20 research outputs found

    Online Load Balancing on Uniform Machines with Limited Migration

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    In the problem of online load balancing on uniformly related machines with bounded migration, jobs arrive online one after another and have to be immediately placed on one of a given set of machines without knowledge about jobs that may arrive later on. Each job has a size and each machine has a speed, and the load due to a job assigned to a machine is obtained by dividing the first value by the second. The goal is to minimize the maximum overall load any machine receives. However, unlike in the pure online case, each time a new job arrives it contributes a migration potential equal to the product of its size and a certain migration factor. This potential can be spend to reassign jobs either right away (non-amortized case) or at any later time (amortized case). Semi-online models of this flavor have been studied intensively for several fundamental problems, e.g., load balancing on identical machines and bin packing, but uniformly related machines have not been considered up to now. In the present paper, the classical doubling strategy on uniformly related machines is combined with migration to achieve an (8/3+ε)(8/3+\varepsilon)-competitive algorithm and a (4+ε)(4+\varepsilon)-competitive algorithm with O(1/ε)O(1/\varepsilon) amortized and non-amortized migration, respectively, while the best known competitive ratio in the pure online setting is roughly 5.8285.828

    A 4/3 Approximation for 2-Vertex-Connectivity

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    The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA\u2717] (improving on earlier results by Khuller and Vishkin [STOC\u2792] and Garg, Vempala and Singla [SODA\u2793]). Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work

    An Empirical Evaluation of k-Means Coresets

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    Coresets are among the most popular paradigms for summarizing data. In particular, there exist many high performance coresets for clustering problems such as k-means in both theory and practice. Curiously, there exists no work on comparing the quality of available k-means coresets. In this paper we perform such an evaluation. There currently is no algorithm known to measure the distortion of a candidate coreset. We provide some evidence as to why this might be computationally difficult. To complement this, we propose a benchmark for which we argue that computing coresets is challenging and which also allows us an easy (heuristic) evaluation of coresets. Using this benchmark and real-world data sets, we conduct an exhaustive evaluation of the most commonly used coreset algorithms from theory and practice

    Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

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    Dual Node and Edge Fairness-Aware Graph Partition

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    Fair graph partition of social networks is a crucial step toward ensuring fair and non-discriminatory treatments in unsupervised user analysis. Current fair partition methods typically consider node balance, a notion pursuing a proportionally balanced number of nodes from all demographic groups, but ignore the bias induced by imbalanced edges in each cluster. To address this gap, we propose a notion edge balance to measure the proportion of edges connecting different demographic groups in clusters. We analyze the relations between node balance and edge balance, then with line graph transformations, we propose a co-embedding framework to learn dual node and edge fairness-aware representations for graph partition. We validate our framework through several social network datasets and observe balanced partition in terms of both nodes and edges along with good utility. Moreover, we demonstrate our fair partition can be used as pseudo labels to facilitate graph neural networks to behave fairly in node classification and link prediction tasks

    Fair Correlation Clustering in General Graphs

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    We consider the family of Correlation Clustering optimization problems under fairness constraints. In Correlation Clustering we are given a graph whose every edge is labeled either with a + or a -, and the goal is to find a clustering that agrees the most with the labels: + edges within clusters and - edges across clusters. The notion of fairness implies that there is no over, or under, representation of vertices in the clustering: every vertex has a color and the distribution of colors within each cluster is required to be the same as the distribution of colors in the input graph. Previously, approximation algorithms were known only for fair disagreement minimization in complete unweighted graphs. We prove the following: (1) there is no finite approximation for fair disagreement minimization in general graphs unless P = NP (this hardness holds also for bicriteria algorithms); and (2) fair agreement maximization in general graphs admits a bicriteria approximation of ? 0.591 (an improved ? 0.609 true approximation is given for the special case of two uniformly distributed colors). Our algorithm is based on proving that the sticky Brownian motion rounding of [Abbasi Zadeh-Bansal-Guruganesh-Nikolov-Schwartz-Singh SODA\u2720] copes well with uncut edges

    Parameterized Complexity of Fair Bisection: FPT-Approximation meets Unbreakability

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    In the Minimum Bisection problem, input is a graph GG and the goal is to partition the vertex set into two parts AA and BB, such that ∣∣A∣−∣B∣∣≤1||A|-|B|| \le 1 and the number kk of edges between AA and BB is minimized. This problem can be viewed as a clustering problem where edges represent similarity, and the task is to partition the vertices into two equally sized clusters, while minimizing the number of pairs of similar objects that end up in different clusters. In this paper, we initiate the study of a fair version of Minimum Bisection. In this problem, the vertices of the graph are colored using one of c≥1c \ge 1 colors. The goal is to find a bisection (A,B)(A, B) with at most kk edges between the parts, such that for each color i∈[c]i\in [c], AA has exactly rir_i vertices of color ii. We first show that Fair Bisection is WW[1]-hard parameterized by cc even when k=0k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class ii has {\em exactly} rir_i vertices in AA. In particular, we give an f(k,c,ϵ)nO(1)f(k,c,\epsilon)n^{O(1)} time algorithm that finds a balanced partition (A,B)(A, B) with at most kk edges between them, such that for each color i∈[c]i\in [c], there are at most (1±ϵ)ri(1\pm \epsilon)r_i vertices of color ii in AA. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP '18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing '14]). Along the way, we also derive a combinatorial result regarding tree decompositions of graphs.Comment: Full version of ESA 2023 paper. Abstract shortened to meet the character limi

    A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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    Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

    A New Coreset Framework for Clustering

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    Given a metric space, the (k,z)(k,z)-clustering problem consists of finding kk centers such that the sum of the of distances raised to the power zz of every point to its closest center is minimized. This encapsulates the famous kk-median (z=1z=1) and kk-means (z=2z=2) clustering problems. Designing small-space sketches of the data that approximately preserves the cost of the solutions, also known as \emph{coresets}, has been an important research direction over the last 15 years. In this paper, we present a new, simple coreset framework that simultaneously improves upon the best known bounds for a large variety of settings, ranging from Euclidean space, doubling metric, minor-free metric, and the general metric cases
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