Code and Data Repository for A New Approximation Algorithm for Minimum-Weight (1,m)–Connected Dominating Set

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of O(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.

Community search on attributed networks has recently attracted great deal of research interest. However, most of existing works require query users to specify some community structure parameters. This may not be always practical as sometimes a user does not have the knowledge and experience to decide the suitable parameters. In this paper, we propose a novel parameter-free contextual community model for attributed community search. The proposed model only requires a query context, i.e., a set of keywords describing the desired matching community context, while the community returned is both structure and attribute cohesive w.r.t. the provided query context. We theoretically show that both our exact and approximate contextual community search algorithms can be executed in worst case polynomial time. The exact algorithm is based on an elegant parametric maximum flow technique and the approximation algorithm that significantly improves the search efficiency is analyzed to have an approximation factor of 1/3. In the experiment, we use six real networks with ground-truth communities to evaluate the effectiveness of our contextual community model. Experimental results demonstrate that the proposed model can find near ground-truth communities. We also test both our exact and approximate algorithms using eight large real networks to demonstrate the high efficiency of the proposed algorithms.

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17].

We investigate a generalization of classical shop scheduling where n jobs are located at the vertices of a general undirected graph and m machines must travel between the vertices to process the jobs. The aim is to minimize the makespan. For the open shop problem, we develop an O(logmloglogm)-approximation algorithm that significantly improves upon the best known $O(\sqrt{m})$ -approximation algorithm. For the flow shop problem, we present an O(m 2/3)-approximation algorithm that improves upon the best known $\max\{\frac{m+1}{2},\rho\}$ -approximation algorithm, where ρ is the approximation factor for metric TSP.

Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produces polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2/(d-1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.

Approximate reasoning for the Semantic Web is based on the idea of sacrificing soundness or completeness for a significant speed-up of reasoning. This is to be done in such a way that the number of introduced mistakes is at least outweighed by the obtained speed-up. When pursuing such approximate reasoning approaches, however, it is important to be critical not only about appropriate application domains, but also about the quality of the resulting approximate reasoning procedures. With different approximate reasoning algorithms discussed and developed in the literature, it needs to be clarified how these approaches can be compared, i.e. what it means that one approximate reasoning approach is better than some other. In this paper, we will formally define such a foundation for approximate reasoning research. We will clarify --- by means of notions from statistics --- how different approximate algorithms can be compared, and ground the most fundamental notions in the field formally. We will also exemplify what a corresponding statistical comparison of algorithms would look like.

On the optimality of exact and approximation algorithms for scheduling problems

Suppose we are given a set of objects that cover a region and a duration associated with each object. Viewing the objects as jobs, can we schedule their beginning times to maximize the length of time that the original region remains covered? We call this problem the SENSOR COVER PROBLEM. It arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence (interval) by sensors placed at various fixed locations. Each sensor has a range (also an interval) and limited battery life. The problem is then to schedule when to turn on the sensors so that the fence is fully monitored for as long as possible.This one-dimensional problem involves intervals on the real line. Associating a duration to each yields a set of rectangles in space and time, each specified by a pair of fixed horizontal endpoints and a height. The objective is to assign a bottom position to each rectangle (by moving them up or down) so as to maximize the height at which the spanning interval is fully covered. We call this one-dimensional problem RESTRICTED STRIP COVERING. If we replace the covering constraint by a packing constraint (rectangles may not overlap, and the goal is to minimize the highest point covered), then the problem becomes identical to DYNAMIC STORAGE ALLOCATION, a well-studied scheduling problem, which is in turn a restricted case of the well known problem STRIP PACKING.We present a collection of algorithms for RESTRICTED STRIP COVERING. We show that the problem is NP-hard and present an O(log log log n)-approximation algorithm. We also present better approximation or exact algorithms for some special cases, including when all intervals have equal width. For the general SENSOR COVER PROBLEM, we distinguish between cases in which elements have uniform or variable durations. The results depend on the structure of the region to be covered: We give a polynomial-time, exact algorithm for the uniform-duration case of RESTRICTED STRIP COVERING but prove that the uniform-duration case for higher-dimensional regions is NP-hard. We give some more specific results for two-dimensional regions. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm for the most general case.

Linear and non-linear data-detection and precoding algorithms for wideband massive multi-user (MU) multiple-input multiple-output (MIMO) wireless systems that rely on orthogonal frequency-division multiplexing (OFDM) or single-carrier frequency-division multiple access (SC-FDMA) often require computation of the Gram matrix for each active subcarrier. Computing the Gram matrix for each active subcarrier, however, results in excessively high computational complexity. In this paper, we propose novel, approximate algorithms that significantly reduce the complexity of Gram-matrix computation by simultaneously exploiting channel hardening and correlation across subcarriers. We show analytically that a small fraction of Gram-matrix computations in combination with approximate interpolation schemes are sufficient to achieve near-optimal error-rate performance at low computational complexity in massive MU-MIMO systems. We furthermore demonstrate that our approximate interpolation algorithms are more robust against channel-estimation errors than exact Gram-matrix interpolation algorithms that require high computational complexity.

On tightness of Tsaknakis-Spirakis descent methods for approximate Nash equilibria

A federated kernel k -means (FedKKM) algorithm is developed in this article to conduct distributed clustering with low memory consumption on user devices. In FedKKM, a federated eigenvector approximation (FEA) algorithm is designed to iteratively determine the low-dimensional approximate vectors of the transformed feature vectors, using only low-dimensional random feature vectors. To maintain high communication efficiency in each iteration of FEA, a communication-efficient Lanczos algorithm (CELA) is further designed in FEA to reduce the communication cost. Based on the low-dimensional approximate vectors, the clustering result is obtained by leveraging a distributed linear k -means algorithm. A theoretical analysis shows that: 1) FEA has a convergence rate of O(1/T) , where T is the number of iterations; 2) the scalability of FedKKM is not affected by the dataset size since the communication cost of FedKKM is independent of the number of users' data; and 3) FedKKM is a (1+ϵ) approximation algorithm. The experimental results show that FedKKM achieves the comparable clustering quality to that of a centralized kernel k -means. Compared with state-of-the-art schemes, FedKKM reduces the memory consumption on user devices by up to 94% and also reduces the communication cost by more than 40%.

In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the Bandwidth problem. For graphs with treewidth t we present a O(n O(t) 2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for any α> 0 there exists a (1 + α)-approximation algorithm running in \(O(\exp(c(t + \sqrt{n/\alpha})\log n))\) time where c is a universal constant. These results seem interesting since Unger has proved that Bandwidth does not belong to APX even when the input graph is a tree (assuming P ≠ NP). So somewhat surprisingly, despite Unger’s result it turns out that not only a subexponential constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r, we present a (4r − 1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in \(O^*(2^{n\over r})\) time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n) time and space algorithm.In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O *(2n) time and space, what matches the best known results.KeywordsFast Fourier TransformApproximation AlgorithmChromatic NumberExact AlgorithmInput GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given an n-point metric space (M, d), metric 1-median asks for a point \(p\in M\) minimizing \(\sum _{x\in M}\,d(p,x)\). We show that for each computable function \(f:\mathbb {Z}^+\rightarrow \mathbb {Z}^+\) satisfying \(f(n)=\omega (1)\), metric 1-median has a deterministic, o(n)-query, \(o(f(n)\cdot \log n)\)-approximation and nonadaptive algorithm. Previously, no deterministic o(n)-query o(n)-approximation algorithms are known for metric 1-median.

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding method leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies 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 the three algorithms for problem instances arising in the context of real-world gas transport networks.