Bicriteria Approximation Algorithm Research Articles

Parameterized approximation algorithms for weighted vertex cover

A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study Weighted Vertex Cover with solution size as a parameter. Formally, in the (k,W)-Vertex Cover problem, given a graph G, an integer k, a positive rational W, and a weight function w:V(G)→Q+, the question is whether G has a vertex cover of size at most k of weight at most W, with k being the parameter. An (a,b)-bi-criteria approximation algorithm for (k,W)-Vertex Cover either produces a vertex cover S such that |S|≤ak and w(S)≤bW, or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.•A simple (2,2)-bi-criteria approximation algorithm for (k,W)-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.•A simple exact parameterized algorithm for (k,W)-Vertex Cover running in O⁎(1.4656k) time1.•A (1+ϵ,2)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.4656(1−ϵ)k) time.•A (1.5,1.5)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.414k) time.•A (2−δ,2−δ)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(∑i=δk(1−2δ)1+2δδk(1−2δ)2δ(δk+iδk−2iδ1−2δ)) time for any δ<0.5. For example, for (1.75,1.75) and (1.9,1.9)-approximation algorithms, we get running times of O⁎(1.272k) and O⁎(1.151k) respectively.Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.

Approximation algorithms for orthogonal line centers

The problem of k orthogonal line center is about computing a set of k axis-parallel lines for a given set of points in ℜ2 such that the maximum among the distances between each point to its nearest line is minimized. A 2-factor approximation algorithm and a (53,32) bi-criteria approximation algorithm is presented for the problem. Both of them are deterministic approximation algorithms using combinatorial techniques and having sub-quadratic running times.

A bicriteria approximation algorithm for the minimum hitting set problem in measurable range spaces

We consider the problem of finding a minimum-size hitting set in a range space F=(Q,R) defined by a measure on a family of subsets of an infinite set R. We observe that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any δ>0, a set of size O(αFzF⁎) that hits a (1−δ)-fraction of the ranges in R (with respect to the given measure) in time proportional to log⁡1δ, where zF⁎ is the value of the fractional optimal solution and αF is the integrality gap of the standard LP relaxation of the hitting set problem for the restriction of F on a set of points of size O(zF⁎log⁡1δ). Some applications of this result in Computational Geometry are given.

Stackelberg Security Games with Contagious Attacks on a Network: Reallocation to the Rescue

In the classic network security games, the defender distributes defending resources to the nodes of the network, and the attacker attacks a node, with the objective of maximizing the damage caused. In this paper, we consider the network defending problem against contagious attacks, e.g., the attack at a node u spreads to the neighbors of u and can cause damage at multiple nodes. Existing works that study shared resources assume that the resource allocated to a node can be shared or duplicated between neighboring nodes. However, in the real world, sharing resource naturally leads to a decrease in defending power of the source node, especially when defending against contagious attacks. Therefore, we study the model in which resources allocated to a node can only be transferred to its neighboring nodes, which we refer to as a reallocation process. We show that the problem of computing optimal defending strategy is NP-hard even for some very special cases. For positive results, we give a mixed integer linear program formulation for the problem and a bi-criteria approximation algorithm. Our experimental results demonstrate that the allocation and reallocation strategies our algorithm computes perform well in terms of minimizing the damage due to contagious attacks.

A note for approximating the submodular cover problem over integer lattice with low adaptive and query complexities

This paper studies the Diminishing Return Submodular Cover (DRSC) problem as follows: given a diminishing return submodular function f:Z+E↦R+ over the ground set E of size n, a positive integer B as the maximum value of a coordinate of a vector in Z+E and a threshold α>0, the problem asks to find the vector with the smallest size x so that f(x)≥α. This problem has been attracted by much research recently due to its importance in machine learning and combinatorial optimization. However, because of high query and adaptive complexities, the existing algorithms might still not be efficient enough when working with large input data. This paper proposes a randomized and bicriteria approximation algorithm that has O(log⁡n) adaptive complexity and O((n+log⁡nlog⁡B)log⁡(nB)) query complexity. Our algorithm significantly reduces both adaptive and query complexities compared to some state-of-the-art algorithms by factors of Ω(log⁡(m)log2⁡(mnlog⁡B)(1+log⁡(log⁡B)/log⁡(n))), Ω(min⁡{n/log⁡(n),log⁡(B)}) respectively, where m=f(B⋅1).

Red–Blue k-Center Clustering with Distance Constraints

We consider a variant of the k-center clustering problem in IRd, where the centers can be divided into two subsets—one, the red centers of size p, and the other, the blue centers of size q, such that p+q=k, and each red center and each blue center must be a distance of at least some given α≥0 apart. The aim is to minimize the covering radius. We provide a bi-criteria approximation algorithm for the problem and a polynomial time algorithm for the constrained problem where all centers must lie on a given line ℓ. Additionally, we present a polynomial time algorithm for the case where only the orientation of the line is fixed in the plane (d=2), although the algorithm works even in IRd by constraining the line to lie in a plane and with a fixed orientation.

Approximating ( k,ℓ )-Median Clustering for Polygonal Curves

In 2015, Driemel, Krivošija, and Sohler introduced the k,ℓ -median clustering problem for polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find k median curves of at most ℓ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this article, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in ℝ d and achieves approximation factor (1+ɛ) with respect to the clustering costs. The algorithm has worst-case running time linear in the number of curves, polynomial in the maximum number of vertices per curve (i.e., their complexity), and exponential in d , ℓ, 1/ɛ and 1/δ (i.e., the failure probability). We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity ℓ, but of complexity at most 2ℓ -2, and whose vertices can be computed efficiently. We combine this lemma with the superset sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest.

Bi-Criteria Approximation for a Multi-Origin Multi-Channel Auto-Scaling Live Streaming Cloud

Live video traffic has been widely observed to vary significantly within short timescale. In order to manage such traffic dynamic of overlay live streaming, the Content Provider (CP) may deploy a set of geo-dispersed auto-scaling servers where the pay-as-you-go deployment cost is charged by the amount of resources used due to server uploading and data transmission between servers. To support geo-distributed user demands, we study a novel multi-origin multi-channel auto-scaling live streaming cloud that pushes each channel stream in the core network overlay as a tree covering the end servers who have local demand for the channel. The Origin-to-End (O2E) delay from an origin to an end server is due to the Server-to-Server (S2S) delays of the overlay links along the path. By optimizing the overlay of the core network, we seek to minimize the deployment cost and O2E delays of the channels (i.e., a bi-criteria problem), which can be equivalently phrased as minimizing the deployment cost while meeting certain given maximum O2E delay constraints. We formulate a realistic problem capturing the major cost and delay components, and show its NP-hardness. We propose Cost-optimized Multi-Origin Multi-Channel Overlay Streaming (COCOS), a novel, efficient and near-optimal bi-criteria approximation algorithm with proven approximation ratio. Trace-driven extensive experimental results based on real-world live streaming service data validate that COCOS outperforms other state-of-the-art schemes by a wide margin (cutting the cost in general by more than 50%).

Happiness maximizing sets under group fairness constraints

Finding a happiness maximizing set (HMS) from a database, i.e., selecting a small subset of tuples that preserves the best score with respect to any nonnegative linear utility function, is an important problem in multi-criteria decision-making. When an HMS is extracted from a set of individuals to assist data-driven algorithmic decisions such as hiring and admission, it is crucial to ensure that the HMS can fairly represent different groups of candidates without bias and discrimination. However, although the HMS problem was extensively studied in the database community, existing algorithms do not take group fairness into account and may provide solutions that under-represent some groups. In this paper, we propose and investigate a fair variant of HMS (FairHMS) that not only maximizes the minimum happiness ratio but also guarantees that the number of tuples chosen from each group falls within predefined lower and upper bounds. Similar to the vanilla HMS problem, we show that FairHMS is NP-hard in three and higher dimensions. Therefore, we first propose an exact interval cover-based algorithm called IntCov for FairHMS on two-dimensional databases. Then, we propose a bicriteria approximation algorithm called BiGreedy for FairHMS on multi-dimensional databases by transforming it into a submodular maximization problem under a matroid constraint. We also design an adaptive sampling strategy to improve the practical efficiency of BiGreedy. Extensive experiments on real-world and synthetic datasets confirm the efficacy and efficiency of our proposal.

Algorithms for covering multiple submodular constraints and applications

We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a weight function w: N rightarrow mathbb {R}_+, r monotone submodular functions f_1,f_2,ldots ,f_r over N and requirements k_1,k_2,ldots ,k_r the goal is to find a minimum weight subset S subseteq N such that f_i(S) ge k_i for 1 le i le r. We refer to this problem as Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with r=1Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced to Submod-SC. A simple greedy algorithm gives an O(log (kr)) approximation where k = sum _i k_i and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for Multi-Submod-Cover that covers each constraint to within a factor of (1-1/e-varepsilon ) while incurring an approximation of O(frac{1}{epsilon }log r) in the cost. Second, we consider the special case when each f_i is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.

The maximum exposure problem

Given a set of points P and axis-aligned rectangles R in the plane, a point p∈P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For range space defined by general rectangles, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k2) rectangles, we can expose at least Ω(1/k) of the optimal number of points.

On Approximating Degree-Bounded Network Design Problems

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph \(G=(V, E)\) with edge costs \(c \in {\mathbb {R}}_{\ge 0}^E\), a root \(r \in V\) and k terminals \(K\subseteq V\), we need to output the minimum-cost arborescence in G that contains an \(r \rightarrow t\) path for every \(t \in K\). Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time \(O(\log ^2k/\log \log k)\)-approximation Algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound \(d_v\) on each vertex \(v \in V\), and we require that every vertex v in the output tree has at most \(d_v\) children. We give a quasi-polynomial time \((O(\log n \log k), O(\log ^2 n))\)-bicriteria approximation: The Algorithm produces a solution with cost at most \(O(\log n\log k)\) times the cost of the optimum solution that violates the degree constraints by at most a factor of \(O(\log ^2n)\). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of \(O(\log ^2n)\) is an \(O(\log n)\)-factor away from the approximation lower bound of \(\Omega (\log n)\) from the set-cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an \((O(\log n\log k), O(\log n))\)-bicriteria approximation Algorithm for DB-GST-T.

Approximation Algorithms for the Capacitated Min–Max Correlation Clustering Problem

In this paper, we propose a so-called capacitated min–max correlation clustering model, a natural variant of the min–max correlation clustering problem. As our main contribution, we present an integer programming and its integrality gap analysis for the proposed model. Furthermore, we provide two approximation algorithms for the model, one of which is a bi-criteria approximation algorithm and the other is based on LP-rounding technique.

Online Makespan Scheduling with Job Migration on Uniform Machines

In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and approx 1.4659. They show that k = O(m) is sufficient to achieve this bound and no k = o(n) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a delta = varTheta (1) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than 1.4659 + delta with k = o(n). We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and approx 1.7992 with k = O(m). We also show that k = varOmega (m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

Minimum Robust Multi-Submodular Cover for Fairness

In this paper, we study a novel problem, Minimum Robust Multi-Submodular Cover for Fairness (MinRF), as follows: given a ground set V; m monotone submodular functions f_1,...,f_m; m thresholds T_1,...,T_m and a non-negative integer r; MinRF asks for the smallest set S such that f_i(S \ X) ≥ T_i for all i ∈ [m] and |X| ≤ r. We prove that MinRF is inapproximable within (1- ε) ln m; and no algorithm, taking fewer than exponential number of queries in term of r, is able to output a feasible set to MinRF with high certainty. Three bicriteria approximation algorithms with performance guarantees are proposed: one for r = 0, one for r = 1, and one for general r. We further investigate our algorithms' performance in two applications of MinRF, Information Propagation for Multiple Groups and Movie Recommendation for Multiple Users. Our algorithms have shown to outperform baseline heuristics in both solution quality and the number of queries in most cases.

Defending against Contagious Attacks on a Network with Resource Reallocation

In classic network security games, the defender distributes defending resources to the nodes of the network, and the attacker attacks a node, with the objective to maximize the damage caused. Existing models assume that the attack at node u causes damage only at u. However, in many real-world security scenarios, the attack at a node u spreads to the neighbors of u and can cause damage at multiple nodes, e.g., for the outbreak of a virus. In this paper, we consider the network defending problem against contagious attacks. Existing works that study shared resources assume that the resource allocated to a node can be shared or duplicated between neighboring nodes. However, in real world, sharing resource naturally leads to a decrease in defending power of the source node, especially when defending against contagious attacks. To this end, we study the model in which resources allocated to a node can only be transferred to its neighboring nodes, which we refer to as a reallocation process. We show that this more general model is difficult in two aspects: (1) even for a fixed allocation of resources, we show that computing the optimal reallocation is NP-hard; (2) for the case when reallocation is not allowed, we show that computing the optimal allocation (against contagious attack) is also NP-hard. For positive results, we give a mixed integer linear program formulation for the problem and a bi-criteria approximation algorithm. Our experimental results demonstrate that the allocation and reallocation strategies our algorithm computes perform well in terms of minimizing the damage due to contagious attacks.

An approximation algorithm for the spherical k-means problem with outliers by local search

We consider the spherical k-means problem with outliers, an extension of the k-means problem. In this clustering problem, all sample points are on the unit sphere. Given two integers k and z, we can ignore at most z points (outliers) and need to find at most k cluster centers on the unit sphere and assign remaining points to these centers to minimize the k-means objective. It has been proved that any algorithm with a bounded approximation ratio cannot return a feasible solution for this problem. Our contribution is to present a local search bi-criteria approximation algorithm for the spherical k-means problem.

Structured Robust Submodular Maximization: Offline and Online Algorithms

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. Although these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids and knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem are known in advance and the online setting where the input data are revealed over time. For the offline setting, we give a general (nearly) optimal bicriteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bicriteria solution with sublinear regret.Summary of Contribution: Constrained submodular maximization is one of the core areas in combinatorial optimization with a wide variety of applications in operations research and computer science. Over the last decades, both communities have been interested on the design and analysis of new algorithms with provable guarantees. Sensor location, influence maximization and data summarization are some of the applications of submodular optimization that lie at the intersection of the aforementioned communities. Particularly, our work focuses on optimizing several submodular functions simultaneously. We provide new insights and algorithms to the offline and online variants of the problem which significantly expand the related literature. At the same time, we provide a computational study that supports our theoretical results.

Multi-Resource VNF Deployment in a Heterogeneous Cloud

The emerging paradigm of Network Function Virtualization (NFV) promises to shorten the renewal cycles of network functions and reduce the capital expenses by flexibly deploying virtualized network functions (VNFs) implementation on commodity servers. However, the required resource of each type (CPU, memory, etc.) for the running VNF should be provisioned to guarantee the performance when processing packets. This comes with different deployment cost, especially in a heterogeneous cloud consisting of a large number of network function platforms from various vendors. To optimally operate VNFs, it is necessary for the network operator to dynamically deploy VNFs in the expensive cloud infrastructures. In this article, we initiate the study of minimizing the deployment cost under multi-resource constraints in a heterogeneous cloud. We formulate multi-resource VNF deployment problem (MVDP) as an optimization program and prove its hardness. We propose an offline <inline-formula><tex-math notation="LaTeX">$(1, d+1)$</tex-math></inline-formula> -bicriteria approximation algorithm and an <inline-formula><tex-math notation="LaTeX">$(\mathcal {O}(1), \mathcal {O}(n \cdot \log n))$</tex-math></inline-formula> -competitive online algorithm to deploy VNFs in a scalable manner, where <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> is the number of resource types and <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of required VNFs. Large-scale simulations and DPDK-based OpenNetVM implementation show that our algorithms can reduce the overall cost by 34% and improve the performance in terms of multi-resource allocation.

Best Bang for the Buck: Cost-Effective Seed Selection for Online Social Networks

We study the min-cost seed selection problem in online social networks for viral marketing, where the goal is to select a set of seed nodes with the minimum total cost such that the expected number of influenced nodes in the network exceeds a predefined threshold. We propose several algorithms that outperform the previous studies both on the theoretical approximation ratio and on the experimental performance. In the case where the nodes have heterogeneous costs, our algorithms are the first bi-criteria approximation algorithms with polynomial running time and provable approximation ratio. In the case where the users have uniform costs, our algorithms achieve logarithmic approximation ratio and provable time complexity which is smaller than that of the existing algorithms in orders of magnitude. We conduct extensive experiments using real social networks. The experimental results show that, our algorithms significantly outperform the existing algorithms both on the total cost and on the running time, and also scale well to billion-scale networks.