Deterministic Approximation Algorithm Research Articles

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.

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

On extensions of the deterministic online model for bipartite matching and max-sat

The surprising results of Karp, Vazirani and Vazirani [39] and (respectively) Buchbinder et al. [18] are examples where rather simple randomization provides provably better approximations than the corresponding deterministic counterparts for online bipartite matching and (respectively) unconstrained non-monotone submodular. We show that seemingly strong extensions of the deterministic online computation model can at best match the performance of naive randomization. More specifically, for bipartite matching, we show that in the priority model (allowing very general ways to order the input stream), we cannot improve upon the trivial 12 approximation achieved by any greedy maximal matching algorithm and likewise cannot improve upon this approximation by any lognlog⁡log⁡n number of online algorithms running in parallel. The latter result yields an improved log⁡log⁡n−log⁡log⁡log⁡n lower bound for the number of advice bits needed to beat the simple deterministic greedy algorithm. For max-sat, we adapt the recent de-randomization approach of Buchbinder and Feldman [16] applied to the Buchbinder et al. [17] algorithm for max-sat to obtain a deterministic 34 approximation algorithm using width 2n parallelism. In order to improve upon this approximation, we show that exponential width parallelism of online algorithms is necessary (in a model that is more general than what is needed for the width 2n algorithm). We relate our results to previous work concerning the priority Branching Tree (pBT) model of Alekhnovich et al. [2].

Approximate and incremental network function placement

The virtualization of modern computer networks introduces interesting new opportunities for a more flexible placement of network functions and middleboxes (firewalls, proxies, traffic optimizers, virtual switches, etc.). This paper studies approximation algorithms for the incremental deployment of a minimum number of middleboxes, such that capacity constraints at the middleboxes and length constraints on the communication routes are respected.Based on a new, purely combinatorial and rigorous proof of submodularity, we obtain our main result: a deterministic greedy approximation algorithm which only employs augmenting paths to serve future communication pairs. Hence, our algorithm does not require any changes to the locations of existing middleboxes or the preemption of previously served communication pairs when additional middleboxes are deployed. It is hence particularly attractive for incremental deployments. We prove that the achieved polynomial-time approximation bound is optimal, unless P=NP holds.This paper also initiates the study of a weighted problem variant, in which entire groups of nodes need to communicate via a middlebox, possibly at different rates. We present an LP relaxation and randomized rounding algorithm for this problem, leveraging an interesting connection to scheduling.We complement our formal results with a simulation study of a large set of synthetically generated instances. Our results indicate that the presented algorithms yield near-optimal solutions in practice.

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

We show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. Our approach works for the Tutte polynomial, the independence polynomial, and partition functions of complex-valued spin- and edge-coloring models.

On Regularity Lemmas and their Algorithmic Applications

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

Throughput-Efficient Channel Allocation Algorithms in Multi-Channel Cognitive Vehicular Networks

Many studies show that the dedicated short range communication band allocated to vehicular communications is insufficient to carry the wireless traffic generated by emerging vehicular applications. A promising bandwidth expansion possibility presents itself through the release of large TV band spectra (i.e., the TV white space spectrum) by the Federal Communications Commission for cognitive access. One primary challenge of the so-called TV white space (TVWS) spectrum access in vehicular networks is the design of efficient channel allocation mechanisms in face of spatial-temporal variations of TVWS channels. In this paper, we address the channel allocation problem for multi-channel cognitive vehicular networks with the objective of system-wide throughput maximization. We show that the problem is an NP-hard non-linear integer programming problem, to which we present three efficient algorithms. We first propose a probabilistic polynomial-time (1-1/e)-approximation algorithm based on linear programming. Next, we prove that the objective function can be written as a submodular set function, based on which we develop a deterministic constant-factor approximation algorithm with a more favorable time complexity. Then, we further modify the second algorithm to improve its approximation ratio without increasing its time complexity. Finally, we show the efficacy of our algorithms through numerical examples.

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

A fast network-decomposition algorithm and its applications to constant-time distributed computation

A partition (C1,C2,…,Cq) of G=(V,E) into clusters of strong (respectively, weak) diameter d, such that the supergraph obtained by contracting each Ci is ℓ-colorable is called a strong (resp., weak) (d,ℓ)-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong (d,ℓ)-network-decompositions with d=ℓ=exp⁡{O(log⁡nlog⁡log⁡n)} can be computed in distributed deterministic time O(d). Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing.The result of Awerbuch et al. was improved by Panconesi and Srinivasan in 1992: in the latter result d=ℓ=exp⁡{O(log⁡n)}, and the running time is O(d) as well. In another remarkable breakthrough Linial and Saks (in 1992) showed that weak (O(log⁡n),O(log⁡n))-network-decompositions can be computed in distributed randomized time O(log2⁡n). Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d=O(1). However, the parameter ℓ in his result is O(n1/2+ϵ).In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong (O(1),O(nϵ))-network-decompositions. As a corollary we derive a constant-time randomized O(nϵ)-approximation algorithm for the distributed minimum coloring problem, improving the previously best-known O(n1/2+ϵ) approximation guarantee. We also derive other improved distributed algorithms for a variety of problems.Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic-time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).

On the distributed complexity of the semi-matching problem

We consider the problem of matching clients with servers, each of which can process a subset of clients. It is known as the semi-matching or load balancing problem in a bipartite graph G=(V,U,E), where U corresponds to the clients and V to the servers. The goal is to find a set of edges M⊆E such that every vertex in U is incident to exactly one edge in M. The load of a server v∈V is defined as (degM⁡(v)+12), and the problem is to find a semi-matching M that minimizes the sum of the loads of the servers. We show that to find an optimal solution in a distributed setting Ω(|V|) rounds are needed and propose distributed deterministic approximation algorithms for the problem. It yields 2-approximation and has time complexity O(Δ5), where Δ is the maximum degree in V. We also give some greedy algorithms.

Reconstruction of sparse Legendre and Gegenbauer expansions

We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony-like methods. The method proposed is robust with respect to noisy sampled data. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order.

An FPTAS for the Volume Computation of 0-1 Knapsack Polytopes Based on Approximate Convolution

Computing high dimensional volumes is a hard problem, even for approximation. Several randomized approximation techniques for #P-hard problems have been developed in the three decades, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a new technique for a deterministic approximation, this paper is concerned with the volume computation of 0-1 knapsack polytopes, which is known to be #P-hard. This paper presents a new technique based on approximate convolutions for a deterministic approximation of volume computations, and provides a fully polynomial-time approximation scheme for the volume computation of 0-1 knapsack polytopes. We also give an extension of the result to multi-constrained knapsack polytopes with a constant number of constraints.

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.

Expanders with respect to Hadamard spaces and random graphs

It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. $\{G_n\}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

A polynomial time approximation scheme for computing the supremum of Gaussian processes

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop {\mathbb{E}}_{X\leftarrow\mathcal{N}^d}[\sup_{v\in V}|\langle v,X\rangle|]$ deterministically in time $\operatorname {poly}(d)\cdot|V|^{O_{\varepsilon}(1)}$. Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471]. This answers an open question of Lee (2010) and Ding [Ann. Probab. 42 (2014) 464-496]. The study of supremum of Gaussian processes is of considerable importance in probability with applications in functional analysis, convex geometry, and in light of the recent breakthrough work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471], to random walks on finite graphs. As such our result could be of use elsewhere. In particular, combining with the work of Ding [Ann. Probab. 42 (2014) 464-496], our result yields a PTAS for computing the cover time of bounded-degree graphs. Previously, such algorithms were known only for trees. Along the way, we also give an explicit oblivious estimator for semi-norms in Gaussian space with optimal query complexity. Our algorithm and its analysis are elementary in nature, using two classical comparison inequalities, Slepian's lemma and Kanter's lemma.

Algoritmo aleatorizado basado en distribuciones deslizantes para el problema de planificación en Sistemas Grid

In this paper we present a randomized algo- rithm for the online version of the Job Shop problem where jobs are composed of processes with preceden- ce constraints and processors are organized in a Grid topology. The proposed randomized algorithm is based on a new technique that we have denominated as sliding distributions, which aims at combining the advantages of the deterministic approximation algorithms with those of the Montecarlo randomized algorithms. The objective is to provide an algorithm that delivers -approximated solutions with high probability, but at the same time, is able to investigate an extended neighborhood of such solutions so that it can escape from local extrema. We formally characterize the temporal complexity of the pro- posed algorithm and show that it is correct. We also evaluate the performance of the proposed algorithm by means of a series of simulation-based experiments. The results show that the proposed algorithm outperforms the traditional state of the art algorithms for scheduling in Grid systems. The performance metrics are average delay, maximum delay, and Grid utilization.

A Structure-driven Randomized Algorithm for the k-center Problem

In this paper we present a new randomized approximation algorithm for the metric discrete k-center problem. The main idea is to apply random perturbations to the decisions made by a deterministic approximation algorithm in such a way as to keep the approximation guarantees with high probability, but at the same time explore an extended neighborhood of the solutions produced by the deterministic approximation algorithm. We formally characterize the proposed algorithm and show that it produces 2-approximated solutions with probability of at least 1-1/N when it is repeated at least αlnN times. α,N∈Z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> are user-defined parameters where α measures the size of the perturbations. Experimental results show that the proposed algorithm performs similar or better than a representative set of algorithms for the k-center problem and a GRASP algorithm, which is a popular state-of-the-art technique for randomizing deterministic algorithms. Our experiments also show that the quality of the solutions found by the proposed algorithm increases faster with the number of iterations and hence, is better suited for big instances where the execution of each iteration is very expensive.

On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods

We study optimization problems for the Euclidean Minimum Spanning Tree (MST) problem on imprecise data. To model imprecision, we accept a set of disjoint disks in the plane as input. From each member of the set, one point must be selected, and the MST is computed over the set of selected points. We consider both minimizing and maximizing the weight of the MST over the input. The minimum weight version of the problem is known as the Minimum Spanning Tree with Neighborhoods (MSTN) problem, and the maximum weight version (max-MSTN) has not been studied previously to our knowledge. We provide deterministic and parameterized approximation algorithms for the max-MSTN problem, and a parameterized algorithm for the MSTN problem. Additionally, we present hardness of approximation proofs for both settings.

Fast Approximation Algorithm for Restricted Euclidean Bottleneck Steiner Tree Problem

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. We first show that the problem is NP-hard and cannot be approximated within unless P=NP, and provide a fast polynomial time deterministic approximation algorithm with performance ratio .