Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$ vertices into a given number $k$ of clusters such that every cluster forms a connected subgraph of $G$ and a given clustering objective gets minimized. The constraint that the clusters are connected has applications in many different fields, like for example community detection and geodesy.
So far, $k$-center and $k$-median have been studied in this setting. It has been shown that connected $k$-median is $Ω(n^{1- ε})$-hard to approximate which also carries over to the connected $k$-means problem, while for connected $k$-center it remained an open question whether one can find a constant approximation in polynomial time. We answer this question by providing an $Ω(\log^*(k))$-hardness result for the problem. Given these hardness results, we study the problems on graphs with bounded treewidth. We provide exact algorithms that run in polynomial time if the treewidth $w$ is a constant. Furthermore, we obtain constant approximation algorithms that run in FPT time with respect to the parameter $\max(w,k)$.
Additionally, we consider the min-sum-radii (MSR) and min-sum-diameter (MSD) objective. We prove that on general graphs connected MSR can be approximated with an approximation factor of $(3 + ε)$ and connected MSD with an approximation factor of $(4 + ε)$. The latter also directly improves the best known approximation guarantee for unconstrained MSD from $(6 + ε)$ to $(4 + ε)$.
We consider min-max $r$-gather clustering problem and min-max $r$-gathering problem. In the min-max $r$-gather clustering problem, we are given a set of users and divide them into clusters with size at least $r$; the goal is to minimize the maximum diameter of clusters. In the min-max $r$-gathering problem, we are additionally given a set of facilities and assign each cluster to a facility; the goal is to minimize the maximum distance between the users and the assigned facility. In this study, we consider the case that the users and facilities are located on a ``spider'' and propose the first …
We present massively parallel (MPC) algorithms and hardness of approximation results for computing Single-Linkage Clustering of $n$ input $d$-dimensional vectors under Hamming, $\ell_1, \ell_2$ and $\ell_\infty$ distances. All our algorithms run in $O(\log n)$ rounds of MPC for any fixed $d$ and achieve $(1+ε)$-approximation for all distances (except Hamming for which we show an exact algorithm). We also show constant-factor inapproximability results for $o(\log n)$-round algorithms under standard MPC hardness assumptions (for sufficiently large dimension depending on the distance used). Effic…
In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $α$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $α= 1$ by Ding et al. For any constant $α> 1$, we give an $O(n^{1-\frac…
We present hardness of approximation results for Correlation Clustering with local objectives and for Hierarchical Clustering with dissimilarity information. For the former, we study the local objective of Puleo and Milenkovic (ICML '16) that prioritizes reducing the disagreements at data points that are worst off and for the latter we study the maximization version of Dasgupta's cost function (STOC '16). Our APX hardness results imply that the two problems are hard to approximate within a constant of 4/3 ~ 1.33 (assuming P vs NP) and 9159/9189 ~ 0.9967 (assuming the Unique Games Conjecture) r…
The Consensus Clustering problem has been introduced as an effective way to analyze the results of different microarray experiments. The problem consists of looking for a partition that best summarizes a set of input partitions (each corresponding to a different microarray experiment) under a simple and intuitive cost function. The problem admits polynomial time algorithms on two input partitions, but is APX-hard on three input partitions. We investigate the restriction of Consensus Clustering when the output partition is required to contain at most k sets, giving a polynomial time approximati…
In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is NP-hard and approximable within O(log t) in polynomial time. In this paper, we further investigate the …
We consider the problem of determining the maximal $α\in (0,1]$ such that every matching $M$ of size $k$ (or at most $k$) in a bipartite graph $G$ contains an induced matching of size at least $α|M|$. This measure was recently introduced in Alon et al. (NIPS 2018) and is motivated by connectionist models of cognition as well as modeling interference in wireless and communication networks. We prove various hardness results for computing $α$ either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching $N$ in a…
In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are b…
We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph $G$ is $H$-free for some graph $H$ if $G$ contains no induced subgraph isomorphic to $H$. It is known that Connected Vertex Cover is NP-complete even for $H$-free graphs if $H$ contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if $H$ contains a cycle or claw. In the…
The $H$-Free Edge Deletion problem asks, for a given graph $G$ and an integer $k$, whether it is possible to delete at most $k$ edges from $G$ to make it $H$-free, that is, not containing $H$ as an induced subgraph. The $H$-Free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) fo…
We study two generalizations of classic clustering problems called dynamic ordered $k$-median and dynamic $k$-supplier, where the points that need clustering evolve over time, and we are allowed to move the cluster centers between consecutive time steps. In these dynamic clustering problems, the general goal is to minimize certain combinations of the service cost of points and the movement cost of centers, or to minimize one subject to some constraints on the other. We obtain a constant-factor approximation algorithm for dynamic ordered $k$-median under mild assumptions on the input. We give a…
Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta's objective (introduced by Moseley-Wang), we present polynomial-time .4246 approximation algorithms that use Max-Uncut B…
The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph $G$ whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of $G$ having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in $O(n^3)$ time for trees and $O(n + m)$ time for split graphs and properly colored bipartite graphs, where $n$ is the number of vertices and…
We consider the min-max $r$-gathering problem described as follows: We are given a set of users and facilities in a metric space. We open some of the facilities and assign each user to an opened facility such that each facility has at least $r$ users. The goal is to minimize the maximum distance between the users and the assigned facility. We also consider the min-max $r$-gather clustering problem, which is a special case of the $r$-gathering problem in which the facilities are located everywhere. In this paper, we study the tractability and the hardness when the underlying metric space is a s…
Google Research（谷歌研究院）；New York University（纽约大学）；University of Michigan（密歇根大学）
The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this …
We consider the approximability of center-based clustering problems where the points to be clustered lie in a metric space, and no candidate centers are specified. We call such problems "continuous", to distinguish from "discrete" clustering where candidate centers are specified. For many objectives, one can reduce the continuous case to the discrete case, and use an $α$-approximation algorithm for the discrete case to get a $βα$-approximation for the continuous case, where $β$ depends on the objective: e.g. for $k$-median, $β= 2$, and for $k$-means, $β= 4$. Our motivating question is whether …
Hardness of Burning Number Problem on Regular Graphs
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；Indian Institute of Science, Bengaluru（印度科学学院班加罗尔分校）；Manipal Academy of Higher Education（马尼帕尔高等教育学院）；Indian Institute of Technology Delhi（德里印度理工学院）；Kyoto University（京都大学）
A graph has \emph{diameter} D if every pair of vertices are connected by a path of at most D edges. The Diameter-D Augmentation problem asks how to add the a number of edges to a graph in order to make the resulting graph have diameter D. It was previously known that this problem is NP-hard \cite{GJ}, even in the D=2 case. In this note, we give a simpler reduction to arrive at this fact and show that this problem is W[2]-hard.
A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most $p$ edges. Nevertheless we show that COLOURFU…
MPI for Biological Cybernetics（马克斯·普朗克生物控制论研究所）；Univ. of Minnesota（明尼苏达大学）
In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor clustering [8,34]. Like k-means, these more general problems also suffer from the NP-hardness of the associated optimization. Researchers have developed approximation algorithms of varying degrees of sophistication for k-means, k-medians, and more recently also for Bregman clustering [2]. However, there seem to be no approximation algorithms for Bregman co- and tens…
We study geometrical properties of the complete set of solutions of the random 3-satisfiability problem. We show that even for moderate system sizes the number of clusters corresponds surprisingly well with the theoretic asymptotic prediction. We locate the freezing transition in the space of solutions which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems.
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Google Zurich（谷歌苏黎世）；EPFL（洛桑联邦理工学院）；Microsoft Research（微软研究院）
Submodular maximization has become established as the method of choice for the task of selecting representative and diverse summaries of data. However, if datapoints have sensitive attributes such as gender or age, such machine learning algorithms, left unchecked, are known to exhibit bias: under- or over-representation of particular groups. This has made the design of fair machine learning algorithms increasingly important. In this work we address the question: Is it possible to create fair summaries for massive datasets? To this end, we develop the first streaming approximation algorithms fo…
In mobile wireless sensor networks (MWSNs), each sensor has the ability not only to sense and transmit data but also to move to some specific location. Because the movement of sensors consumes much more power than that in sensing and communication, the problem of scheduling mobile sensors to cover all targets and maintain network connectivity such that the total movement distance of mobile sensors is minimized has received a great deal of attention. However, in reality, due to a limited budget or numerous targets, mobile sensors may be not enough to cover all targets or form a connected networ…
We consider communities whose vertices are predominantly connected, i.e., the vertices in each community are stronger connected to other community members of the same community than to vertices outside the community. Flake et al. introduced a hierarchical clustering algorithm that finds such predominantly connected communities of different coarseness depending on an input parameter. We present a simple and efficient method for constructing a clustering hierarchy according to Flake et al. that supersedes the necessity of choosing feasible parameter values and guarantees the completeness of the …
A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800（波尔多大学）；University of Innsbruck（因斯布鲁克大学）；School of Computer Science, Georgia Institute of Technology（佐治亚理工学院计算机学院）；School of Computer Science and School of Mathematics, Georgia Institute of Technology（佐治亚理工学院计算机学院与数学学院）；Collegium de Lyon, ENS de Lyon, LIP（里昂学院，巴黎高等师范学院里昂分校，计算与语言研究所）；Kahlert School of Computing, University of Utah（犹他大学卡尔赫特计算机学院）；Univ. Lyon, ENS de Lyon, UCBL, CNRS, LIP（里昂大学，巴黎高等师范学院里昂分校，里昂第一大学，法国国家科学研究中心，计算与语言研究所）
We study the {\em verification} problem in distributed networks, stated as follows. Let $H$ be a subgraph of a network $G$ where each vertex of $G$ knows which edges incident on it are in $H$. We would like to verify whether $H$ has some properties, e.g., if it is a tree or if it is connected. We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification, and give almost tight lower bo…
We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm that, using submodularity, obtains an approximation ratio of $(1 - 1/e - ε)$ in the exponent of the number of spanning trees for any $ε> 0$ in time $\tilde{O}(m ε^{-1} + (n + q) ε^{-3})$, where $m$ and $q$ is the number of edges in the original graph and the candidate edge set, respectively. Our running time is optimal with respect to the input size up to log…
Ben-Gurion University（本-古里安大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over $W$ of the $\ell_p$-norms of the weighted distances of points in $P$ from the centers. This generalizes various clustering problems, including Socially Fair $k$-Median and $k$-Means, and is closely connected to other problems such as Densest $k$-Subgraph and Min $k$-Union. We utilize convex programming techniques to approximate the $(p,q)$-Fair Clustering pr…
We study a variant of classical clustering formulations in the context of algorithmic fairness, known as diversity-aware clustering. In this variant we are given a collection of facility subsets, and a solution must contain at least a specified number of facilities from each subset while simultaneously minimizing the clustering objective ($k$-median or $k$-means). We investigate the fixed-parameter tractability of these problems and show several negative hardness and inapproximability results, even when we afford exponential running time with respect to some parameters. Motivated by these resu…
Massachusetts Institute of Technology（麻省理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
We consider the model introduced by Bilu and Linial (2010), who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations, e.g. on the order of $O(\sqrt{n})$ for max-cut clustering. Awasthi et al. (2010) consider center-based objectives, and Balcan and Liang (2011) analyze the $k$-median and min-sum objectives, giving efficient algorithms for instances resilient to certain constant multiplicati…
Motivated by the fact that distances between data points in many real-world clustering instances are often based on heuristic measures, Bilu and Linial~\cite{BL} proposed analyzing objective based clustering problems under the assumption that the optimum clustering to the objective is preserved under small multiplicative perturbations to distances between points. The hope is that by exploiting the structure in such instances, one can overcome worst case hardness results. In this paper, we provide several results within this framework. For center-based objectives, we present an algorithm that c…
Given a large graph G = (V,E) with millions of nodes and edges, how do we compute its connected components efficiently? Recent work addresses this problem in map-reduce, where a fundamental trade-off exists between the number of map-reduce rounds and the communication of each round. Denoting d the diameter of the graph, and n the number of nodes in the largest component, all prior map-reduce techniques either require d rounds, or require about n|V| + |E| communication per round. We propose two randomized map-reduce algorithms -- (i) Hash-Greater-To-Min, which provably requires at most 3log(n) …
In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$ \sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{ij})b_{ij}. $$ We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question pos…
