arXiv subjects

Mingyu Xiao

Publications and source records attributed to Mingyu Xiao.

At least 19 recordsLinked to original sources

arXiv2026

A Near-Linear Element Kernel for $d$-Hitting Set

In $d$-\textsc{Hitting Set}, the input consists of a finite universe $U$, a family $\mathcal S$ of subsets of $U$ with size at most $d$, and an integer $k$. The task is to decide whether at most $k$ elements of $U$ can intersect every set in $\mathcal S$. For every fixed $d\geq3$, we give a one-sided randomized kernel with $O(k\log^3k)$ elements and a deterministic kernel with $O(k^2\log k)$ elements for $d$-\textsc{Hitting Set}. In the one-sided randomized kernel, every NO-instance is always mapped to a NO-instance, and a YES-instance is mapped to a YES-instance with constant probability. The previously known kernels for $d$-\textsc{Hitting Set} contain $O(k^{d-1})$ elements and $O(k^d)$ sets. It has been asked in the literature whether $d$-\textsc{Hitting Set} allows kernels with $O(k^{d-1-\varepsilon})$ elements for some constant $\varepsilon>0$. In this paper, we answer this question affirmatively by giving near-linear element-kernels through a re-encoding of the instance. On the other hand, our kernel may still contain $k^{O(d)}$ sets and the parameter $k$ may grow polynomially.

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arXiv2026

Linear Kernels for $l$-Exact Component Order Connectivity

The \textsc{$l$-Exact Component Order Connectivity} problem asks whether, given an input graph $G$ and an integer $k$, there exists a vertex subset $S\subseteq V(G)$ of size at most $k$ such that every connected component in $G - S$ has exactly $l$ vertices. In this paper, we present an $O(kl)$-vertex kernel for this problem, computable in $|V(G)|^{O(l)}$ time. This is the first known linear kernel for each fixed $l\geq 3$. For $l=1$, this problem reduces to the classical \textsc{Vertex Cover}, and our result matches the best-known $2k$-vertex kernel. For $l=2$ (known as \textsc{Deletion to Induced Matching}), we can get a $(3k + 1)$-vertex kernel, improving the previously known result of $6k$ vertices. Our kernelization algorithm is built upon on an extended crown decomposition combined with linear programming and other techniques.

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arXiv2026

Deterministic Single Exponential Time Algorithms for Co-Path Packing and Co-Path Set Parameterized by Treewidth

The \textsc{Co-Path Packing} (resp., \textsc{Co-Path Set}) problem asks whether a given graph can be edited to a collection of induced paths by deleting at most $k$ vertices (resp., $k$ edges). Both are fundamental problems with significant applications in bioinformatics and have been extensively studied within the framework of exact and parameterized algorithms. Currently, the state-of-the-art approach utilizes the randomized ``Cut \& Count'' technique, which solves \textsc{Co-Path Set} in $O^*(4^{\mathbf{tw}})$ time and \textsc{Co-Path Packing} in $O^*(5^{\mathbf{pw}})$ time, where $\mathbf{tw}$ is treewidth and $\mathbf{pw}$ is pathwidth. However, as there is no known method to derandomize the ``Cut \& Count'' technique, the existence of deterministic single exponential time algorithms for these problems parameterized by treewidth has remained an open question. In this paper, we resolve this gap by providing deterministic single exponential time algorithms for both problems when parameterized by treewidth.

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arXiv2026

New Algorithms for Parity-SAT and Its Bounded-Occurrence Versions

Parity-SAT is the problem of determining whether a given CNF formula has an odd number of satisfying assignments. As a canonical $\oplus$P-complete problem, it represents a fundamental variant of the exact model counting problem (#SAT). Under the Strong Exponential Time Hypothesis (SETH), Parity-SAT admits no $O^*((2-\varepsilon)^n)$-time or $O^*((2-\varepsilon)^m)$-time algorithm for any constant $\varepsilon>0$, where $n$ and $m$ denote the numbers of variables and clauses, respectively. Thus, breaking the $2^n$ or $2^m$ barrier appears impossible in full generality. In this work, we revisit this barrier through structural restrictions and a refined exploitation of parity. We study Parity-$d$-occ-SAT, where each variable appears in at most $d$ clauses, and obtain three main results. First, we design a randomized $O^*(2^{m(1-1/O(d))})$-time algorithm, thereby breaking the $2^m$ barrier for every fixed $d$. Second, for the special case $d=2$, we develop a significantly sharper branching algorithm running in $O^*(1.1193^n)$ time or $O^*(1.3248^m)$ time. Third, leveraging the structural insights underlying the $d=2$ case, we obtain an $O^*(1.1052^L)$-time algorithm for general Parity-SAT, where $L$ denotes the formula length. All algorithms use only polynomial space. Notably, our running-time bounds are better than the best known bounds for the corresponding exact counting counterparts, highlighting a genuine algorithmic advantage of parity over counting. Conceptually, our results demonstrate that parity admits finer structural reductions and more efficient branching than exact model counting, and that bounded occurrence can be systematically leveraged to circumvent classical exponential barriers.

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arXiv2026

Feedback Set Problems on Bounded-Degree (Planar) Graphs

The feedback set problems are about removing the minimum number of vertices or edges from a graph to break all its cycles. Much effort has gone into understanding their complexity on planar graphs as well as on graphs of bounded degree. We obtain a complete complexity classification for these problems on bounded-degree digraphs, including the planar case. In particular, we show that both problems are $\NP$-complete on digraphs of maximum degree three, while on planar digraphs the feedback vertex set problem is polynomial-time solvable when each vertex has either indegree at most one or outdegree at most one, and $\NP$-complete otherwise. We also give tight degree bounds for the connected feedback vertex set problem on undirected graphs, both planar and non-planar. We close the paper with a historical account of results for feedback vertex set on undirected graphs of bounded degree.

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arXiv2026

A Faster Deterministic Algorithm for Kidney Exchange via Representative Set

The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchange -- comprising cycles or chains starting from altruistic donors -- such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. Due to constraints in medical resources, some limits are often imposed on the lengths of these cycles and chains. These exchanges create a network of transplants aimed at maximizing the total number, $t$, of successful transplants. Recently, this problem was deterministically solved in $O^*(14.34^t)$ time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in $O^*(6.855^t)$ time.

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arXiv2026

Faster Parameterized Vertex Multicut

In the {\sc Vertex Multicut} problem the input consists of a graph $G$, integer $k$, and a set $\mathbf{T} = \{(s_1, t_1), \ldots, (s_p, t_p)\}$ of pairs of vertices of $G$. The task is to find a set $X$ of at most $k$ vertices such that, for every $(s_i, t_i) \in \mathbf{T}$, there is no path from $s_i$ to $t_i$ in $G - X$. Marx and Razgon [STOC 2011 and SICOMP 2014] and Bousquet, Daligault, and Thomass\'{e} [STOC 2011 and SICOMP 2018] independently and simultaneously gave the first algorithms for {\sc Vertex Multicut} with running time $f(k)n^{O(1)}$. The running time of their algorithms is $2^{O(k^3)}n^{O(1)}$ and $2^{O(k^{O(1)})}n^{O(1)}$, respectively. As part of their result, Marx and Razgon introduce the {\em shadow removal} technique, which was subsequently applied in algorithms for several parameterized cut and separation problems. The shadow removal step is the only step of the algorithm of Marx and Razgon which requires $2^{O(k^3)}n^{O(1)}$ time. Chitnis et al. [TALG 2015] gave an improved version of the shadow removal step, which, among other results, led to a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Vertex Multicut}. We give a faster algorithm for the {\sc Vertex Multicut} problem with running time $k^{O(k)}n^{O(1)}$. Our main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time. The new shadow removal step implies a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Directed Subset Feedback Vertex Set} and a $k^{O(k)}n^{O(1)}$ time algorithm for {\sc Directed Multiway Cut}, improving over the previously best known algorithms of Chitnis et al. [TALG 2015].

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arXiv2026

The Complexity of Tournament Fixing: Subset FAS Number and Acyclic Neighborhoods

The \textsc{Tournament Fixing Problem} (TFP) asks whether a knockout tournament can be scheduled to guarantee that a given player $v^*$ wins. Although TFP is NP-hard in general, it is known to be \emph{fixed-parameter tractable} (FPT) when parameterized by the feedback arc/vertex set number, or the in/out-degree of $v^*$ (AAAI 17; IJCAI 18; AAAI 23; AAAI 26). However, it remained open whether TFP is FPT with respect to the \emph{subset FAS number of $v^*$} -- the minimum number of arcs intersecting all cycles containing $v^*$ -- a parameter that is never larger than the aforementioned ones (AAAI 26). In this paper, we resolve this question negatively by proving that TFP stays NP-hard even when the subset FAS number of $v^*$ is constant $\geq 1$ and either the subgraph induced by the in-neighbors $D[N_{\mathrm{in}}(v^*)]$ or the out-neighbors $D[N_{\mathrm{out}}(v^*)]$ is acyclic. Conversely, when both $D[N_{\mathrm{in}}(v^*)]$ and $D[N_{\mathrm{out}}(v^*)]$ are acyclic, we show that TFP becomes FPT parameterized by the subset FAS number of $v^*$. Furthermore, we provide sufficient conditions under which $v^*$ can win even when this parameter is unbounded.

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arXiv2026

Improved Approximations for Dial-a-Ride Problems

The multi-vehicle dial-a-ride problem (mDaRP) is a fundamental vehicle routing problem with pickups and deliveries, widely applicable in ride-sharing, economics, and transportation. Given a set of $n$ locations, $h$ vehicles of identical capacity $\lambda$ located at various depots, and $m$ ride requests each defined by a source and a destination, the goal is to plan non-preemptive routes that serve all requests while minimizing the total travel distance, ensuring that no vehicle carries more than $\lambda$ passengers at any time. The best-known approximation ratio for the mDaRP remains $\mathcal{O}(\sqrt{\lambda}\log m)$. We propose two simple algorithms: the first achieves the same approximation ratio of $\mathcal{O}(\sqrt{\lambda}\log m)$ with improved running time, and the second attains an approximation ratio of $\mathcal{O}(\sqrt{\frac{m}{\lambda}})$. A combination of them yields an approximation ratio of $\mathcal{O}(\sqrt[4]{n}\log^{\frac{1}{2}}n)$ under $m=\Theta(n)$. Moreover, for the case $m\gg n$, by extending our algorithms, we derive an $\mathcal{O}(\sqrt{n\log n})$-approximation algorithm, which also improves the current best-known approximation ratio of $\mathcal{O}(\sqrt{n}\log^2n)$ for the classic (single-vehicle) DaRP, obtained by Gupta et al. (ACM Trans. Algorithms, 2010).

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arXiv2026

Improved Approximations for the Unsplittable Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. In this problem, we are given a depot and a set of customers, each with a demand, embedded in a metric space. The objective is to find a set of tours, each starting and ending at the depot, operated by the capacititated vehicle at the depot to serve all customers, such that all customers are served, and the total travel cost is minimized. We consider the unplittable variant, where the demand of each customer must be served entirely by a single tour. Let $\alpha$ denote the current best-known approximation ratio for the metric traveling salesman problem. The previous best approximation ratio was $\alpha+1+\ln 2+\delta<3.1932$ for a small constant $\delta>0$ (Friggstad et al., Math. Oper. Res. 2025), which can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023). In this paper, we propose two improved approximation algorithms. The first algorithm focuses on the case of fixed vehicle capacity and achieves an approximation ratio of $\alpha+1+\ln\bigl(2-\frac{1}{2}y_0\bigr)<3.0897$, where $y_0>0.39312$ is the unique root of $\ln\bigl(2-\frac{1}{2}y\bigr)=\frac{3}{2}y$. The second algorithm considers general vehicle capacity and achieves an approximation ratio of $\alpha+1+y_1+\ln\left(2-2y_1\right)+\delta<3.1759$ for a small constant $\delta>0$, where $y_1>0.17458$ is the unique root of $\frac{1}{2} y_1+ 6 (1-y_1)\bigl(1-e^{-\frac{1}{2} y_1}\bigr) =\ln\left(2-2y_1\right)$. Both approximations can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023).

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arXiv2026

How Hard Is It to Rig a Tournament When Few Players Can Beat or Be Beaten by the Favorite?

In knockout tournaments, players compete in successive rounds, with losers eliminated and winners advancing until a single champion remains. Given a tournament digraph $D$, which encodes the outcomes of all possible matches, and a designated player $v^* \in V(D)$, the \textsc{Tournament Fixing} problem (TFP) asks whether the tournament can be scheduled in a way that guarantees $v^*$ emerges as the winner. TFP is known to be NP-hard, but is fixed-parameter tractable (FPT) when parameterized by structural measures such as the feedback arc set (fas) or feedback vertex set (fvs) number of the tournament digraph. In this paper, we introduce and study two new structural parameters: the number of players who can defeat $v^*$ (i.e., the in-degree of $v^*$, denoted by $k$) and the number of players that $v^*$ can defeat (i.e., the out-degree of $v^*$, denoted by $\ell$). A natural question is that: can TFP be efficiently solved when $k$ or $\ell$ is small? We answer this question affirmatively by showing that TFP is FPT when parameterized by either the in-degree or out-degree of $v^*$. Our algorithm for the in-degree parameterization is particularly involved and technically intricate. Notably, the in-degree $k$ can remain small even when other structural parameters, such as fas or fvs, are large. Hence, our results offer a new perspective and significantly broaden the parameterized algorithmic understanding of the \textsc{Tournament Fixing} problem.

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arXiv2026

Improved Approximation Algorithms for the Multiple-Depot Split Delivery Vehicle Routing Problem

The Multiple-Depot Split Delivery Vehicle Routing Problem (MD-SDVRP) is a challenging problem with broad applications in logistics. The goal is to serve customers' demand using a fleet of capacitated vehicles located in multiple depots, where each customer's demand can be served by more than one vehicle, while minimizing the total travel cost of all vehicles. We study approximation algorithms for this problem. Previously, the only known result was a $6$-approximation algorithm for a constant number of depots (INFORMS J. Comput. 2023), and whether this ratio could be improved was left as an open question. In this paper, we resolve it by proposing a $(6-2\cdot 10^{-36})$-approximation algorithm for this setting. Moreover, we develop constant-factor approximation algorithms that work beyond a constant number of depots, improved parameterized approximation algorithms related to the vehicle capacity and the number of depots, as well as bi-factor approximation algorithms.

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arXiv2025

An Improved Approximation Algorithm for Maximum Weight 3-Path Packing

Given a complete graph with $n$ vertices and non-negative edge weights, where $n$ is divisible by 3, the maximum weight 3-path packing problem is to find a set of $n/3$ vertex-disjoint 3-paths such that the total weight of the 3-paths in the packing is maximized. This problem is closely related to the classic maximum weight matching problem. In this paper, we propose a $10/17$-approximation algorithm, improving the best-known $7/12$-approximation algorithm (ESA 2015). Our result is obtained by making a trade-off among three algorithms. The first is based on the maximum weight matching of size $n/2$, the second is based on the maximum weight matching of size $n/3$, and the last is based on an approximation algorithm for star packing. Our first algorithm is the same as the previous $7/12$-approximation algorithm, but we propose a new analysis method -- a charging method -- for this problem, which is not only essential to analyze our second algorithm but also may be extended to analyze algorithms for some related problems.

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arXiv2025

An Improved Approximation Algorithm for the Capacitated Arc Routing Problem

The Capacitated Arc Routing Problem (CARP), introduced by Golden and Wong in 1981, is an important arc routing problem in Operations Research, which generalizes the famous Capacitated Vehicle Routing Problem (CVRP). When every customer has a unit demand, the best known approximation ratio for CARP, given by Jansen in 1993, remains $\frac{5}{2}-\frac{1.5}{k}$, where $k$ denotes the vehicle capacity. Based on recent progress in approximating CVRP, we improve this result by proposing a $(\frac{5}{2}-Θ(\frac{1}{\sqrt{k}}))$-approximation algorithm, which to the best of our knowledge constitutes the first improvement over Jansen's bound.

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arXiv2025

Strategyproofness and Monotone Allocation of Auction in Social Networks

Strategyproofness in network auctions requires that bidders not only report their valuations truthfully, but also do their best to invite neighbours from the social network. In contrast to canonical auctions, where the value-monotone allocation in Myerson's Lemma is a cornerstone, a general principle of allocation rules for strategyproof network auctions is still missing. We show that, due to the absence of such a principle, even extensions to multi-unit network auctions with single-unit demand present unexpected difficulties, and all pioneering researches fail to be strategyproof. For the first time in this field, we identify two categories of monotone allocation rules on networks: Invitation-Depressed Monotonicity (ID-MON) and Invitation-Promoted Monotonicity (IP-MON). They encompass all existing allocation rules of network auctions as specific instances. For any given ID-MON or IP-MON allocation rule, we characterize the existence and sufficient conditions for the strategyproof payment rules, and show that among all such payment rules, the revenue-maximizing one exists and is computationally feasible. With these results, the obstacle of combinatorial network auction with single-minded bidders is now resolved.

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arXiv2025

New Algorithms for #2-SAT and #3-SAT

The #2-SAT and #3-SAT problems involve counting the number of satisfying assignments (also called models) for instances of 2-SAT and 3-SAT, respectively. In 2010, Zhou et al. proposed an $\mathcal{O}^*(1.1892^m)$-time algorithm for #2-SAT and an efficient approach for #3-SAT, where $m$ denotes the number of clauses. In this paper, we show that the weighted versions of #2-SAT and #3-SAT can be solved in $\mathcal{O}^*(1.1082^m)$ and $\mathcal{O}^*(1.4423^m)$ time, respectively. These results directly apply to the unweighted cases and achieve substantial improvements over the previous results. These advancements are enabled by the introduction of novel reduction rules, a refined analysis of branching operations, and the application of path decompositions on the primal and dual graphs of the formula.

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arXiv2025

Contracting with a Mechanism Designer

This paper explores the economic interactions within modern crowdsourcing markets. In these markets, employers issue requests for tasks, platforms facilitate the recruitment of crowd workers, and workers complete tasks for monetary rewards. Recognizing that these roles serve distinct functions within the ecosystem, we introduce a three-party model that distinguishes among the principal (the requester), the intermediary (the platform), and the pool of agents (the workers). The principal, unable to directly engage with agents, relies on the intermediary to recruit and incentivize them. This interaction unfolds in two stages: first, the principal designs a profit-sharing contract with the intermediary; second, the intermediary implements a mechanism to select an agent to complete the delegated task. We analyze the proposed model as an extensive-form Stackelberg game. Our contributions are threefold. First, we fully characterize the subgame perfect equilibrium of our model. In particular, the principal's contract design problem can be represented as virtual value pricing, a novel auction-theoretic formulation. We identify the optimality of linear contracts, even when the task has multiple outcomes and agents' cost distributions are asymmetric. Second, to quantify the principal's utility loss from delegation and information asymmetry, we introduce the price of double marginalization (PoDM) and the classical price of anarchy (PoA). We derive tight or nearly tight bounds on both ratios under regular and monotone hazard rate distributions. Finally, we extend our analysis to two natural variants of the base model: (i) the intermediary is restricted to anonymous pricing mechanisms, and (ii) the principal lacks precise information about the market size.

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arXiv2025

Approximation Algorithms for the Cumulative Vehicle Routing Problem with Stochastic Demands

In the Cumulative Vehicle Routing Problem (Cu-VRP), we need to find a feasible itinerary for a capacitated vehicle located at the depot to satisfy customers' demand, as in the well-known Vehicle Routing Problem (VRP), but the goal is to minimize the cumulative cost of the vehicle, which is based on the vehicle's load throughout the itinerary. If the demand of each customer is unknown until the vehicle visits it, the problem is called Cu-VRP with Stochastic Demands (Cu-VRPSD). Assume that the approximation ratio of metric TSP is $1.5$. In this paper, we propose a randomized $3.456$-approximation algorithm for Cu-VRPSD, improving the best-known approximation ratio of $6$ (Discret. Appl. Math. 2020). Since VRP with Stochastic Demands (VRPSD) is a special case of Cu-VRPSD, as a corollary, we also obtain a randomized $3.25$-approximation algorithm for VRPSD, improving the best-known approximation ratio of $3.5$ (Oper. Res. 2012). For Cu-VRP, we give a randomized $3.194$-approximation algorithm, improving the best-known approximation ratio of $4$ (Oper. Res. Lett. 2013). Moreover, if each customer is allowed to be satisfied by using multiple tours, we obtain further improvements for Cu-VRPSD and Cu-VRP.

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