arXiv subjects

Xiao Mao

Publications and source records attributed to Xiao Mao.

14 recordsLinked to original sources

arXiv2026

Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time

We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $\epsilon>0$, compute a $(1+\epsilon)$-approximation for ED and a $(1-\epsilon)$-approximation for LCS in time $n^2 / 2^{\log^{\Omega(1)}(n)}$ for two strings of total length at most $n$. This running time improves upon the classical quadratic-time dynamic programming algorithms by a quasi-polynomial factor. Our results yield significant insights into fine-grained complexity: Firstly, for ED, prior work indicates that any exact algorithm cannot be improved beyond a few logarithmic factors without refuting established complexity assumptions [Abboud, Hansen, Vassilevska Williams, Williams, 2016]; our quasi-polynomial speed-up shows a separation the complexity of approximate ED from that of exact ED, even for approximation factor arbitrarily close to $1$. Secondly, for LCS, obtaining similar approximation-time tradeoffs via deterministic algorithms would imply breakthrough circuit lower bounds [Chen, Goldwasser, Lyu, Rothblum, Rubinstein, 2019]; our randomized algorithm demonstrates derandomization hardness for LCS approximation.

cs.DS↗

arXiv2026

Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection

We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation $\odot: S\times S\to S$ in optimal time $O(|S|^2)$, they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations $\odot,\oplus: S\times S\to S$. Our results are as follows: * We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time $O(|S|^\omega)$, together with a matching conditional lower bound based on the Triangle Detection Hypothesis. * We propose arithmetic progression detection in small universes as a consequential algorithmic challenge: We show that unless we can detect $4$-term arithmetic progressions in a set $X\subseteq\{1,\dots, N\}$ in time $O(N^{2-\epsilon})$, then (a) the 3-uniform 4-hyperclique hypothesis is true, and (b) verifying certain identities requires running time~$|S|^{3-o(1)}$. * A careful combination of our algorithmic and hardness ideas allows us to \emph{fully classify} a natural subclass of identities: Specifically, any 3-variable identity over binary operations in which no side is a subexpression of the other is either: (1) verifiable in randomized time $O(|S|^2)$, (2) verifiable in randomized time $O(|S|^\omega)$ with a matching lower bound from triangle detection, or (3) trivially verifiable in time $O(|S|^3)$ with a matching lower bound from hardness of 4-term arithmetic progression detection. * We obtain near-optimal algorithms for verifying whether a given algebraic structure forms a field or ring, and show that \emph{counting} the number of distributive triples is conditionally harder than verifying distributivity.

cs.DS↗

arXiv2026

CAMO: A Conditional Neural Solver for the Multi-objective Multiple Traveling Salesman Problem

Robotic systems often require a team of robots to collectively visit multiple targets while optimizing competing objectives, such as total travel cost and makespan. This setting can be formulated as the Multi-Objective Multiple Traveling Salesman Problem (MOMTSP). Although learning-based methods have shown strong performance on the single-agent TSP and multi-objective TSP variants, they rarely address the combined challenges of multi-agent coordination and multi-objective trade-offs, which introduce dual sources of complexity. To bridge this gap, we propose CAMO, a conditional neural solver for MOMTSP that generalizes across varying numbers of targets, agents, and preference vectors, and yields high-quality approximations to the Pareto front (PF). Specifically, CAMO consists of a conditional encoder to fuse preferences into instance representations, enabling explicit control over multi-objective trade-offs, and a collaborative decoder that coordinates all agents by alternating agent selection and node selection to construct multi-agent tours autoregressively. To further improve generalization, we train CAMO with a REINFORCE-based objective over a mixed distribution of problem sizes. Extensive experiments show that CAMO outperforms both neural and conventional heuristics, achieving a closer approximation of PFs. In addition, ablation results validate the contributions of CAMO's key components, and real-world tests on a mobile robot platform demonstrate its practical applicability.

cs.RO↗

arXiv2026

A Faster Directed Single-Source Shortest Path Algorithm

This paper presents a new deterministic algorithm for single-source shortest paths (SSSP) on real non-negative edge-weighted directed graphs, with running time $O(m\sqrt{\log n}+\sqrt{mn\log n\log \log n})$, which is $O(m\sqrt{\log n\log \log n})$ for sparse graphs. This improves the recent breakthrough result of $O(m\log^{2/3} n)$ time for directed SSSP algorithm [Duan, Mao, Mao, Shu, Yin 2025].

cs.DS↗

arXiv2025

Improving After-sales Service: Deep Reinforcement Learning for Dynamic Time Slot Assignment with Commitments and Customer Preferences

Problem definition: For original equipment manufacturers (OEMs), high-tech maintenance is a strategic component in after-sales services, involving close coordination between customers and service engineers. Each customer suggests several time slots for their maintenance task, from which the OEM must select one. This decision needs to be made promptly to support customers' planning. At the end of each day, routes for service engineers are planned to fulfill the tasks scheduled for the following day. We study this hierarchical and sequential decision-making problem-the Dynamic Time Slot Assignment Problem with Commitments and Customer Preferences (DTSAP-CCP)-in this paper. Methodology/results: Two distinct approaches are proposed: 1) an attention-based deep reinforcement learning with rollout execution (ADRL-RE) and 2) a scenario-based planning approach (SBP). The ADRL-RE combines a well-trained attention-based neural network with a rollout framework for online trajectory simulation. To support the training, we develop a neural heuristic solver that provides rapid route planning solutions, enabling efficient learning in complex combinatorial settings. The SBP approach samples several scenarios to guide the time slot assignment. Numerical experiments demonstrate the superiority of ADRL-RE and the stability of SBP compared to both rule-based and rollout-based approaches. Furthermore, the strong practicality of ADRL-RE is verified in a case study of after-sales service for large medical equipment. Implications: This study provides OEMs with practical decision-support tools for dynamic maintenance scheduling, balancing customer preferences and operational efficiency. In particular, our ADRL-RE shows strong real-world potential, supporting timely and customer-aligned maintenance scheduling.

cs.LG↗

arXiv2025

$(1-ε)$-Approximation of Knapsack in Nearly Quadratic Time

Knapsack is one of the most fundamental problems in theoretical computer science. In the $(1 - ε)$-approximation setting, although there is a fine-grained lower bound of $(n + 1 / ε) ^ {2 - o(1)}$ based on the $(\min, +)$-convolution hypothesis ([K{ü}nnemann, Paturi and Stefan Schneider, ICALP 2017] and [Cygan, Mucha, Wegrzycki and Wlodarczyk, 2017]), the best algorithm is randomized and runs in $\tilde O\left(n + (\frac{1}ε)^{11/5}/2^{Ω(\sqrt{\log(1/ε)})}\right)$ time [Deng, Jin and Mao, SODA 2023], and it remains an important open problem whether an algorithm with a running time that matches the lower bound (up to a sub-polynomial factor) exists. We answer the question positively by showing a deterministic $(1 - ε)$-approximation scheme for knapsack that runs in $\tilde O(n + (1 / ε) ^ {2})$ time. We first extend a known lemma in a recursive way to reduce the problem to $n ε$-additive approximation for $n$ items with profits in $[1, 2)$. Then we give a simple efficient geometry-based algorithm for the reduced problem.

cs.DS↗

arXiv2025

Breaking the Sorting Barrier for Directed Single-Source Shortest Paths

We give a deterministic $O(m\log^{2/3}n)$-time algorithm for single-source shortest paths (SSSP) on directed graphs with real non-negative edge weights in the comparison-addition model. This is the first result to break the $O(m+n\log n)$ time bound of Dijkstra's algorithm on sparse graphs, showing that Dijkstra's algorithm is not optimal for SSSP.

cs.DS↗

arXiv2024

Fully-Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update Time

The All-Pairs Shortest Paths (APSP) problem is one of the fundamental problems in theoretical computer science. It asks to compute the distance matrix of a given $n$-vertex graph. We revisit the classical problem of maintaining the distance matrix under a fully dynamic setting undergoing vertex insertions and deletions with a fast worst-case running time and efficient space usage. Although an algorithm with amortized update-time $\tilde O(n ^ 2)$ has been known for nearly two decades [Demetrescu and Italiano, STOC 2003], the current best algorithm for worst-case running time with efficient space usage runs is due to [Gutenberg and Wulff-Nilsen, SODA 2020], which improves the space usage of the previous algorithm due to [Abraham, Chechik, and Krinninger, SODA 2017] to $\tilde O(n ^ 2)$ but fails to improve their running time of $\tilde O(n ^ {2 + 2 / 3})$. It has been conjectured that no algorithm in $O(n ^ {2.5 - ε})$ worst-case update time exists. For graphs without negative cycles, we meet this conjectured lower bound by introducing a Monte Carlo algorithm running in randomized $\tilde O(n ^ {2.5})$ time while keeping the $\tilde O(n ^ 2)$ space bound from the previous algorithm. Our breakthrough is made possible by the idea of ``hop-dominant shortest paths,'' which are shortest paths with a constraint on hops (number of vertices) that remain shortest after we relax the constraint by a constant factor.

cs.DS↗

arXiv2023

DL-DRL: A double-level deep reinforcement learning approach for large-scale task scheduling of multi-UAV

Exploiting unmanned aerial vehicles (UAVs) to execute tasks is gaining growing popularity recently. To solve the underlying task scheduling problem, the deep reinforcement learning (DRL) based methods demonstrate notable advantage over the conventional heuristics as they rely less on hand-engineered rules. However, their decision space will become prohibitively huge as the problem scales up, thus deteriorating the computation efficiency. To alleviate this issue, we propose a double-level deep reinforcement learning (DL-DRL) approach based on a divide and conquer framework (DCF), where we decompose the task scheduling of multi-UAV into task allocation and route planning. Particularly, we design an encoder-decoder structured policy network in our upper-level DRL model to allocate the tasks to different UAVs, and we exploit another attention based policy network in our lower-level DRL model to construct the route for each UAV, with the objective to maximize the number of executed tasks given the maximum flight distance of the UAV. To effectively train the two models, we design an interactive training strategy (ITS), which includes pre-training, intensive training and alternate training. Experimental results show that our DL-DRL performs favorably against the learning-based and conventional baselines including the OR-Tools, in terms of solution quality and computation efficiency. We also verify the generalization performance of our approach by applying it to larger sizes of up to 1000 tasks. Moreover, we also show via an ablation study that our ITS can help achieve a balance between the performance and training efficiency.

cs.LG↗

arXiv2023

Approximating Knapsack and Partition via Dense Subset Sums

Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.

cs.DS↗

arXiv2022

On Problems Related to Unbounded SubsetSum: A Unified Combinatorial Approach

Unbounded SubsetSum is a classical textbook problem: given integers $w_1,w_2,\cdots,w_n\in [1,u],~c,u$, we need to find if there exists $m_1,m_2,\cdots,m_n\in \mathbb{N}$ satisfying $c=\sum_{i=1}^n w_im_i$. In its all-target version, $t\in \mathbb{Z}_+$ is given and answer for all integers $c\in[0,t]$ is required. In this paper, we study three generalizations of this simple problem: All-Target Unbounded Knapsack, All-Target CoinChange and Residue Table. By new combinatorial insights into the structures of solutions, we present a novel two-phase approach for such problems. As a result, we present the first near-linear algorithms for CoinChange and Residue Table, which runs in $\tilde{O}(u+t)$ and $\tilde{O}(u)$ time deterministically. We also show if we can compute $(\min,+)$ convolution for $n$-length arrays in $T(n)$ time, then All-Target Unbounded Knapsack can be solved in $\tilde{O}(T(u)+t)$ time, thus establishing sub-quadratic equivalence between All-Target Unbounded Knapsack and $(\min,+)$ convolution.

cs.DS↗

arXiv2021

Breaking the Cubic Barrier for (Unweighted) Tree Edit Distance

The (unweighted) tree edit distance problem for $n$ node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in $O(n ^ 3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007]. The same paper also showed that $O(n ^ 3)$ is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly sub-cubic time under the APSP conjecture [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. In this paper, we break the cubic barrier by showing an $O(n ^ {2.9546})$ time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly sub-cubic time [Bringmann, Grandoni, Saha, and Vassilevska Williams, FOCS 2016].

cs.DS↗

arXiv2021

Shortest non-separating st-path on chordal graphs

Many NP-Hard problems on general graphs, such as maximum independence set, maximal cliques and graph coloring can be solved efficiently on chordal graphs. In this paper, we explore the problem of non-separating st-paths defined on edges: for a connected undirected graph and two vertices, a non-separating path is a path between the two vertices such that if we remove all the edges on the path, the graph remains connected. We show that on general graphs, checking the existence of non-separating st-paths is NP-Hard, but the same problem can be solved in linear time on chordal graphs. In the case that such path exists, we introduce an algorithm that finds the shortest non-separating st-path on a connected chordal graph of $n$ vertices and $m$ edges with positive edge lengths that runs in $O(n\log{n} + m)$ time.

cs.DS↗

arXiv2016

Knapsack problem for automaton groups

The knapsack problem is a classic optimisation problem that has been recently extended in the setting of groups. Its study reveals to be interesting since it provides many different behaviours, depending on the considered class of groups. In this paper we deal with groups generated by Mealy automata-a class that is often used to study group-theoretical conjectures-and prove that the knapsack problem is undecidable for this class. In a second time, we construct a graph that, if finite, provides a solution to the knapsack problem. We deduce that the knapsack problem is decidable for the so-called bounded automaton groups, a class where the order and conjugacy problems are already known to be decidable.

math.GR↗