Deterministic Approximation Algorithm Research Articles

Improved deterministic algorithms for non-monotone submodular maximization

Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization.In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283−o(1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1/6−ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints.

An efficient 3-approximation algorithm for the Steiner tree problem with the minimum number of Steiner points and bounded edge length.

We present improved algorithms for the Steiner tree problem with the minimum number of Steiner points and bounded edge length. Given n terminal points in a 2D Euclidean plane and an edge length bound, the problem asks to construct a spanning tree of n terminal points with minimal Steiner points such that every edge length of the spanning tree is within the given bound. This problem is known to be NP-hard and has practical applications such as relay node placements in wireless networks, wavelength-division multiplexing(WDM) optimal network design, and VLSI design. The best-known deterministic approximation algorithm has O(n3) running time with an approximation ratio of 3. This paper proposes an efficient approximation algorithm using the Voronoi diagram that guarantees an approximation ratio of 3 in O(n log n) time. We also present the first exact algorithm to find an optimal Steiner tree for given three terminal points in constant time. Using this exact algorithm, we improve the 3-approximation algorithm with better performance regarding the number of required Steiner points in O(n log n) time.

Deterministic Near-Optimal Approximation Algorithms for Dynamic Set Cover

Deterministic Near-Optimal Approximation Algorithms for Dynamic Set Cover

Approximation algorithms for orthogonal line centers

The problem of k orthogonal line center is about computing a set of k axis-parallel lines for a given set of points in ℜ2 such that the maximum among the distances between each point to its nearest line is minimized. A 2-factor approximation algorithm and a (53,32) bi-criteria approximation algorithm is presented for the problem. Both of them are deterministic approximation algorithms using combinatorial techniques and having sub-quadratic running times.

Quasipolynomial-time algorithms for Gibbs point processes

AbstractWe demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval $[0,\lambda ]$ . As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all $\lambda \lt 1/(e^{B + 1} \hat C_\phi )$ where $\hat C_\phi$ is a temperedness parameter and $B$ is the stability constant of $\phi$ . In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $\lambda \lt e/\Delta _\phi$ , where $\Delta _\phi$ is the potential-weighted connective constant of the potential $\phi$ . Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

Service Home Identification of Multiple-Source IoT Applications in Edge Computing

The real-time communication requirement of the Internet of Things (IoT) applications promotes the convergence of IoT and Mobile Edge Computing (MEC). The MEC paradigm greatly shortens the IoT service delay by leveraging cloudlets (edge servers) of MEC in the proximity of IoT devices. Considering limited computing and storage resources in an MEC network, it is challenging to provide efficient IoT-enabled service provisioning in such a network. In this article, we study the service home identification problem of service provisioning for multi-source IoT applications in an MEC network, by identifying a service home (cloudlet) of each multi-source IoT application for its data processing, querying and storage. Each multi-source IoT application consists of multiple sources located at different geographical locations and each source uploads its data stream via a gateway (its nearby access point) to the MEC network and the uploaded data then is aggregated with the stream data of the other sources of the IoT application at the service home. We here focus on two novel service home identification problems: the service operational cost minimization problem with the aim to minimize the total service operational cost by admitting as many multi-source IoT applications as possible, and the online throughput maximization problem with the aim to maximize the number of multi-source IoT application requests admitted. We first show that both the problems are NP-hard. We then formulate an Integer Linear Programming (ILP) solution to the service operational cost minimization problem, and propose a randomized algorithm with high probability and a deterministic approximation algorithm respectively, at moderate resource capacity violations. We third develop an efficient heuristic algorithm for the problem without any resource violation. Furthermore, we deal with the online throughput maximization problem under an assumption that multi-source IoT application requests arrive one by one without the knowledge of future arrivals, for which we formulate an Integer Linear Programming (ILP) solution to its offline version, followed by devising an online algorithm with competitive ratio. We finally evaluate the performance of the proposed algorithms through experimental simulations. Simulation results demonstrate that the proposed algorithms are promising, and outperform their comparison counterparts.

Correlation decay and the absence of zeros property of partition functions

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok for designing deterministic approximation algorithms for various graph counting and related problems. An earlier method used for the same problem is one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply often coincide or nearly coincide. In this article we show that this is not a coincidence. We establish that if the interpolation method is valid for a family of graphs, then this family exhibits a form of the correlation decay property which is asymptotic strong spatial mixing at superlogarithmic distances. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the design of the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs. Indeed this conjecture was recently confirmed by Regts. See the body of the article for details.

Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM

The Expectation Maximisation (EM) algorithm is widely used to optimise non-convex likelihood functions with latent variables. Many authors modified its simple design to fit more specific situations. For instance, the Expectation (E) step has been replaced by Monte Carlo (MC), Markov Chain Monte Carlo or tempered approximations, etc. Most of the well-studied approximations belong to the stochastic class. By comparison, the literature is lacking when it comes to deterministic approximations. In this paper, we introduce a theoretical framework, with state-of-the-art convergence guarantees, for any deterministic approximation of the E step. We analyse theoretically and empirically several approximations that fit into this framework. First, for intractable E-steps, we introduce a deterministic version of MC-EM using Riemann sums. A straightforward method, not requiring any hyper-parameter fine-tuning, useful when the low dimensionality does not warrant a MC-EM. Then, we consider the tempered approximation, borrowed from the Simulated Annealing literature and used to escape local extrema. We prove that the tempered EM verifies the convergence guarantees for a wider range of temperature profiles than previously considered. We showcase empirically how new non-trivial profiles can more successfully escape adversarial initialisations. Finally, we combine the Riemann and tempered approximations into a method that accomplishes both their purposes.

Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where $[k]:=\{1,\ldots,k\}$. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $\Delta\in \mathbb{N}$ and any $k\geq e\Delta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${\bf Z}(G;k,w)\neq 0$ for any $w\in U$ and any graph $G$ of maximum degree at most $\Delta$. (Here $e$ denotes the base of the natural logarithm.) For small values of $\Delta$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.

Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

In this paper, we study the generalized submodular maximization problem with a non-negative monotone submodular set function as the objective function and subject to a matroid constraint. The problem is generalized through the curvature parameter α∈[0,1] which measures how far a set function deviates from linearity to submodularity. We propose a deterministic approximation algorithm which uses the approximation algorithm proposed by Buchbinder et al. [2] as a building block and inherits the approximation guarantee for α=1. For general value of the curvature parameter α∈[0,1], we present an approximation algorithm with a factor of 1+hα(y)+Δ⋅[3+α−(2+α)y−(1+α)hα(y)]2+α+(1+α)(1−y), where y∈[0,1] is a predefined parameter for tuning the ratio. In particular, when α=1 we obtain a ratio 0.5008 when setting y=0.9, coinciding with the renowned state-of-art approximate ratio; when α=0 that the object is a linear function, the approximation factor equals one and our algorithm is indeed an exact algorithm that always produces optimum solutions.

Minimum budget for misinformation detection in online social networks with provable guarantees

Misinformation detection in Online Social Networks has recently become a critical topic due to its important role in restraining misinformation. Recent studies have showed that machine learning methods can be used to detect misinformation/fake news/rumors by detecting user’s behaviour. However, we can not implement this strategy for all users on a social network due to the limitation of budget. Therefore, it is critical to optimize the monitor/sensor placement to effectively detect misinformation. In this paper, we investigate Minimum Budget for Misinformation Detection problem which aims to find the smallest set of nodes to place monitors in a social network so that detection function is at least a given threshold. Beside showing the inapproximability of the problem under the well-known Independent Cascade diffusion model, we then propose three approximation algorithms including: Greedy, Sampling-based Misinformation Detection and Importance Sampling-based Misinformation Detection. Greedy is a deterministic approximation algorithm which utilizes the properties of monotone and submodular of the detection function. The rest is two randomized algorithms with provable guarantees based on developing two novel techniques (1) estimating detection function by using the concepts of influence sample and importance influence sample with proof of correctness, and (2) an algorithmic framework to select the solution with theoretical analysis. Experiments on real social networks show the effectiveness and scalability of our algorithms.

A generalization of permanent inequalities and applications in counting and optimization

A polynomial p∈R[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems.Here, we study a more general question: Given a stable multilinear polynomial p with nonnegative coefficients and a set of monomials S, we show that if the polynomial obtained by summing up all monomials in S is real stable, then we can lowerbound the sum of coefficients of monomials of p that are in S. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijver's inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh [23], and give deterministic polynomial time approximation algorithms for several counting problems.

Maximum Target Coverage Problem in Mobile Wireless Sensor Networks.

We formulate and analyze a generic coverage optimization problem arising in wireless sensor networks with sensors of limited mobility. Given a set of targets to be covered and a set of mobile sensors, we seek a sensor dispatch algorithm maximizing the covered targets under the constraint that the maximal moving distance for each sensor is upper-bounded by a given threshold. We prove that the problem is NP-hard. Given its hardness, we devise four algorithms to solve it heuristically or approximately. Among the approximate algorithms, we first develop randomized -optimal algorithm. We then employ a derandomization technique to devise a deterministic -approximation algorithm. We also design a deterministic approximation algorithm with nearly approximation ratio by using a colouring technique, where denotes the maximal number of subsets covering the same target. Experiments are also conducted to validate the effectiveness of the algorithms in a variety of parameter settings.

Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

For a family of graphs ℱ, the W<scp;>eighted</scp;> ℱ V<scp;>ertex</scp;> D<scp;>eletion</scp;> problem, is defined as follows: given an n -vertex undirected graph G and a weight function w : V ( G )࢐ ℝ, find a minimum weight subset S ⊆ V ( G ) such that G - S belongs to ℱ. We devise a recursive scheme to obtain O(log O(1) n )-approximation algorithms for such problems, building upon the classical technique of finding balanced separators . We obtain the first O(log O(1) n )-approximation algorithms for the following problems. • Let F be a finite set of graphs containing a planar graph, and ℱ= G ( F ) be the maximal family of graphs such that every graph H ∈ G ( F ) excludes all graphs in F as minors. The vertex deletion problem corresponding to ℱ= G ( F ) is the W eighted P lanar F -M inor -F ree D eletion (WP F -MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F -MFD with ratios O(log 1.5 n ) and O(log 2 n ), respectively. Prior to our work, a randomized constant factor approximation algorithm for the unweighted version was known [FOCS 2012]. After our work, a deterministic constant factor approximation algorithm for the unweighted version was also obtained [SODA 2019]. • We give an O(log 2 n )-factor approximation algorithm for W eighted C hordal V ertex D eletion , the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for M ulticut on chordal graphs. • We give an O(log 3 n )-factor approximation algorithm for W eighted D istance H ereditary V ertex D eletion . We believe that our recursive scheme can be applied to obtain O(log O(1) n )-approximation algorithms for many other problems as well.

Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms

The bootstrap method is one of the major developments in statistics in the 20th century for computing confidence intervals directly from data. However, the bootstrap method is traditionally approximated with a randomized algorithm, which can sometimes produce inaccurate confidence intervals. In “Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms,” Bertsimas and Sturt present a new perspective of the bootstrap method through the lens of counting integer points in a polyhedron. Through this perspective, the authors develop the first computational complexity results and efficient deterministic approximation algorithm (fully polynomial time approximation scheme) for bootstrap confidence intervals, which unlike traditional methods, has guaranteed bounds on its error. In experiments on real and synthetic data sets from clinical trials, the proposed deterministic algorithms quickly produce reliable confidence intervals, which are significantly more accurate than those from randomization.

A Method of Correcting Estimation Failure in Latent Differential Equations with Comparisons to Kalman Filtering

Studies have used the latent differential equation (LDE) model to estimate the parameters of damped oscillation in various phenomena, but it has been shown that correct, non-zero parameter estimates are only obtained when the latent series exhibits little or no process noise. Consequently, LDEs are limited to modeling deterministic processes with measurement error rather than those with random behavior in the true latent state. The reasons for these limitations are considered, and a piecewise deterministic approximation (PDA) algorithm is proposed to treat process noise outliers as functional discontinuities and obtain correct estimates of the damping parameter. Comprehensive, random-effects simulations were used to compare results with those obtained using a state-space model (SSM) based on the Kalman filter. The LDE with the PDA algorithm (LDEPDA) successfully recovered the simulated damping parameter under a variety of conditions when process noise was present in the latent state. The LDEPDA had greater precision and accuracy than the SSM when estimating parameters from data with sparse jump discontinuities, but worse performance for diffusion processes overall. All three methods were applied to a sample of postural sway data. The basic LDE estimated zero damping, while the LDEPDA and SSM estimated moderate to high damping. The SSM estimated the smallest standard errors for both frequency and damping parameter estimates.

Distributed Dominating Set Approximations beyond Planar Graphs

The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O (log * n )-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Generalizing expectation propagation with mixtures of exponential family distributions and an application to Bayesian logistic regression

Expectation propagation (EP) is a widely used deterministic approximate inference algorithm in Bayesian machine learning. Traditional EP approximates an intractable posterior distribution through a set of local approximations which are updated iteratively. In this paper, we propose a generalized version of EP called generalized EP (GEP), which is a new method based on the minimization of KL divergence for approximate inference. However, when the variance of the gradient is large, the algorithm may need a long time to converge. We use control variates and develop a variance reduced version of this method called GEP-CV. We evaluate our approach on Bayesian logistic regression, which provides faster convergence and better performance than other state-of-the-art approaches.