Achieves Approximation Ratio Research Articles

Structural iterative rounding for generalized k-median problems

AbstractThis paper considers approximation algorithms for generalized k-median problems. These problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for k-median with outliers as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018). The main technical innovation is allowing richer constraint sets in the iterative rounding and using the structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios $$6.994 + \epsilon $$ 6.994 + ϵ and $$6.387 + \epsilon $$ 6.387 + ϵ for k-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio $$7.081 + \epsilon $$ 7.081 + ϵ for both problems as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018).

The obnoxious facility location game with dichotomous preferences

We consider a facility location game in which n agents reside at known locations on a path, and k heterogeneous facilities are to be constructed on the path. Each agent is adversely affected by some subset of the facilities, and is unaffected by the others. We design two classes of mechanisms for choosing the facility locations given the reported agent preferences: utilitarian mechanisms that strive to maximize social welfare (i.e., to be efficient), and egalitarian mechanisms that strive to maximize the minimum welfare. For the utilitarian objective, we present a weakly group-strategyproof efficient mechanism for up to three facilities, we give strongly group-strategyproof mechanisms that achieve approximation ratios of 5/3 and 2 for k=1 and k>1, respectively, and we prove that no strongly group-strategyproof mechanism achieves an approximation ratio less than 5/3 for the case of a single facility. For the egalitarian objective, we present a strategyproof egalitarian mechanism for arbitrary k, and we prove that no weakly group-strategyproof mechanism achieves a o(n) approximation ratio for two facilities. We extend our egalitarian results to the case where the agents are located on a cycle, and we extend our first egalitarian result to the case where the agents are located in the unit square.

(In)approximability of maximum minimal FVS

We study the approximability of the NP-complete Maximum Minimal Feedback Vertex Set problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: Maximum Minimal Vertex Cover, for which the best achievable approximation ratio is n, and Upper Dominating Set, which does not admit any n1−ϵ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for Maximum Minimal Feedback Vertex Set with a ratio of O(n2/3), as well as a matching hardness of approximation bound of n2/3−ϵ, improving the previously known hardness of n1/2−ϵ. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r, produces an r-approximate solution in time nO(n/r3/2). This time-approximation trade-off is essentially tight under the ETH.

Parallel approximation for partial set cover

In a minimum partial set cover problem (MinPSC), given a ground set E with n elements, a collection S of subsets of E with |S|=m, a cost function c:S→R+, and an integer k≤n, the goal of MinPSC is to find a minimum cost sub-collection of S that covers at least k elements of E. In this paper, we design a parallel algorithm for MinPSC which yields a solution with approximation ratio at most f1−2ε in O(1εlogmnε) rounds, where f is the maximum number of sets containing a common element, and 0<ε<1/2 is a constant. We also design a parallel algorithm for a special MinPSC problem, the minimum power partial cover problem (MinPPC), which achieves approximation ratio at most (3+2ε)α1−2ε in O(1εlogmnεlog2m) rounds, where α≥1 is the attenuation factor of power.

Efficient Algorithms for Max-Weighted Point Sweep Coverage on Lines.

As an important application of wireless sensor networks (WSNs), deployment of mobile sensors to periodically monitor (sweep cover) a set of points of interest (PoIs) arises in various applications, such as environmental monitoring and data collection. For a set of PoIs in an Eulerian graph, the point sweep coverage problem of deploying the fewest sensors to periodically cover a set of PoIs is known to be Non-deterministic Polynomial Hard (NP-hard), even if all sensors have the same velocity. In this paper, we consider the problem of finding the set of PoIs on a line periodically covered by a given set of mobile sensors that has the maximum sum of weight. The problem is first proven NP-hard when sensors are with different velocities in this paper. Optimal and approximate solutions are also presented for sensors with the same and different velocities, respectively. For M sensors and N PoIs, the optimal algorithm for the case when sensors are with the same velocity runs in time; our polynomial-time approximation algorithm for the case when sensors have a constant number of velocities achieves approximation ratio ; for the general case of arbitrary velocities, and approximation algorithms are presented, respectively, where integer is the tradeoff factor between time complexity and approximation ratio.

Natural evolution strategies and variational Monte Carlo

A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. The recent work of Gomes et al (2019 arXiv:1910.10675) on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies (NES). The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular, it is found that NES can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.

Combinatorial Auctions Do Need Modest Interaction

We study the necessity of interaction for obtaining efficient allocations in combinatorial auctions with subadditive bidders . This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC’14) as the following simple market scenario: m items are to be allocated among n bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: In each round, each bidder, simultaneously with others, broadcasts a message to all parties involved. At the end, the central planner computes an allocation solely based on the communicated messages. Dobzinski et al. showed that (at least some) interaction is necessary for obtaining even an approximately efficient allocation: No non-interactive (1-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than Ω( m 1/4 ), while for any r ≥ 1, there exists r -round protocols that achieve Õ ( r · m 1/( r +1) ) approximation with polynomial communication. This in particular implies that O (log m ) rounds of interaction suffice to obtain an approximately efficient allocation—namely, a polylog(m)-approximation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain approximately efficient allocations. In this article, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: We show that for any r ≥ 1, any r -round protocol that uses poly( m , n ) bits of communication can only approximate the social welfare up to a factor of Ω(1/ r · m 1/(2 r +1) ). This in particular implies that Ω(log m /log log m ) rounds of interaction are necessary for obtaining any allocation with a reasonable welfare approximation—namely, a constant or even a polylog( m )-approximation. Our work builds on the multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS’15)—used to prove similar-in-spirit lower bounds for round-approximation tradeoff in unit-demand markets—and settles an open question posed initially by Dobzinski et al. (STOC’14) and subsequently by Alon et al. (FOCS’15).

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. Durr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the approximation ratio converges to the inverse of the golden ratio $2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is extremely fast --- the $5$-pass Category-Advice algorithm is already within $0.01\%$ of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than $1-1/e$, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.

Streaming Communication Protocols

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. Input arrives as a stream, spread between several agents across a network. Each agent has a bounded memory, which can be updated upon receiving a new bit, or a message from another agent. We provide tight tradeoffs between the necessary resources, i.e., communication between agents and memory, for some of the canonical problems from communication complexity by proving a strong general lower bound technique. Second, we analyze the Approximate Matching problem and show that the complexity of this problem (i.e., the achievable approximation ratio) in the one-way variant of our model is strictly different both from the streaming complexity and the one-way communication complexity thereof.

A Family of Scheduling Algorithms for Hybrid Parallel Platforms

More and more parallel computing platforms are built upon hybrid architectures combining multi-core processors (CPUs) and hardware accelerators like General Purpose Graphics Processing Units (GPGPUs). We present in this paper a new method for scheduling efficiently parallel applications with [Formula: see text] CPUs and [Formula: see text] GPGPUs, where each task of the application can be processed either on an usual core (CPU) or on a GPGPU. We consider the problem of scheduling [Formula: see text] independent tasks with the objective to minimize the time for completing the whole application (makespan). This problem is NP-hard, thus, we present two families of approximation algorithms that can achieve approximation ratios of [Formula: see text] or [Formula: see text] for any integer [Formula: see text] when only one GPGPU is considered, and [Formula: see text] or [Formula: see text] for [Formula: see text] GPGPUs, where [Formula: see text] is an arbitrary small value which corresponds to the target accuracy of a binary search. The proposed method is based on a dual approximation scheme that uses a dynamic programming algorithm. The associated computational costs are for the first (resp. second) family in [Formula: see text] (resp. [Formula: see text]) per step of dual approximation. The greater the value of parameter [Formula: see text], the better the approximation, but the more expensive the computational cost. Finally, we propose a relaxed version of the algorithm which achieves a running time in [Formula: see text] with a constant approximation bound of [Formula: see text]. This last result is compared to the state-of-the-art algorithm HEFT. The proposed solving method is the first general purpose algorithm for scheduling on hybrid machines with a theoretical performance guarantee that can be used for practical purposes.

Distributed backup placement in networks

We consider the backup placement problem, defined as follows. Some nodes (processors) in a given network have objects (e.g., files, tasks) whose backups should be stored in additional nodes for increased fault resilience. To minimize the disturbance in case of a failure, it is required that a backup copy should be located at a neighbor of the primary node. The goal is to find an assignment of backup copies to nodes which minimizes the maximum load (number or total size of backup copies) over all nodes in the network. It is known that a natural selfish local improvement policy has approximation ratio $$\varOmega (\log n/\log \log n)$$ ; we show that it may take this policy $$\varOmega (\sqrt{n})$$ time to reach equilibrium in the distributed setting. Our main result in this paper is a randomized distributed algorithm which finds a placement in polylogarithmic time and achieves approximation ratio $$O\left(\frac{\log n}{\log \log n}\right)$$ . We obtain this result using a randomized distributed approximation algorithm for f-matching in bipartite graphs that may be of independent interest.

Distributed algorithms for content placement in hierarchical cache networks

The growing popularity of mobile multimedia content and the increase of wireless access bitrates are straining backhaul capacity in mobile networks. A cost-effective solution to reduce the strain, enabled by emerging all-IP 4G and 5G mobile backhaul architectures, could be in-network caching of popular content during times of peak demand. Motivated by the potential benefits of caching in mobile backhaul networks, in this paper we formulate the problem of content placement in a hierarchical cache network as a binary integer programming problem. We provide a polynomial time solution when the link costs are induced by a potential and we propose a 2-approximation algorithm for the general case. The 2-approximation requires full information about the network topology and the link costs, as well as about the content demands at the different caches, we thus propose two distributed algorithms that are based on limited information on the content demands. We show that the distributed algorithms terminate in a finite number of steps, and we provide analytical results on their approximation ratios. We use simulations to evaluate the proposed algorithms in terms of the achieved approximation ratio and computational complexity on hierarchical cache network topologies as a model of mobile backhaul networks.

Relationship between superstring and compression measures: New insights on the greedy conjecture

A superstring of a set of words is a string that contains each input word as a substring. Given such a set, the Shortest Superstring Problem (SSP) asks for a superstring of minimum length. SSP is an important theoretical problem related to the Asymmetric Travelling Salesman Problem, and also has practical applications in data compression and in bioinformatics. Indeed, it models the question of assembling a genome from a set of sequencing reads. Unfortunately, SSP is known to be NP-hard even on a binary alphabet and also hard to approximate with respect to the superstring length or to the compression achieved by the superstring. Even the variant in which all words share the same length r, called r-SSP, is NP-hard whenever r>2. Numerous involved approximation algorithms achieve approximation ratio above 2 for the superstring, but remain difficult to implement in practice. In contrast the greedy conjecture asked in 1988 whether a simple greedy algorithm achieves ratio of 2 for SSP. Here, we present a novel approach to bound the superstring approximation ratio with the compression ratio, which, when applied to the greedy algorithm, shows a 2 approximation ratio for 3-SSP, and also that greedy achieves ratios smaller than 2. This leads to a new version of the greedy conjecture.

Greedy Algorithms for the Maximum Satisfiability Problem: Simple Algorithms and Inapproximability Bounds

We give a simple, randomized greedy algorithm for the maximum satisfiability problem (MAX SAT) that obtains a $\frac{3}{4}$-approximation in expectation. In contrast to previously known $\frac{3}{4}$-approximation algorithms, our algorithm does not use flows or linear programming. Hence we provide a positive answer to a question posed by Williamson in 1998 on whether such an algorithm exists. Moreover, we show that Johnson's greedy algorithm cannot guarantee a $\frac{3}{4}$-approximation, even if the variables are processed in a random order. Thereby we partially solve a problem posed by Chen, Friesen, and Zheng in 1999. In order to explore the limitations of the greedy paradigm, we use the model of priority algorithms of Borodin, Nielsen, and Rackoff. Since our greedy algorithm works in an online scenario where the variables arrive with their set of undecided clauses, we wonder if a better approximation ratio can be obtained by further fine-tuning its random decisions. For a particular information model we show that no priority algorithm can approximate Online MAX SAT within $\frac{3}{4} + \varepsilon$ (for any $\varepsilon > 0$). We further investigate the strength of deterministic greedy algorithms that may choose the variable ordering. Here we show that no adaptive priority algorithm can achieve approximation ratio $\frac{3}{4}$. We propose two ways in which this inapproximability result can be bypassed. First we show that if our greedy algorithm is additionally given the variable assignments of an optimal solution to the canonical LP relaxation, then we can derandomize its decisions while preserving the overall approximation guarantee. Second we give a simple, deterministic algorithm that performs an additional pass over the input. We show that this 2-pass algorithm satisfies clauses with a total weight of at least $\frac{3}{4} {OPT}_{LP}$, where ${OPT}_{LP}$ is the objective value of the canonical linear program. Moreover, we demonstrate that our analysis is tight and detail how each pass can be implemented in linear time.

Revisiting a randomized algorithm for the minimum rainbow subgraph problem

The minimum rainbow subgraph problem arises in bioinformatics. The graph is given as an edge-colored undirected graph. Our goal is to find a subgraph with minimum number of vertices such that there is exactly one edge from each color class. The currently best approximation ratio achieved by a deterministic approximation algorithm is O(Δ). (Here Δ is the max degree of a graph.) In [4] he proposes a randomized algorithm which achieves an approximation ratio of O(Δln⁡Δ). However, we find that there is a flaw in his probability analysis which renders this approximation ratio invalid. We present a simple example to show why his analysis does not work. Instead, we propose an alternative analysis for his randomized algorithm. Our estimate shows that this randomized algorithm may achieve approximation ratio of O(Δ) in general. However, if the number of colors is Θ(nΔr) for some positive r≤1, his randomized algorithm can beat the bound of O(Δ). Moreover, through our analysis, we also find that if we impose an extra constraint on the color function, the bound O(Δln⁡Δ) still holds.

Submodular maximization meets streaming: matchings, matroids, and more

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching. We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range--they store only $$O(n)$$O(n) edges, using $$O(n\log n)$$O(nlogn) working memory--that achieve approximation ratios of 7.75 in a single pass and $$(3+\varepsilon )$$(3+?) in $$O(\varepsilon ^{-3})$$O(?-3) passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. Our framework is not specific to matchings. Rather, we identify a general pattern for algorithms that maximize linear weight functions over and prove that such algorithms can be adapted to maximize a submodular function. The notion of independence here is very general; in particular, appealing to known weight-maximization algorithms, we obtain results for submodular maximization over hypermatchings in hypergraphs as well as independent sets in the intersection of multiple matroids.

Maximum Capacity Overlapping Channel Assignment Based on Max-Cut in 802.11 Wireless Mesh Networks

By exploiting multi-radio multi-channel technology, wireless mesh networks can effectively provide wireless broadband access to the Internet for mobile users. Due to the limited number of orthogonal channels, overlapping channel assignment is one of the main factors that greatly affect the network capacity. However, current results in this area are not so satisfying. In this paper, we first propose a model for measuring achieved network capacity in MR-WMNs. Then we prove that finding an optimal overlapping channel assignment in a given MR-WMN with odd number of channels, is equivalent to finding an optimal assignment by only using its orthogonal channels. This theory allows us to use fewer channels to solve complicated channel assignment problems. Third, we prove that in 802.11b/g MR-WMN the simplified optimization problem is a Max-3-Cut problem. Although this problem is NP-hard, it has an efficient approximation algorithm that achieves approximation ratio of 1.19616 probabilistically by using the algorithm for Max-Cut whose approximation ratio is 1.1383 probabilistically. Based on the algorithm for Max-Cut, this paper proposes Max-Cut based channel assignment (MCCA) which uses a heuristic method to adjust the result produced by the Max-Cut algorithm to achieve an even better result. Finally, we perform extensive simulations to compare the MCCA with a state-of-the-art Tabu-Search based algorithm. The results show that the Max-Cut based overlapping channel assignment algorithm effectively and efficiently improves on the network capacity compared with existing algorithms.

Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ (log n ), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O (log 3 n ), or O (log 2 n ) when the host graph is a tree. We obtain an O (log n polyloglog n ) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group.

A probabilistic PTAS for shortest common superstring

We consider approximation algorithms for the shortest common superstring problem (SCS). It is well known that there is a constant f>1 such that there is no efficient approximation algorithm for SCS achieving a factor of at most f in the worst case, unless P=NP. We study SCS on random inputs and present an approximation scheme that achieves, for every ε>0, a (1+ε)-approximation in expected polynomial time. We also show that the greedy algorithm achieves approximation ratio 1+ε with probability exponentially close to 1. These results apply not only if the letters are chosen independently at random, but also to the more realistic mixing model, which allows dependencies among the letters of the random strings. Our results are based on a sharp tail bound on the optimal compression, which improves a previous result by Frieze and Szpankowski (1998).

Efficient Coordination Mechanisms for Unrelated Machine Scheduling

We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game.Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process the jobs in an uninterrupted way and may introduce some idle time. Our first coordination mechanism has approximation ratio Θ(logm) and always guarantees that the induced game has pure Nash equilibria to which the system converges in at most n rounds. This result improves a bound of O(log2 m) due to Azar, Jain, and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs that have different load characteristics. Our second mechanism handles anonymous jobs and has approximation ratio \(O(\frac{\log m}{\log\log m})\) although the game induced is not a potential game and, hence, the existence of pure Nash equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles anonymous jobs and has a nice “cost-revealing” potential function. We use this potential function in order, not only to prove the existence of equilibria, but also to upper-bound the price of stability of the induced game by O(logm) and the price of anarchy by O(log2 m). Our third coordination mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy. In order to obtain the above bounds, our coordination mechanisms use m as a parameter. Slight variations of these mechanisms in which this information is not necessary achieve approximation ratios of O(m ϵ), for any constant ϵ>0.