and establish a recursive formula on maximum integral multiflow values of these instances to design a dynamic programming that computes the maximum integral multiflow values of all $O(n)$ rooted instances in linear time. We can prove that the algorithm implicitly maintains a cut-system so that not only a maximum integral multiflow but also a minimum cut-system can be constructed in linear time for any rooted instance whenever it is necessary. The resulting algorithm is rather compact and succinct.

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arXiv2016

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.

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arXiv2016

A Parameterized Algorithm for Bounded-Degree Vertex Deletion

The $d$-bounded-degree vertex deletion problem, to delete at most $k$ vertices in a given graph to make the maximum degree of the remaining graph at most $d$, finds applications in computational biology, social network analysis and some others. It can be regarded as a special case of the $(d+2)$-hitting set problem and generates the famous vertex cover problem. The $d$-bounded-degree vertex deletion problem is NP-hard for each fixed $d\geq 0$. In terms of parameterized complexity, the problem parameterized by $k$ is W[2]-hard for unbounded $d$ and fixed-parameter tractable for each fixed $d\geq 0$. Previously, (randomized) parameterized algorithms for this problem with running time bound $O^*((d+1)^k)$ are only known for $d\leq2$. In this paper, we give a uniform parameterized algorithm deterministically solving this problem in $O^*((d+1)^k)$ time for each $d\geq 3$. Note that it is an open problem whether the $d'$-hitting set problem can be solved in $O^*((d'-1)^k)$ time for $d'\geq 3$. Our result answers this challenging open problem affirmatively for a special case. Furthermore, our algorithm also gets a running time bound of $O^*(3.0645^k)$ for the case that $d=2$, improving the previous deterministic bound of $O^*(3.24^k)$.

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arXiv2016

Linear Kernels for Separating a Graph into Components of Bounded Size

Graph separation and partitioning are fundamental problems that have been extensively studied both in theory and practice. The \textsc{$p$-Size Separator} problem, closely related to the \textsc{Balanced Separator} problem, is to check whether we can delete at most $k$ vertices in a given graph $G$ such that each connected component of the remaining graph has at most $p$ vertices. This problem is NP-hard for each fixed integer $p\geq 1$ and it becomes the famous \textsc{Vertex Cover} problem when $p=1$. It is known that the problem with parameter $k$ is W[1]-hard for unfixed $p$. In this paper, we prove a kernel of $O(pk)$ vertices for this problem, i.e., a linear vertex kernel for each fixed $p \geq 1$. In fact, we first obtain an $O(p^2k)$ vertex kernel by using a nontrivial extension of the expansion lemma. Then we further reduce the kernel size to $O(pk)$ by using some `local adjustment' techniques. Our proofs are based on extremal combinatorial arguments and the main result can be regarded as a generalization of the Nemhauser and Trotter's theorem for the \textsc{Vertex Cover} problem. These techniques are possible to be used to improve kernel sizes for more problems, especially problems with kernelization algorithms based on techniques similar to the expansion lemma or crown decompositions.

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