Constant Factor Approximation Research Articles

Distributed distance-r covering problems on sparse high-girth graphs

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

Strengthening ties towards a highly-connected world

Online social networks provide a forum where people make new connections, learn more about the world, get exposed to different points of view, and access information that were previously inaccessible. It is natural to assume that content-delivery algorithms in social networks should not only aim to maximize user engagement but also to offer opportunities for increasing connectivity and enabling social networks to achieve their full potential. Our motivation and aim is to develop methods that foster the creation of new connections, and subsequently, improve the flow of information in the network. To achieve our goal, we propose to leverage the strong triadic closure principle, and consider violations to this principle as opportunities for creating more social links. We formalize this idea as an algorithmic problem related to the densest k-subgraph problem. For this new problem, we establish hardness results and propose approximation algorithms. We identify two special cases of the problem that admit a constant-factor approximation. Finally, we experimentally evaluate our proposed algorithm on real-world social networks, and we additionally evaluate some simpler but more scalable algorithms.

An Optimal Approximation for Submodular Maximization Under a Matroid Constraint in the Adaptive Complexity Model

An Exponentially Faster Algorithm for Submodular Maximization Under a Matroid Constraint This paper studies the problem of submodular maximization under a matroid constraint. It is known since the 1970s that the greedy algorithm obtains a constant-factor approximation guarantee for this problem. Twelve years ago, a breakthrough result by Vondrák obtained the optimal 1 − 1/e approximation. Previous algorithms for this fundamental problem all have linear parallel runtime, which was considered impossible to accelerate until recently. The main contribution of this paper is a novel algorithm that provides an exponential speedup in the parallel runtime of submodular maximization under a matroid constraint, without loss in the approximation guarantee.

A constant-factor approximation for directed latency in quasi-polynomial time

We consider the directed minimum latency problem (DirLat ), wherein we seek a path P visiting all points (or clients) in a given asymmetric metric starting at a given root node r, so as to minimize the sum of the client waiting times along P. We give the first constant-factor approximation guarantee for DirLat, but in quasi-polynomial time. A key ingredient of our result, and our chief technical contribution, is an extension of a recent result of Köhne et al. (2019) [17] showing that the integrality gap of the natural Held-Karp relaxation for asymmetric TSP-Path (ATSPP ) is at most a constant. We also give a better approximation guarantee for the minimum total-regret problem, where the goal is to find a path P that minimizes the total time that nodes spend in excess of their shortest-path distances from r, which can be cast as a special case of DirLat involving so-called regret metrics.

Efficient Link Scheduling Solutions for the Internet of Things Under Rayleigh Fading

Link scheduling is an appealing solution for ensuring the reliability and latency requirements of Internet of Things (IoT). Most existing results on the link scheduling problem were based on the graph or SINR (Signal-to-Interference-plus-Noise-Ratio) models, which ignored the impact of the random fading gain of the signals strength. In this paper, we address the link scheduling problem under the Rayleigh fading model. Both Shortest Link Scheduling (SLS) and Maximum Link Scheduling (MLS) problems are studied. In particular, we show that a set of links can be activated simultaneously under Rayleigh fading model if all link SINR constraints are satisfied. Based on the analysis of previous Link Diversity Partition (LDP) algorithm, we propose an Improved LDP (ILDP) algorithm and a centralized algorithm by localizing the global interference (denoted by CLT), building on which we design a distributed CLT algorithm (denoted by RCRDCLT) that converges to a constant approximation factor of the optimum with the time complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\ln n)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the number of links. Furthermore, executing repeatedly RCRDCLT can solve the SLS with an approximation factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta (\ln n)$ </tex-math></inline-formula> . Extensive simulations indicate that CLT is more effective than previous six popular link scheduling algorithms, and RCRDCLT has the lowest time complexity while only losses a constant fraction of the optimum schedule.

Visual Monitoring of Points of Interest on a 2.5D Terrain Using a UAV With Limited Field-of-View Constraint

We study the problem of visually monitoring a set of points on a 2.5D terrain using an unmanned aerial vehicle (UAV) with a downward-facing camera. The goal is to find a tour of minimum length for the UAV offline to visually inspect all points of interest. Varying terrain and limited field of view of the camera restrict the visibility of the UAV and can create obstacles in the flight path, making the problem challenging. The problem is NP-hard and generalizes the traveling salesperson problem (TSP). We present several algorithms to solve this problem. Our main theoretical contribution is a constant-factor approximation algorithm (assuming fixed parameters for the terrain). We also present a practical algorithm that uses the solution to a generalized TSP (GTSP) subinstance. We benchmark the GTSP-based algorithm using a branch-and-cut integer linear programming formulation and find that the proposed algorithm scales to much larger instances and is computationally fast. We also show proof-of-concept using field deployment of a UAV to visually monitor points of interest in the environment.

Near-optimal clustering in the k-machine model

The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The k-machine model (Klauck et al., (SODA 2015) [8]) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the p-median problem, and the p-center problem and presents O(1)-factor approximation algorithms for these problems running in O˜(n/k) rounds in the k-machine model. These algorithms are optimal up to polylogarithmic factors because this paper also shows Ω˜(n/k) lower bounds for obtaining polynomial-factor approximation algorithms for these problems. These are the first results for clustering problems in the k-machine model.We assume that the metric provided as input for these clustering problems is only implicitly provided, as an edge-weighted graph and in a nutshell, our main technical contribution is to show that constant-factor approximation algorithms for all three clustering problems can be obtained by learning only a small portion of the input metric.

Bayesian active learning with abstention feedbacks

We study pool-based active learning with abstention feedbacks where a labeler can abstain from labeling a queried example with some unknown abstention rate. This is an important problem with many useful applications. We take a Bayesian approach to the problem and develop two new greedy algorithms that learn both the classification problem and the unknown abstention rate at the same time. These are achieved by simply incorporating the estimated average abstention rate into the greedy criteria. We prove that both algorithms have near-optimality guarantees: they respectively achieve a (1-1e) constant factor approximation of the optimal expected or worst-case value of a useful utility function. Our experiments show the algorithms perform well in various practical scenarios.

On Communication Complexity of Fixed Point Computation

Brouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1]^n to [0,1]^n , and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2^{\Omega (n)} for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. Each player is given a function from [0,1]^n to [0,1]^{n/2} , and their goal is to find an approximate fixed point of the concatenation of the functions. Each player is given a function from [0,1]^n to [0,1]^{n} , and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of 2^{\Omega (n)} for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation T of the d -simplex is publicly known and one player is given a set S_A\subset T and a coloring function from S_A to \lbrace 0,\ldots ,d/2\rbrace , and the other player is given a set S_B\subset T and a coloring function from S_B to \lbrace d/2+1,\ldots ,d\rbrace , such that S_A\dot{\cup }S_B=T , and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T|^{\Omega (1)} for the aforementioned problem as well (when d is large). On the positive side, we show that if d\le 4 then there is a deterministic protocol for the Sperner problem with O((\log |T|)^2) bits of communication.

Approximate CVPp in time 20.802n

We show that a constant factor approximation of the shortest and closest lattice vector problem in any ℓp-norm can be computed in time 2(0.802+ε)n. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem in the ℓ2 norm. To obtain our result, we combine the latter algorithm for ℓ2 with geometric insights related to coverings.

A greedy algorithm for the social golfer and the Oberwolfach problem

Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the number of rounds that can be guaranteed in a Swiss-system tournament. Matches of these tournaments are usually determined in a myopic round-based way dependent on the results of previous rounds. Together with the hard constraint that no two players meet more than once during the tournament, at some point it might become infeasible to schedule a next round. For tournaments with n players and match sizes of k≥2 players, we prove that we can always guarantee ⌊nk(k−1)⌋ rounds. We show that this bound is tight. This provides a simple polynomial time constant factor approximation algorithm for the social golfer problem.We extend the results to the Oberwolfach problem. We show that a simple greedy approach guarantees at least ⌊n+46⌋ rounds for the Oberwolfach problem. This yields a polynomial time 13+ε-approximation algorithm for any fixed ε>0 for the Oberwolfach problem. Assuming that El-Zahar’s conjecture is true, we improve the bound on the number of rounds to be essentially tight.

(Almost) efficient mechanisms for bilateral trading

We study the bilateral trade problem: one seller, one buyer and a single, indivisible item for sale. It is well known that there is no fully-efficient and incentive compatible mechanism for this problem that maintains a balanced budget. We design simple and robust mechanisms that obtain approximate efficiency with these properties. We show that even minimal use of statistical data can yield good approximation results. We then demonstrate how a mechanism for this simple bilateral-trade problem can be used as a “black-box” for constructing mechanisms in more general environments. Finally, we show that dominant-strategy incentive-compatible mechanisms cannot guarantee any constant-factor approximation to the optimal gains from trade.

Dynamic Programming Approach to the Generalized Minimum Manhattan Network Problem

We study the generalized minimum Manhattan network (GMMN) problem: given a set $$P$$ of pairs of points in the Euclidean plane $${\mathbb{R}}^2$$ , we are required to find a minimum-length geometric network which consists of axis-aligned segments and contains a shortest path in the $$L_1$$ metric (a so-called Manhattan path) for each pair in $$P$$ . This problem commonly generalizes several NP-hard network design problems that admit constant-factor approximation algorithms, such as the rectilinear Steiner arborescence (RSA) problem, and it is open whether so does the GMMN problem. As a bottom-up exploration, Schnizler (2015) focused on the intersection graphs of the rectangles defined by the pairs in $$P$$ , and gave a polynomial-time dynamic programming algorithm for the GMMN problem whose input is restricted so that both the treewidth and the maximum degree of its intersection graph are bounded by constants. In this paper, as the first attempt to remove the degree bound, we provide a polynomial-time algorithm for the star case, and extend it to the general tree case based on an improved dynamic programming approach.

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter $$d\ge 1$$ , in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For $$d=1$$ , this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor $$n^{1-\epsilon }$$ , for any $$\epsilon >0$$ . Moreover, it is known that, for any $$d\ge 2$$ , it is NP-hard to approximate MaxDBS within a factor $$n^{1/2-\epsilon }$$ , for any $$\epsilon >0$$ . In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

Algorithms for fair k-clustering with multiple protected attributes

We study fair center based clustering problems. In an influential paper, Chierichetti, Kumar, Lattanzi and Vassilvitskii (NIPS 2017) consider the problem of finding a good clustering, say of women and men, such that every cluster contains an equal number of women and men. They were able to obtain a constant factor approximation for this problem for most center based k-clustering objectives such as k-median, k-means, and k-center. Despite considerable interest in extending this problem for multiple protected attributes (e.g. women and men, with or without citizenship), so far constant factor approximations for these problems have remained elusive except in special cases. We settle this question in the affirmative by giving the first constant factor approximation for a wide range of center based k-clustering objectives.

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f(1/varepsilon )cdot mathrm {poly}(|I|). Previously, only constant factor approximations of 5/3 and 4/3 + varepsilon , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

Deals or No Deals: Contract Design for Online Advertising

We present a formal study of first-look and preferred deals that are a recently introduced generation of contracts for selling online advertisements, which generalize traditional reservation contracts and are suitable for advertisers with advanced targeting capabilities. Under these deals, one or more advertisers gain priority access to an inventory of impressions before others and can choose to purchase in real time. In particular, we propose constant-factor approximation algorithms for maximizing the revenue that can be obtained from these deals when offered to all or a subset of the advertisers, whose valuation distributions can be independent or correlated through a common value component. We evaluate our algorithms using data from Google’s ad exchange platform and show they perform better than the approximation guarantees and obtain significantly higher revenue than auctions; in certain cases, the observed revenue is 85%–96% of the optimal revenue achievable. We also prove the NP-hardness of designing deals when advertisers’ valuations are arbitrarily correlated and the optimality of menus of deals among a certain class of selling mechanisms in an incomplete distributional information setting.

Packing under convex quadratic constraints

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are mathsf {APX}-hard to approximate and present constant-factor approximation algorithms based upon two different algorithmic techniques: a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation, and a greedy strategy. We further show that a combination of these techniques can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of these algorithms for problem instances arising in the context of real-world gas transport networks.

Reversals and transpositions distance with proportion restriction.

In the field of comparative genomics, one way of comparing two genomes is through the analysis of how they distinguish themselves based on a set of mutations called rearrangement events. When considering that genomes undergo different types of rearrangements, it can be assumed that some events are more common than others. To model this assumption, one can assign different weights to different events, where common events tend to cost less than others. However, this approach, called weighted, does not guarantee that the rearrangement assumed to be the most frequent will be also the most frequently returned by proposed algorithms. To overcome this issue, we investigate a new problem where we seek the shortest sequence of rearrangement events able to transform one genome into the other, with a restriction regarding the proportion between the events returned. Here, we consider two rearrangement events: reversal, that inverts the order and the orientation of the genes inside a segment of the genome, and transposition, that moves a segment of the genome to another position. We show the complexity of this problem for any desired proportion considering scenarios where the orientation of the genes is known or unknown. We also develop an approximation algorithm with a constant approximation factor for each scenario and, in particular, we describe an improved (asymptotic) approximation algorithm for the case where the gene orientation is known. At last, we present the experimental tests comparing the proposed algorithms with others from the literature for the version of the problem without the proportion restriction.

Real-time Approximate Routing for Smart Transit Systems

The advent of ride-hailing platforms such as Lyft and Uber has revolutionized urban mobility in the past decade. Given their increasingly important role in today's society, recent years have seen growing interest in integrating these services with mass transit options, in order to leverage the on-demand and flexible nature of the former, with the sustainability and cost-effectiveness of the latter. Our work explores a set of operational questions that are critical to the success of any such integrated marketplace: Given a set of trip requests, and the ability to utilize ride-hailing services, which mass-transit routes should the transit agency operate? How frequently should it operate each route? And how can ride-hailing trips be used to both help connect passengers to these routes, and also cover trips which are not efficiently served by mass transit? We study these under a model in which a Mobility-on-Demand provider (the platform ) has control of a vehicle fleet comprising both single-occupancy and high-capacity vehicles (e.g., cars and buses, respectively). The platform is faced with a number of trip requests to fill during a short time window, and can service these trip requests via different trip options : it can dispatch a car to transport the passenger from source to destination; it can use a car for the first and last legs of the trip, and have the passenger travel by bus in between; or it can use more complicated trips comprising of multiple car and bus legs. Given a set of rewards for matching passengers to trip options, and costs for operating each line (i.e., a bus route and associated frequency), the goal of the platform is to determine the optimal set of lines to operate (given a fixed budget for opening lines), as well as the assignment of passengers to trip options utilizing these lines, in order to maximize the total reward. We refer to this as the Real-Time Line Planning Problem (RLpp). We first demonstrate the computational limits of RLpp by showing that no constant-factor approximation is possible if we relax either one of two assumptions: (i) access to a pre-specified set of feasible lines, and (ii) no bus-to-bus transfers. These assumptions are practically motivated and common in the literature, but our work is the first to formally demonstrate their necessity.