APX-hard Research Articles

An Improved Approximation Algorithm for The Asymmetric Traveling Salesman Problem

We revisit the constant-factor approximation algorithm for the asymmetric traveling salesman problem by Svensson, Tarnawski, and Végh [J. ACM, 67 (2020), 37]. We improve on each part of this algorithm. We avoid the reduction to irreducible instances and thus obtain a simpler and much better reduction to vertebrate pairs. We also show that a slight variant of their algorithm for vertebrate pairs has a much smaller approximation ratio. Overall we improve the approximation ratio from 506 to $22+\epsilon$ for any $\epsilon > 0$. This also improves the upper bound on the integrality ratio from 319 to 22.

Budget-Feasible Mechanisms in Two-Sided Crowdsensing Markets: Truthfulness, Fairness, and Efficiency

In a crowdsensing platform, users are invited to provide data services, and multiple requesters compete for desired services. Due to users' costs of providing services, it is critical to design incentive mechanisms to incentivize users with (monetary) rewards. Meanwhile, requesters may have individual budgets and compete for services with different procurement abilities. Such a setting falls into the budget-feasible mechanism design. However, most of the existing budget-feasible mechanisms focus on one-sided markets with a single requester rather than the two-sided markets with multiple requesters having different procurement abilities. Moreover, requesters and users can be selfish and strategic with their private information, which requires preventing information manipulation on both requesters' and users' sides. In this paper, we investigate budget-feasible mechanisms in two-sided crowdsensing markets where multiple strategic requesters come with private budgets to obtain services from the strategic users. We also consider the fairness on the requesters' side, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , a requester with more budget should obtain more service. We propose budget-feasible mechanisms for two models by distinguishing the types of services, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , the homogeneous or heterogeneous services. All proposed mechanisms satisfy fairness, budget feasibility, truthfulness on both users' and requesters' sides, and the constant approximation ratio. Numerical experiment results further demonstrate the efficiency of our proposed mechanisms.

Taming the knight's tour: Minimizing turns and crossings

We introduce two new metrics of “simplicity” for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with 9.25n+O(1) turns and 12n+O(1) crossings on an n×n board, and we show lower bounds of (6−ϵ)n and 4n−O(1) on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of 9.25/6+o(1) and 3+o(1). Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in O(n2/p) time using p processors in the CREW PRAM model. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for (1,4)-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.

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.

An auction based task dispatching and pricing mechanism in bike-sharing

As an economical, low-carbon and convenient travel model, bike-sharing has become common in many cities around the world. However, the daily usage of shared bikes results in the dispatching problem, i.e., dispatching bikes to the specific destinations to satisfy riding demands. The bike-sharing platform can hire riders as workers and pay to incentivize them to accomplish the dispatching tasks. However, there exist multiple workers competing for the dispatching tasks, and they may strategically report their task accomplishing costs (which are usually private information only known by themselves) in order to make more profits, which may result in inefficient task dispatching results. In this paper, we first design a dispatching algorithm named GDY-MAX to allocate tasks to workers, which can achieve good performance. However, it cannot prevent workers strategically misreporting their task accomplishing costs. Regarding this issue, we further design a strategy proof mechanism under the budget constraint, which consists of a task dispatching algorithm and a worker pricing algorithm. We theoretically prove that our mechanism can satisfy incentive compatibility, individual rationality, budget constraint and a constant approximation ratio. Furthermore, we run extensive experiments to evaluate our mechanism based on a Mobike dataset. The results show that the performance of the proposed strategy proof mechanism and GDY-MAX is similar to the optimal algorithm in terms of the coverage ratio of accomplished task regions and the sum of task region value, and our mechanism has better performance than the uniform algorithm in terms of the total payment and the unit cost value.

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.

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.

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.

Approximating nash social welfare under rado valuations

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ∑Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.

Scheduling Stochastic Jobs - Complexity and Approximation Algorithms

Uncertainty is an omnipresent issue in real-world optimization problems. This paper studies a fundamental problem concerning uncertainty, known as the beta-robust scheduling problem. Given a set of identical machines and a set of jobs whose processing times follow a normal distribution, the goal is to assign jobs to machines such that the probability that all the jobs are completed by a given common due date is maximized. We give the first systematic study on the complexity and algorithms for this problem. A strong negative result is shown by ruling out the existence of any polynomial-time algorithm with a constant approximation ratio for the general problem unless P=NP. On the positive side, we provide the first FPT-AS (fixed parameter tractable approximation scheme) parameterized by the number of different kinds of jobs, which is a common parameter in scheduling problems. It returns a solution arbitrarily close to the optimal solution, provided that the job processing times follow a few different types of distributions. We further complement the theoretical results by implementing our algorithm. The experiments demonstrate that by choosing an appropriate approximation ratio, the algorithm can efficiently compute a near-optimal solution.

Tight Localizations of Feedback Sets

The classical NP–hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs ε ⊆ E or vertices ν ⊆ V whose removal G ∖ ε, G ∖ ν makes a given multi–digraph G =( V , E ) acyclic, respectively. Though both problems are known to be APX–hard, constant ratio approximations or proofs of inapproximability are unknown. We propose a new universal O (| V || E | 4 )–heuristic for the directed FASP. While a ratio of r ≈ 1.3606 is known to be a lower bound for the APX–hardness, at least by empirical validation we achieve an approximation of r ≤ 2. Most of the relevant applications, such as circuit testing , ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds with our approach.

Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint

Arising from practical problems such as in sensor placement and influence maximization in social network, submodular and non-submodular maximization on the integer lattice has attracted much attention recently. In this work, we consider the problem of maximizing the sum of a monotone non-negative diminishing return submodular (DR-submodular) function and a supermodular function on the integer lattice subject to a cardinality constraint. By exploiting the special combinatorial structures in the problem, we introduce a decreasing threshold greedy algorithm with a binary search as its subroutine to solve the problem. To avoid introducing the diminishing return ratio and submodularity ratio of the objective function, we generalize the total curvatures of submodular functions and supermodular functions to the integer lattice version. We show that our algorithm has a constant approximation ratio parameterized by the new introduced total curvatures on integer lattice with a polynomial query complexity.

Coverage-Aware Stable Task Assignment in Opportunistic Mobile Crowdsensing

In opportunistic mobile crowdsensing, the objective of service requesters is to have as many of their sensing tasks completed as possible within their budget constraints, whereas that of participants (workers) is to collect the highest monetary reward possible on their trajectories. However, these objectives can conflict and may result in unhappy service requesters or workers if the matching between them is not handled carefully. In this paper, we study the problem of finding task assignments that fulfill both coverage-aware preferences of service requesters and profit-based preferences of workers in a budget-constrained, opportunistic mobile crowdsensing system. Since this is a matching problem with bilateral preferences, we aim to find a matching in which everyone is satisfied with their assignment based on their preference profile. We first propose a polynomial-time approximation algorithm for general settings, and then show that a slightly modified version of this algorithm has a constant approximation ratio when the rewards offered to workers by service requesters are proportional to the coverage capability of workers for corresponding tasks. Through extensive simulations, we evaluate the performance of our algorithms in different settings, and show that they mostly provide substantially better task assignments in terms of user happiness and coverage quality while having a few orders of magnitude lower running times compared to the benchmark algorithms.

Non-Submodular Maximization with Matroid and Knapsack Constraints

We investigate the problem of maximizing a [Formula: see text]-submodular function subject to one or multiple matroid constraints and one knapsack constraint. By the greedy local search technique, we present approximation algorithms with constant approximation ratios. When [Formula: see text], our model reduces to the regular submodular maximization problem investigated in the literature under the same type of constraints. Our results therefore contribute towards the line of research on constrained non-submodular function maximization.

An approximation algorithm for General Energy Restricted Sweep Coverage problem

Considerable attention has been paid to sweep coverage in Wireless Sensor Networks (WSNs) for the past few years. In sweep coverage, mobile sensor nodes are scheduled to visit Points of Interests (POIs) periodically. Due to the limited energy of mobile sensors, energy consumption poses an important challenge in sweep coverage. In this paper, we extend the Energy Restricted Sweep Coverage to the General Energy Restricted Sweep Coverage (GERSC) by assuming the unit energy consumption of mobile sensors varies in different road sections. The goal of GERSC is also to minimize the required number of mobile sensors in sweep coverage under the energy constraint. In GERSC, the approximation ratio of the state-of-the-art algorithm named ERSweepCoverage is no longer guaranteed. Therefore, we devise a constant-factor approximation algorithm named IERSC for GERSC. The approximation ratio of IERSC is β[3αmin(θ,1−θ)+1]. θ is the predefined parameter of IERSC ranging from 0 to 1. With θ being 12, IERSC has the best approximation ratio of (6α+1)β. Finally, the promising experimental results demonstrate the effectiveness and efficiency of the proposed algorithm.