We consider the classic correlation clustering problem in the hierarchical setting. Given a complete graph $G=(V, E)$ and $\ell$ layers of input information, where the input of each layer consists of a non-negative weight and a labeling of the edges with either + or -, this problem seeks to compute for each layer a partition of V such that the partition for any non-top layer subdivides the partition in the upper-layer and the weighted number of disagreements over the layers is minimized, where the disagreement of a layer is the number of + edges across parts plus the number of - edges within parts. Hierarchical correlation clustering is a natural formulation of the classic problem of fitting distances by ultrametrics, which is further known as numerical taxonomy [1]–[3] in the literature. While single-layer correlation clustering received wide attention since it was introduced in [4] and major progress evolved in the past three years [5]–[8], few is known for this problem in the hierarchical setting [9], [10]. The lack of understanding and adequate tools is reflected in the large approximation ratio known for this problem, which originates from 2021. In this work we make both conceptual and technical contributions towards the hierarchical clustering problem. We present a simple paradigm that greatly facilitates LP-rounding in hierarchical clustering, illustrated with a delicate algorithm providing a significantly improved approximation guarantee of 25.7846 for the hierarchical correlation clustering problem. Our techniques reveal surprising new properties and advances the current understanding for the formulation presented and subsequently used in [9] –[12] for hierarchical clustering over the past two decades. This provides a unifying interpretation on the core-technical problem in hierarchical clustering as the problem of finding cuts with prescribed properties regarding the average distance of certain cut pairs. We further illustrate this perspective by showing that a direct application of the paradigm and techniques presented in this work gives a simple alternative to the state-of-the-art result presented in [12] for the ultrametric violation distance problem. -hierarchical correlation clustering, ultrametric embedding, correlation clustering, linear programming rounding, approximation algorithms

The rapid growth of Internet of Things (IoT) applications has spurred the need for efficient data collection mechanisms. Traditional approaches relying on fixed infrastructure have limitations in coverage, scalability, and deployment costs. Unmanned Aerial Vehicles (UAVs) have emerged as a promising alternative due to their mobility and flexibility. In this paper, we aim to minimize the number of UAVs deployed to collect data in IoT networks while considering a delay budget for energy limitation and data freshness. To this end, we propose a novel 3-approximation dynamic-programming-based algorithm called GPUDA to address the challenges of efficient data collection from IoT devices via UAVs for real-world scenarios where the number of UAVs owned by an individual or organization is unlikely to be excessive, improving the best-known approximation ratio of 4. GPUDA is a geometric partition-based method that incorporates data rounding techniques. The experimental results demonstrate that the proposed algorithm requires 35.01% to 58.55% fewer deployed UAVs than the existing algorithms on average.

The Capacitated Facility Location (CFL), a long-standing classic problem with intriguing approximability and literature dated back to the 90s, is considered. Following the open question posted in [Williamson and Shmoys, 2011] and the notable work due to [An et al., FOCS~2014], we present an LP-based approximation algorithm with a guarantee of $(10+\sqrt{67})/2 \approx 9.0927$, a significant improvement upon the previous LP-based ratio of $288$ due to An et al. in 2014. Our contribution for this part is a simple and elegant rounding algorithm that brings clear insights for the MFN relaxation and the CFL problem. For CFL with cardinality facility cost (CFL-CFC), we present an LP-based $4$-approximation algorithm, which improves upon the decades-old ratio of 5 due to Levi et al. that ages up since 2004. Prior to our work, it was not clear whether or not LP-based methods can be used to provide a guarantee better than 5 for the CFL problem, even for restricted versions of this problem, for which natural LPs are already known to have small integrality gaps. Our rounding algorithm provides the first affirmative answer on the case with cadinality facility cost.

The Multicommodity Flow Network (MFN) relaxation, developed in [An, Singh, Svensson, FOCS 2014], is the only polynomial-time solvable relaxation that is known to provide a bounded integrality gap for the classic capacitated facility location (CFL) problem. The best upper-bound known for the integrality gap of this strong LP relaxation, however, is in the order of 288.

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G=(V,E) with a maximum edge size f, a cost function w: V - > Z^+, and edge subsets P_1,P_2,...,P_r of E along with covering requirements k_1,k_2,...,k_r for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset P_i, at least k_i edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an (f * H_r + H_r)-approximation for this problem, where H_r is the r^{th} harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.

We consider the partial vertex cover problem with hard capacity constraints (Partial VC-HC) on hypergraphs. In this problem we are given a hypergraph G=(V,E) with a maximum edge size f and a covering requirement R. Each edge is associated with a demand, and each vertex is associated with a capacity and an (integral) available multiplicity. The objective is to compute a minimum vertex multiset such that at least R units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an f-approximation for this problem, improving over a previous result of (2f+2)(1+epsilon) by Cheung et al to the tight extent possible. Our new ingredient of this work is a generalized analysis on the extreme points of the natural LP, developed from previous works, and a strengthened LP lower-bound obtained for the optimal solutions.

We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an f-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al. (SODA 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor in (FOCS 2002). Our main technical tool for establishing the approximation guarantee is a separation lemma that certifies the existence of a strong partition for solutions that are basic feasible in an extended version of the natural LP. We believe that our rounding technique is of independent interest when hard constraints are considered.

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph G = (V, E) with a maximum edge size f. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an O(f) bi-approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of k geq 2, a cover with a cost ratio of (1+ frac{1}{k - 1})(f - 1) to the optimal cover for the original instance can be obtained. This improves over a previous result, which has a cost ratio of f^2 via augmenting the available multiplicity by a factor of f.

We consider the problem of online dynamic power management that provides hard real-time guarantees for multi-processor systems. In this problem, a set of jobs, each associated with an arrival time, a deadline, and an execution time, arrives to the system in an online fashion. The objective is to compute a non-migrative preemptive schedule of the jobs and a sequence of power on/off operations of the processors so as to minimize the total energy consumption while ensuring that all the deadlines of the jobs are met. We assume that we can use as many processors as necessary. In this paper we examine the complexity of this problem and provide online strategies that lead to practical energy-efficient solutions for real-time multi-processor systems. First, we consider the case for which we know in advance that the set of jobs can be scheduled feasibly on a single processor. We show that, even in this case, the competitive factor of any online algorithm is at least 2.06. On the other hand, we give a 4-competitive online algorithm that uses at most two processors. For jobs with unit execution times, the competitive factor of this algorithm improves to 3.59. Second, we relax our assumption by considering as input multiple streams of jobs, each of which can be scheduled feasibly on a single processor. We present a trade-off between the energy-efficiency of the schedule and the number of processors to be used. More specifically, for k given job streams and h processors with h>k, we give a scheduling strategy such that the energy usage is at most 4.k/(h-k) times that used by any schedule which schedules each of the k streams on a separate processor. Finally, we drop the assumptions on the input set of jobs. We show that the competitive factor of any online algorithm is at least 2.28, even for the case of unit job execution times for which we further derive an O(1)-competitive algorithm.