Linear Time Approximation Research Articles

Neural Network Approximators for Marginal MAP in Probabilistic Circuits

Probabilistic circuits (PCs) such as sum-product networks efficiently represent large multi-variate probability distributions. They are preferred in practice over other probabilistic representations, such as Bayesian and Markov networks, because PCs can solve marginal inference (MAR) tasks in time that scales linearly in the size of the network. Unfortunately, the most probable explanation (MPE) task and its generalization, the marginal maximum-a-posteriori (MMAP) inference task remain NP-hard in these models. Inspired by the recent work on using neural networks for generating near-optimal solutions to optimization problems such as integer linear programming, we propose an approach that uses neural networks to approximate MMAP inference in PCs. The key idea in our approach is to approximate the cost of an assignment to the query variables using a continuous multilinear function and then use the latter as a loss function. The two main benefits of our new method are that it is self-supervised, and after the neural network is learned, it requires only linear time to output a solution. We evaluate our new approach on several benchmark datasets and show that it outperforms three competing linear time approximations: max-product inference, max-marginal inference, and sequential estimation, which are used in practice to solve MMAP tasks in PCs.

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ>0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an (1+ϵ)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ,ϵ and k. This improves the O(n2d)-time algorithm by Eiben et al. (2021) [7] by a factor of n.

Configuring Intelligent Reflecting Surface With Performance Guarantees: Optimal Beamforming

This work proposes linear time strategies to optimally configure the phase shifts for the reflective elements of an intelligent reflecting surface (IRS). Specifically, we show that the binary phase beamforming can be optimally solved in linear time to maximize the received signal-to-noise ratio (SNR). For the general K-ary phase beamforming, we develop a linear time approximation algorithm that guarantees performance within a constant fraction (1+\cos(\pi/K))/2 of the global optimum, e.g., it can attain over 85% of the optimal performance for the quadrature beamforming with K=4. According to the numerical results, the proposed approximation algorithm for discrete IRS beamforming outperforms the existing algorithms significantly in boosting the received SNR.

Complexity of paired domination in AT-free and planar graphs

For a graph G=(V,E), a subset D of vertex set V, is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if the subgraph induced by D in G has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem is NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the other side, the problem is efficiently solvable for many graph classes including interval graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graphs. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NP-complete for planar graphs, which answers an open question asked by Lin et al. (2015) [19] and (2020) [20]. Alvarado et al. (2015) [1] conjectured that given a graph G=(V,E), it is NP-hard to decide whether γ(G)=γpr(G). In this paper we settle this conjecture affirmatively.

A linear time approximation scheme for scheduling unbounded batch machines with delivery times and inclusive processing set restrictions

<abstract><p>We consider the problem of scheduling jobs with delivery times and inclusive processing set restrictions on unbounded batch machines to minimize the maximum delivery completion time, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. We develop a polynomial time approximation scheme for this strongly NP-hard problem that runs in linear time for any fixed accuracy requirement.</p></abstract>

Near-linear Time Approximation Schemes for Clustering in Doubling Metrics

We consider the classic Facility Location, k -Median, and k -Means problems in metric spaces of doubling dimension d . We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is Õ(2 (1/ε) O(d2) n) , making a significant improvement over the state-of-the-art algorithms that run in time n (d/ε) O(d) . Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k -Median and k -Means and efficient bicriteria approximation schemes for k -Median with outliers, k -Means with outliers and k -Center.

A simple method for proving lower bounds in the zero-visibility cops and robber game

Cops and Robbers is a classical pursuit and evasion game in graph theory, which was introduced by Nowakowski and Winkler and independently by Quilliot. In this paper, we study the zero-visibility cops and robber game, which is a variant of Cops and Robbers. In the zero-visibility cops and robber game, the robber is invisible to the cops throughout the game. We introduce a simple method for proving lower bounds on the zero-visibility cop number. This lower bound method is based on a connection between the zero-visibility cop number and the matching number. Using this technique, we investigate graph joins, lexicographic products of graphs, complete multipartite graphs and split graphs. For each of these classes of graphs, we prove lower bounds and upper bounds on the zero-visibility cop number. We also present a linear time approximation algorithm for computing the lexicographic product of a tree and a graph G. The approximation ratio of this algorithm is bounded by $$|V(G)| / (\nu (G) + |V(G) {\setminus } V(\mathcal {M}(G))| )$$ , where V(G) is the vertex set of G, $$\nu (G)$$ is the matching number of G, $$\mathcal {M}(G)$$ is a maximum matching of G and $$V(\mathcal {M}(G))$$ is the vertex set of $$\mathcal {M}(G)$$ .

LinearPartition: linear-time approximation of RNA folding partition function and base-pairing probabilities.

MotivationRNA secondary structure prediction is widely used to understand RNA function. Recently, there has been a shift away from the classical minimum free energy methods to partition function-based methods that account for folding ensembles and can therefore estimate structure and base pair probabilities. However, the classical partition function algorithm scales cubically with sequence length, and is therefore prohibitively slow for long sequences. This slowness is even more severe than cubic-time free energy minimization due to a substantially larger constant factor in runtime.ResultsInspired by the success of our recent LinearFold algorithm that predicts the approximate minimum free energy structure in linear time, we design a similar linear-time heuristic algorithm, LinearPartition, to approximate the partition function and base-pairing probabilities, which is shown to be orders of magnitude faster than Vienna RNAfold and CONTRAfold (e.g. 2.5 days versus 1.3 min on a sequence with length 32 753 nt). More interestingly, the resulting base-pairing probabilities are even better correlated with the ground-truth structures. LinearPartition also leads to a small accuracy improvement when used for downstream structure prediction on families with the longest length sequences (16S and 23S rRNAs), as well as a substantial improvement on long-distance base pairs (500+ nt apart).Availability and implementationCode: http://github.com/LinearFold/LinearPartition; Server: http://linearfold.org/partition.Supplementary information Supplementary data are available at Bioinformatics online.

Task Splitting and Load Balancing of Dynamic Real-Time Workloads for Semi-Partitioned EDF

Many real-time software systems, such as those commonly found in the context of multimedia, cloud computing, robotics, and real-time databases, are characterized by a dynamic workload, where applications can join and leave the system at runtime. Global schedulers can transparently support dynamic workload without requiring any off-line task-allocation phase, thus providing advantages to the system designer. Nevertheless, such schedulers exhibit poor worst-case performance when compared to semi-partitioned schedulers, which instead can achieve near-optimal schedulability performance when used in conjunction with smart task splitting and partitioning techniques, and they are also lighter in terms of run-time overhead. This article proposes an approach to efficiently schedule dynamic real-time workloads on multiprocessor systems by means of semi-partitioned scheduling. A linear-time approximation scheme for the C=D splitting algorithm under partitioned EDF scheduling is proposed. Then, a load-balancing algorithm is presented to admit new real-time workloads with a limited number of re-allocations. The article finally reports on a large-scale experimental study showing that (i) the linear-time approximation is characterized by a very limited utilization loss compared with the corresponding exact approach (that has a much higher complexity), and that (ii) the whole approach allows achieving considerable improvements with respect to global and partitioned EDF scheduling.

Approximation Schemes for Low-rank Binary Matrix Approximation Problems

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constraints. The new constrained clustering problem generalizes a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are L ow GF(2)-R ank A pproximation , L ow B oolean -R ank A pproximation , and various versions of B inary C lustering . For example, for L ow GF(2)-R ank A pproximation problem, where for an m × n binary matrix A and integer r > 0, we seek for a binary matrix B of GF(2) rank at most r such that the ℓ 0 -norm of matrix A−B is minimum, our algorithm, for any ϵ > 0 in time f ( r ,ϵ)⋅ n ⋅ m , where f is some computable function, outputs a (1+ϵ)-approximate solution with probability at least (1−1\ e ). This is the first linear time approximation scheme for these problems. We also give (deterministic) PTASes for these problems running in time n f ( r )1\ϵ 2 log 1\ϵ , where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting on its own.

Approximation algorithms in combinatorial scientific computing

We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted$b$-matching, and minimization versions of weighted edge cover and$b$-edge cover. Exact algorithms for these problems are impractical for massive graphs with several millions of edges. For each problem we discuss theoretical foundations, the design of several linear or near-linear time approximation algorithms, their implementations on serial and parallel computers, and applications. Our focus is on practical algorithms that yield good performance on modern computer architectures with multiple threads and interconnected processors. We also include information about the software available for these problems.

Efficient computation of the kinship coefficients.

For families, kinship coefficients are quantifications of the amount of genetic sharing between a pair of individuals. These coefficients are critical for understanding the breeding habits and genetic diversity of diploid populations. Historically, computations of the inbreeding coefficient were used to prohibit inbred marriages and prohibit breeding of some pairs of pedigree animals. Such prohibitions foster genetic diversity and help prevent recessive Mendelian disease at a population level. This paper gives the fastest known algorithms for computing the kinship coefficient of a set of individuals with a known pedigree, especially for large pedigrees. These algorithms outperform existing methods. In addition, the algorithms given here consider the possibility that the founders of the known pedigree may themselves be inbred and compute the appropriate inbreeding-adjusted kinship coefficients, which has not been addressed in literature. The exact kinship algorithm has running-time O(n2) for an n-individual pedigree. The recursive-cut exact kinship algorithm has running time O(s2m) where s is the number of individuals in the largest segment of the pedigree and m is the number of cuts. The approximate algorithm has running-time O(nd) for an n-individual pedigree on which to estimate the kinship coefficients of n individuals of interest from n founder kinship coefficients and d is the number of samples. The above polynomial-time exact algorithm and the linear-time approximation algorithms are implemented as PedKin in C++ and are available under the GNU GPL v2.0 open source license. The PedKin source code is available at: http://www.intrepidnetcomputing.com/research/code/.

A novel parallel distance metric-based approach for diversified ranking on large graphs

Diversified ranking on graphs (DRG) is an important and challenging issue in researching graph data mining. Traditionally, this problem is modeled by a submodular optimization objective, and solved by applying a cardinality constrained monotone submodular maximization. However, the existing submodular objectives do not directly capture the dis-similarity over pairs of nodes, while most of algorithms cannot easily take full advantage of the power of a distributed cluster computing platform, such as Spark, to significantly promote the efficiency of algorithms. To overcome the deficiencies of existing approaches, in this paper, a generalized distance metric based on a subadditive set function over the symmetry difference of neighbors of pairs of nodes is introduced to capture the pairwise dis-similarity over pairs of nodes. In our approach,DRG is formulated as a Max-Sum k-dispersion problem with metrical edge weights, which is NP-hard, in association with the proposed distance metric, a centralized linear time 2-approximation algorithm GA is then developed to significantly solve the problem ofDRG. Moreover, we develop a highly parallelizable algorithm forDRG, which can be easily implemented in MapReduce style parallel computation models using GA as a basic reducer. Finally, extensive experiments are conducted on real network datasets to verify the effectiveness and efficiency of our proposed approaches.

Nearly linear-time packing and covering LP solvers

Packing and covering linear programs (PC-LP s) form an important class of linear programs (LPs) across computer science, operations research, and optimization. Luby and Nisan (in: STOC, ACM Press, New York, 1993) constructed an iterative algorithm for approximately solving PC-LP s in nearly linear time, where the time complexity scales nearly linearly in N, the number of nonzero entries of the matrix, and polynomially in $$\varepsilon $$ , the (multiplicative) approximation error. Unfortunately, existing nearly linear-time algorithms (Plotkin et al. in Math Oper Res 20(2):257–301, 1995; Bartal et al., in: Proceedings 38th annual symposium on foundations of computer science, IEEE Computer Society, 1997; Young, in: 42nd annual IEEE symposium on foundations of computer science (FOCS’01), IEEE Computer Society, 2001; Koufogiannakis and Young in Algorithmica 70:494–506, 2013; Young in Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs, 2014. arXiv:1407.3015 ; Allen-Zhu and Orecchia, in: SODA, 2015) for solving PC-LP s require time at least proportional to $$\varepsilon ^{-2}$$ . In this paper, we break this longstanding barrier by designing a packing solver that runs in time $$\widetilde{O}(N \varepsilon ^{-1})$$ and covering LP solver that runs in time $$\widetilde{O}(N \varepsilon ^{-1.5})$$ . Our packing solver can be extended to run in time $$\widetilde{O}(N \varepsilon ^{-1})$$ for a class of well-behaved covering programs. In a follow-up work, Wang et al. (in: ICALP, 2016) showed that all covering LPs can be converted into well-behaved ones by a reduction that blows up the problem size only logarithmically.

Provably Good Max–Min- $m$ -Neighbor-TSP-Based Subfield Scheduling for Electron-Beam Photomask Fabrication

Electron beam lithography (EBL) has been used for high-resolution photomask fabrication; its successive heating process in a certain region, however, may cause critical dimension (CD) distortion. As a result, subfield scheduling, which reorders a sequence of subfields in the writing process, is desirable to avoid the heating problem and thus CD distortion. To consider longer range heat dissipation, this paper models a subfield scheduling problem with blocked region consideration as a constrained max–min <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> -neighbor traveling salesman problem (called constrained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> -nTSP). To solve the constrained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> -nTSP which is NP-complete in general, we decompose a constrained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> -nTSP into subproblems conforming to a special case with points on two parallel lines, solve each of them with a provably good linear-time approximation algorithm, and merge them into a complete scheduling solution. In particular, our algorithm can also minimize the distances between successive subfields to alleviate the throughput degradation of EBL writing due to moving a writing head, while minimizing the heating problem. Average reductions of 10% in the maximum temperature and 14% in the distances between successive subfields over the state-of-the-art work can be achieved.

Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs

Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time [Formula: see text]-approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.

Sparse certificates for 2-connectivity in directed graphs

Motivated by the emergence of large-scale networks in today's applications, we show how to compute efficiently smaller subgraphs that maintain some properties of an input graph. In particular, let G be a strongly connected directed graph. We consider the problem of computing the smallest strongly connected spanning subgraph of G that maintains certain connectivity relations of G. Specifically, for 2-edge-connectivity, we consider how to maintain the maximal 2-edge-connected subgraphs (2ECS) or the 2-edge-connected components (2ECC) of G, or both the maximal 2-edge-connected subgraphs and the 2-edge-connected components (2EC). Similarly, for 2-vertex-connectivity, we consider how to maintain the maximal 2-vertex-connected subgraphs (2VCS) or the 2-vertex-connected components (2VCC) of G, or both the maximal 2-vertex-connected subgraphs and the 2-vertex-connected components (2VC). All those problems are NP-hard, and thus we are interested in approximation algorithms. Additionally, we aim at designing algorithms with a good practical performance, so that they are able to scale effectively to very large graphs.While for 2ECS and 2VCS one can obtain an approximation ratio smaller than 2 by combining previously known results, providing good approximations for the 2-edge and the 2-vertex-components case seems more challenging. Here, we present linear-time approximation algorithms that achieve the following approximation guarantees:•4-approximation for 2ECC and 2EC, and•6-approximation for 2VCC and 2VC. Also, augmented versions of our 2VCC algorithm computes a 6-approximation for maintaining both the 2-edge and the 2-vertex-connected components (2CC), and for maintaining all the 2-connectivity relations of G (2C), i.e., both the 2-edge and the 2-vertex-connected subgraphs and components. Moreover, we provide heuristics that improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios.

Maximum flow under proportional delay constraint

Given a network and a set of source destination pairs (connections), we consider the problem of maximizing the sum of the flow under proportional delay constraints. In this paper, the delay for crossing a link is proportional to the total flow crossing this link. If a connection supports non-zero flow, then the sum of the delays along any path corresponding to that connection must be lower than a given bound. The constraints of delay are on–off constraints because if a connection carries zero flow, then there is no constraint for that connection. The difficulty of the problem comes from the choice of the connections supporting non-zero flow. We first prove a general approximation ratio using linear programming for a variant of the problem. We then prove a linear time 2-approximation algorithm when the network is a path. We finally show a Polynomial Time Approximation Scheme when the graph of intersections of the paths has bounded treewidth.

Approximating the DCJ distance of balanced genomes in linear time

BackgroundRearrangements are large-scale mutations in genomes, responsible for complex changes and structural variations. Most rearrangements that modify the organization of a genome can be represented by the double cut and join (DCJ) operation. Given two balanced genomes, i.e., two genomes that have exactly the same number of occurrences of each gene in each genome, we are interested in the problem of computing the rearrangement distance between them, i.e., finding the minimum number of DCJ operations that transform one genome into the other. This problem is known to be NP-hard.ResultsWe propose a linear time approximation algorithm with approximation factor O(k) for the DCJ distance problem, where k is the maximum number of occurrences of any gene in the input genomes. Our algorithm works for linear and circular unichromosomal balanced genomes and uses as an intermediate step an O(k)-approximation for the minimum common string partition problem, which is closely related to the DCJ distance problem.ConclusionsExperiments on simulated data sets show that our approximation algorithm is very competitive both in efficiency and in quality of the solutions.