Clustering is a fundamental unsupervised learning problem with lots of applications in data mining, image classification, and other fields. Although clustering algorithms for k-means and k-median are widely used due to their simplicity and effectiveness on point-based data, they often perform poorly on structured data such as lines, graphs, and time series, where point-based representations fail to capture the underlying structure. Moreover, existing algorithms for structured data typically do not account for fairness constraints, which are increasingly important in modern applications involving sensitive attributes such as gender, race, or user groups. In this paper, we formally introduce the group fair k-median of lines problem (Gf-k-Ml), a new variant of k-median that integrates fairness constraints into the clustering of structured data represented by lines. Given a set L of n lines in ℝ^d , partitioned into t disjoint color groups L_1, … , L_t , the goal of the Gf-k-Ml problem is to partition L into k clusters such that the proportion of lines from each color group in each cluster remains within a specified range, and the sum of distances over each line to its assigned center is minimized. We introduce a group-wise coreset construction algorithm that computes a separate coreset for each group obtained by partitioning the input lines according to sensitive attributes, and prove that these groups satisfy composability under fairness constraints. Our main result is a coreset that satisfies the fairness constraint and has a size of O( td^2klog ^2klog ^2n/ε ^2) with an error parameter ε∈ (0,1) , which can be constructed in nearly linear time with respect to n.
In this paper, we study the problem of testing some first-order logic properties on sparse graphs under the adjacency list model, including k-dominating set property, k-vertex cover property and diameter ≤ k property. (1) For the k-dominating set property, we give a tester with query complexity O( (kC)^k+1/ε ^k+2log( kC/ε) ) on n-vertex graphs with at most Cn edges. Furthermore, if the input graph is planar, we improve the query complexity to O( k^6/ε ^3log (k^2/ε)) . (2) For the k-vertex cover property, we give a tester whose query complexity is O( α ^3k^2log k/ε ^2) on graphs with arboricity bounded by α . Previously, these properties were known to be testable with constant query complexity on general graphs under a stronger model with random edge sampling queries. By leveraging edge sampling simulation techniques, one can achieve poly(log n) query complexity in the adjacency model for bounded arboricity graphs. In contrast, our algorithms achieve constant query complexity (for fixed ε and k) on a broader class of sparse graphs or the same class of bounded arboricity graphs. (3) For the diameter ≤ k property, we can distinguish whether a sparse graph has a diameter at most k or is ε -far from any graph that has a diameter at most k+2 with query complexity O( C^⌊ k/2⌋ +1/ε ^⌊ k/2⌋ +2) . Previously, a tester was known for general graphs that distinguishes between having a diameter at most k and being ε -far from any graph with diameter at most β (k) , where β (k) ranges from k+4 to 4k+2 . We improve the upper bound on β (k) to at most k+2 for sparse graphs, though at the cost of slightly higher query complexity.
In this paper, we delve into the online budget allocation maximization (BAM) problem on two uniform machines. Given two uniform machines, M_1 and M_2 , with speeds s≥ 1 and 1, respectively, all jobs 𝒥={J_1,⋯ ,J_n} are ordered in a list and arrive one by one. Each job J_j has a size p_j and a common due date d for j=1,⋯ ,n . The objective is to determine a schedule that maximizes the total size of jobs processed by these two machines before the due date. For this problem, we propose an online algorithm with a competitive ratio of (√(s^2+10s+9)-s-1)/2 , which is bounded by a constant value of 2 and reaches √(5)-1 when s=1 . For a special case, known as the BAM _≤ 1 problem, we devise an online algorithm with a competitive ratio of (√(9s^2+10s+1)-s-1)/2s , which is no greater than √(5)-1 .
The rapid development of cloud computing, artificial intelligence and Internet of Things has not only led to high-speed growth of data centers, but also put forward a higher level of demand for data centers. Data center networks (DCNs), as crucial components of data centers, play significant roles in the overall operation and function realization of data centers. As a server-centric DCN, BCube connected crossbars (BCCC) offer excellent network performance in terms of great scalability, low communication latency and high robustness to component failures. As the scale of BCCC increases, so does the likelihood of server failures, it is critical to identify and replace faulty servers promptly for network reliability. In this work, we first investigate the intermittent fault diagnosability and local diagnosability of BCCC(n, k). On this basis, we also determine the traditional diagnosability of BCCC(n, k) and the fact that BCCC(n, k) has strong local diagnosability under the PMC model and MM* model. Subsequently, we further obtain that the faulty BCCC(n, k) with min{n- 1, k - 1} missing edges can still maintain strong local diagnosability under the PMC model. In addition, we present a corresponding local diagnosis algorithm to identify the vertex state and evaluate its performance by simulation experiments, which show that the algorithm keeps good diagnosis correctness even though the number of faulty vertices in BCCC(n, k) reaches 30%.
This paper presents an adaptive weighting-based local search (AWLS) algorithm for solving the vehicle routing problem with time windows (VRPTW), focusing on minimizing the number of routes. AWLS improves the ejection pool framework by adopting an adaptive weighting technique to diversify the search. First, it adjusts the penalties on customers when both inserting and ejecting them into and from the routes to prevent frequently moving the same customer. Second, we introduce route weighting in the route repairing procedure to encourage paying more attention to corrupted routes that persist for a long time. Third, we design a new objective function for identifying the best set of customers to eject, along with several acceleration strategies. Experimental results on 300 Gehring and Homberger's benchmarks show that our AWLS algorithm solves 247 instances to optimality, matches the state-ofthe-art route minimization algorithms in significantly less runtime, and improves the best-known results for two instances in the literature.
The influence maximization problem studies how to select a set of nodes as initial seeds in a social network to maximize their influence. A dual problem of the influence maximization problem is the target set selection problem that asks for a minimum-cardinality seed set that can influence all users in the social network. In this paper, we consider the decision problem of deciding if we can select at most k seeds to influence at least t users in expectation, which formulates both the influence maximization problem and the target set selection problem. We study the parameterized complexity of this decision problem and consider the two most studied diffusion models: the independent cascade model (IC) and the linear threshold model (LT). We show that the problem is W[1]-hard under both models even if both k and t are given as parameters. For the special case with t being the number of the vertices in the network (which coincides with the target set selection problem), we show that the problem under the IC model is polynomial-time solvable, and the problem under the LT model, known to be NP-hard, is fixed-parameter tractable parameterized by k.
Comparing genomes based on gene order is a classical combinatorial optimization problem in computational biology, which seeks the minimum number of genome rearrangement operations required to transform one genome into another. The problem of sorting genomes by translocations has been extensively studied over the past few decades. Computing the translocation distance is NP-hard when the input genomes are unsigned, posing significant computational challenges. A widely adopted approach to approximating this problem involves decomposing the breakpoint graph into proper alternating cycles. However, this decomposition step becomes a bottleneck in calculating the corresponding rearrangement distances, hindering the ability to achieve approximation factors better than 1.375 in polynomial time. In this paper, we propose a novel FPT (fixed-parameter tractable) approximation algorithm for the problem of sorting genomes by translocations, improving the approximation factor to 4/3 + ε , thereby surpassing the long-standing best ratio of 1.375, which has held since 2016 [12]. Our algorithm employs a new randomized method for decomposing the breakpoint graph, which succeeds with high probability, 1 - 1/e^O(n) , as guaranteed by the Chernoff Bound. The time complexity of the algorithm is O(2^d^*· n^O(1/ϵ)) , where n represents the length of each genome and d^* denotes the optimal translocation distance.
It is widely recognized that structural variation represents a significant source of genetic variation. As a well-known type of structural variation, reversion is studied drastically by both biologists and computer scientists. Recently, scientists have found that repetitive sequences always appear at the ends of the segment where the reversal occurred on the chromosome, which has inspired new interests in models for sorting unsigned chromosomes by symmetric reversals (abbreviated MUSR). The problem of MUSR asks for a minimum number of unsigned symmetric reversals to transform a chromosome S into another chromosome T, and requires symmetric reversals to be performed on the segment flanked by the same letter. In this paper, we show that MUSR is NP-hard through an intricate reduction from the MAX-(3,B2)-SAT problem. Moreover, we provide an innovative depiction of the optimal solution and then develop an improved approximation algorithm that ensures the approximation factor of 2 ln3+7/8 (approximately 1.15) and a time complexity of O(n(2)).
Motivated by applications in cloud computing, this paper studies a hybrid problem that combines the Parallel Two-stage Flowshop Scheduling problem with the Bin Packing problem, referred to as the Two-stage Bin Packing problem. Given a sequence of two-stage jobs, the problem aims to pack them into the minimum number of two-stage flowshops such that the completion time of each flowshop does not exceed a given time limit. To our best knowledge, the problem has not been studied before. Recognizing its NP-hardness, we investigate several approximation algorithms. First, we present two online algorithms based on the well-known First-fit and Next-fit strategies, which achieve an absolute approximation ratio 4 with a lower bound 3.166 and a tight asymptotic approximation ratio 4, respectively. We then introduce an algorithm that applies Johnson's Order to each individual flowshop and assigns incoming jobs to flowshops using the First-fit strategy, which is shown to have an asymptotic approximation ratio 3.061 with a lower bound 2.66. Besides, we show that applying Johnson's Order in flowshops cannot improve the Next-fit strategy in terms of approximation ratio.
In this work, we study the coding sequence design problem, which involves designing a coding sequence to encode a given amino acid sequence by optimizing both its secondary structure stability and codon usage. The structural stability and codon usage are quantified by minimum free energy and codon adaptation index, respectively. The coding sequence design problem is important since it has significant potential for the development of mRNA-based vaccines. Previously, we proposed an 𝒪(L^3) time and 𝒪(L^2) space dynamic programming algorithm to solve the coding sequencing design problem, where L is the length of the coding sequence to be designed. In this study, we utilize the sparsification technique to further reduce the time complexity of this dynamic programming algorithm from 𝒪(L^3) to 𝒪(L^2+ZP) for the problem under the base pair-based energy model, where Z and P are two sparsity parameters satisfying Z≤ L (6+P) and P≤ 36 L .
Recommender systems utilize Graph Neural Networks (GNNs) to learn vectorized representations of users and items from user-item interactions for predicting recommendations. Recent methods improve recommendations by incorporating item-related entities through a technique known as the Collaborative Knowledge Graph (CKG). However, the theoretical foundation of entity integration remains underexplored, leading to unresolved challenges in maintaining two critical properties for GNN-based recommender models: Local Consistency and the Inclusion of Indispensable Entities. This paper addresses two key research questions: (1) Do CKG-based models align well with these requirements? (2) Can an alternative graph structure better integrate entities into recommender systems? To answer these questions, we analyze CKG-based models and prove their fundamental limitation: they fail to simultaneously satisfy both properties. To resolve this issue, we propose a novel graph structure, the Fusion Graph (FG). We prove a theorem that demonstrates FG-based models meet the requirements of recommender systems. The source code is available at https://github.com/wangyifeibeijing/FGN.
We study the fair allocation of graphical resources, where the resources are the vertices in a graph. Upon receiving a set of resources, an agent's utility equals the weight of a maximum matching in the induced subgraph. We care about maximin share (MMS) fairness and envyfreeness up to one item (EF1). Regarding MMS fairness, the problem does not admit a finite approximation ratio for heterogeneous agents. For homogeneous agents, we design polynomial-time constant-approximation algorithms, and also note that significant amount of social welfare is sacrificed inevitably in order to ensure (approximate) MMS fairness. We then consider EF1 allocations whose existence is guaranteed. However, the social welfare guarantee of EF1 allocations cannot be better than 1/n for the general case, where.n is the number of agents. Fortunately, for three special cases, two-agent, binary-weight and homogeneous-agent, we are able to design polynomial-time algorithms that also ensure a constant fraction of the maximum social welfare.
In this paper, we consider a bilevel adversarial scheduling problem on parallel machines. Given a set of jobs that the leader has to select some jobs within the leader’s budget. Then, the follower next schedules these jobs to minimize the makespan. The goal is to select the jobs so that the optimal (minimum) value of the makespan is maximum. We design a simple (2+ ε )-approximation algorithm and a polynomial time approximation scheme (PTAS) for this problem. We also propose a simple efficient polynomial time approximation scheme (EPTAS) for this problem when the number of machines is fixed.
Motivated by the challenge of maximizing connected coverage with limited UAVs in communication networks, we address the problem within a graph network framework G=(V,E) , where V represents potential UAV deployment positions and E denotes communication links between nodes. A utility function f: 2^V→ℝ_+ is defined to characterize coverage efficiency. Under the constraint of limited field-of-view (FoV) UAVs, the objective is to identify a subset S⊆ V with |S|≤ K that maximizes f(S) while ensuring the induced subgraph G[S] remains connected. We formulate this as the Maximum Connected Submodular function with Cardinality constraint (MCSC) problem and propose a 1-e^-1/2√(K-1)+5 -approximation algorithm, leveraging a novel tree decomposition technique. Additionally, we present a bicriteria ( (1-e^-1)α/2√(K)+3α,α ^2) -approximation algorithm for the problem, where α >1 is a constant. For a special case of the MCSC problem, where the submodular utility exhibits partial additivity when subsets are sufficiently far apart, we define the Maximum Connected h-Hop Submodular function with a Cardinality constraint (MCHSC) problem. We provide an approximation algorithm with a ratio of (1-2ε )( 1-e^-1/5(h+1)+1-δ) when K > 25h(h+1) -5 , where ε , δ are small positive constants and h captures the partial additivity property.
A (k]-tuple common substring (abbr. (k]-CSS) is a sequence of at most k common substrings of two or more strings. A longest (k]-CSS of two strings is known retrievable in quadratic time and linear space and even more, in subquadratic time and space if k is a constant. Motivated by computational biology applications in need of a (k]-CSS with designated number of consecutively matching letters, we propose to find a longest (k]-CSS of two strings whose substrings are of length within [l(1), l(2)]. We present a sliding window based dynamic programming algorithm to find such a longest (k]-CSS of two strings whose lengths are n(1) and n(2) in O(kn(1)n(2)) time and space, the same complexity as without the length bounds l(1) and l(2). Through rolling array based dynamic programming to get the longest (k]-CSS length in advance, we present a divide-and-conquer algorithm to find such a longest (k]-CSS in O(kn(1)n(2)) time and O(n(1) + kl(2)n(2)) space, which is intended to work for two much longer given strings. We also present an algorithm to find such a longest (2]-CSS in O(n log(2) n) time where n is the total length of input strings.
Partial Maximum Satisfiability (PMS) is a generalisation of the well-known Maximum Satisfiability (MaxSAT), incorporating both hard and soft clauses. Weighted Partial Maximum Satisfiability (WPMS) further extends PMS by associating each soft clause with a positive integer weight. WPMS is particularly significant in practical applications, as it can encode numerous industrial optimisation problems involving hard constraints and soft constraints with varying priorities. Stochastic local search (SLS) algorithms have been extensively studied for solving WPMS, which has achieved significant advancements in recent years. In this work, we identify two issues in current SLS solvers and propose a corresponding solution. Firstly, we observe that current SLS solvers typically employ a fixed initialisation procedure at the start of each local search round, which may restrict the diversity of search directions. Secondly, current SLS solvers often fail to effectively utilise historical information. To address these issues, we propose a novel clause initialisation method that dynamically adjusts the weights of soft clauses based on both the current search state and historical information. Based on this method, we develop a new SLS solver for WPMS named HistLS. Extensive experiments on WPMS benchmarks from the incomplete track of MaxSAT Evaluations (MSEs) of the five recent years demonstrate that HistLS outperforms state-of-the-art SLS solvers.
Completely independent spanning trees play an important role in security protection routing and enhancing the robustness of ad hoc networks. In a graph G, a collection of k spanning trees is termed completely independent spanning trees if the paths connecting any pair of vertices x and y in these distinct trees are internally disjoint. Fan (1984) [10] proposed a sufficient condition for the existence of Hamilton cycles. Based on Fan's result, we derive a sufficient condition for the existence of two CISTs and identify the graphs that satisfy Fan's condition but do not contain two CISTs.
We study Pareto optimal matching under multilayer preferences, where each agent has more than one preference list with each list representing a criterion based on which the agents of the opposite side are evaluated. We introduce four intuitive concepts of Pareto optimality with multilayer preferences and study parameterized complexity of them. We obtain W[1]-hardness, W[2]-hardness and para-NP-hardness results for most parameters except n, the number of men/women. Although n is FPT, we show that O* (n!) time algorithm is essentially optimal for most of the concepts. In addition, almost no concept admits polynomial kernels with respect to n. These results even hold for combined parameters. We then consider cases where preferences satisfy certain desirable properties, that is, uniformity, single-layer and master list. We show that if preferences are uniform or single-layered, all of them are simply trivial and can be determined in polynomial time. However, the problem soon becomes NP-hard even when the maximum Hamming distance of preferences is a constant. For the case of master list, we find that even when there are only three layers and the preference lists on each side are all derived from the same single master list, the four problems remain NP-hard.
The two-stage stochastic facility location problem (2-SFLP) involves selecting initial facility locations under uncertainty, given known probabilities for each demand scenario. After the actual demand scenario is realized, additional facilities in the second stage, which incur higher costs, may be added to reduce the overall expected cost, including both opening and connection expenses. We present an improved per-scenario bound of.2.322 for 2-SFLP using the LP-rounding algorithm from prior work. By introducing the integrated distance estimation technique, we offer a more refined analysis. This technique, which involves a detailed estimation through a non-negative linear combination of the maximum and average distances within the neighborhood of an arbitrary clientscenario pair, has potential applications for the analysis of other facility location problems.
