Randomized Approximation Algorithms Research Articles

Approximate NFA universality and related problems motivated by information theory

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is ‘approximately’ maximal, which then boils down to whether a given NFA is ‘approximately’ universal. Here we introduce the notion of a (1−ε)-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable ε.

Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bound of varOmega ( M log min {|varSigma |,M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n/log (1+varepsilon ) for approximating the answer with multiplicative error (1 + varepsilon ). Second, we design four real-time algorithms for this problem. Three of them are Monte Carlo approximation algorithms for additive error, “small” and “big” multiplicative errors, respectively. Each algorithm uses mathcal {O}(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The fourth algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.

An optimal (ϵ,δ)‐randomized approximation scheme for the mean of random variables with bounded relative variance

Randomized approximation algorithms for many #P‐complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a {0,1}‐matrix, and many others) reduce to creating random variables X1,X2,… with finite mean μ and standard deviation σ such that μ is the solution for the problem input, and the relative standard deviation |σ/μ| ≤ c for known c. Under these circumstances, it is known that the number of samples from the {Xi} needed to form an (ϵ,δ)‐approximation that satisfies is at least . We present here an easy to implement (ϵ,δ)‐approximation that uses samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.

Automatic identification of optimal marker genes for phenotypic and taxonomic groups of microorganisms.

Finding optimal markers for microorganisms important in the medical, agricultural, environmental or ecological fields is of great importance. Thousands of complete microbial genomes now available allow us, for the first time, to exhaustively identify marker proteins for groups of microbial organisms. In this work, we model the biological task as the well-known mathematical “hitting set” problem, solving it based on both greedy and randomized approximation algorithms. We identify unique markers for 17 phenotypic and taxonomic microbial groups, including proteins related to the nitrite reductase enzyme as markers for the non-anammox nitrifying bacteria group, and two transcription regulation proteins, nusG and yhiF, as markers for the Archaea and Escherichia/Shigella taxonomic groups, respectively. Additionally, we identify marker proteins for three subtypes of pathogenic E. coli, which previously had no known optimal markers. Practically, depending on the completeness of the database this algorithm can be used for identification of marker genes for any microbial group, these marker genes may be prime candidates for the understanding of the genetic basis of the group's phenotype or to help discover novel functions which are uniquely shared among a group of microbes. We show that our method is both theoretically and practically efficient, while establishing an upper bound on its time complexity and approximation ratio; thus, it promises to remain efficient and permit the identification of marker proteins that are specific to phenotypic or taxonomic groups, even as more and more bacterial genomes are being sequenced.

Using TPA to count linear extensions

A linear extension of a poset P is a permutation of the elements of the set that respects the partial order. Let L(P) be the set of linear extensions. It is a #P complete problem to determine the size of L(P) exactly for an arbitrary poset, and so randomized approximation algorithms are used that employ samples that are uniformly drawn from the set of linear extensions. In this work, a new randomized approximation algorithm is proposed where the linear extensions are not uniformly drawn, but instead come from a distribution indexed by a continuous parameter β. The introduction of β allows for the use of a more efficient method for approximating #L(P) called TPA. Our primary result is that it is possible to sample from this continuous embedding in time that as fast or faster than the best known methods for sampling uniformly from linear extensions. For a poset containing n elements, this means we can approximate L(P) to within a factor of 1+ϵ with probability at least 1−δ using O(n3(ln⁡n)(ln⁡#L(P))2ϵ−2ln⁡δ−1) expected number of random uniforms and comparisons.

Improved approximation algorithms for cumulative VRP with stochastic demands

In this paper, we give randomized approximation algorithms for stochastic cumulative VRPs for the split and unsplit deliveries. The approximation ratios are max{1+1.5α,3} and 6, respectively, where α is the approximation ratio for the metric TSP. The approximation factor is further reduced for trees. These results extend the results in Anupam Gupta et al. (2012) and Daya Ram Gaur et al. (2013). The bounds reported here improve the bounds in Daya Ram Gaur et al. (2016).

Randomized approximation algorithms for planar visibility counting problem

Given a set S of n disjoint line segments in R2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can be solved trivially in O(log⁡n) query time using O(n4log⁡n) preprocessing time and O(n4) space.Gudmundsson and Morin (2010) [10] proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. For any constant 0≤α≤1, their algorithm answers any query in Oϵ(m(1−α)/2) time with Oϵ(m1+α) of preprocessing time and space, where ϵ>0 is a constant that can be made arbitrarily small and Oϵ(f(n))=O(f(n)nϵ) and m=O(n2) is a number that depends on the configuration of the segments.In this paper, we propose two randomized approximation algorithms for VCP. The first algorithm depends on two constants 0≤β≤23 and 0<δ≤1, and the expected preprocessing time, the expected space, and the expected query time are O(m2−3β/2log⁡m), O(m2−3β/2), and O(1δ2mβ/2log⁡m), respectively. The algorithm, in the preprocessing phase, selects a sequence of random samples, whose size and number depend on the tradeoff parameters. When a query point p is given by an adversary unaware of the random sample of our algorithm, it computes the number of visible segments from p, denoted by mp, exactly, if mp≤3δ2mβ/2log⁡(2m). Otherwise, it computes an approximated value, mp′, such that with the probability of at least 1−1m, we have (1−δ)mp≤mp′≤(2+2δ)mp. The preprocessing time and space of the second algorithm are O(n2log⁡n) and O(n2), respectively. This algorithm computes the exact value of mp if mp≤1δ2nlog⁡n, otherwise it returns an approximated value mp″ in expected O(1δ2nlog⁡n) time, such that with the probability at least 1−1log⁡n, we have (1−3δ)mp≤mp″≤(1.5+3δ)mp.

A Bernoulli mean estimate with known relative error distribution

AbstractSuppose that are independent identically distributed Bernoulli random variables with mean p, so and . Any estimate of p has relative error . This paper builds a new estimate of p with the remarkable property that the relative error of the estimate does not depend in any way on the value of p. This allows the easy construction of exact confidence intervals for p of any desired level without needing any sort of limit or approximation. In addition, is unbiased. For ∊ and δ in (0, 1), to obtain an estimate where , the new algorithm takes on average at most samples. It is also shown that any such algorithm that applies whenever requires at least samples on average. The same algorithm can also be applied to estimate the mean of any random variable that falls in . The used here employs randomness external to the sample, and has a small (but nonzero) chance of being above 1. It is shown that any nontrivial where the relative error is independent of p must also have these properties. Applications of this methodology include finding exact p‐values and randomized approximation algorithms for # P complete problems. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 173–182, 2017

Improved approximating $2$-CatSP for $\sigma\geq 0.50$ with an unbalanced rounding matrix

$2$-CatSP is a fundamental NP-Complete optimization problem,which is formulated as given a ground set $U$ of$n$ items, $m$ subsets $S_1,S_2,\cdots,$ $S_m$ of $U$ and an integer$1\leq t\leq n$, find two subsets $A_1$ and $A_2$ of $U$ such that$|A_1|=|A_2|\leq t$ and $\sum_{i=1}^{m}\max\left(|S_i\capA_1|,|S_i\cap A_2|\right)$ is maximized. In this paper, wereconsider randomized approximation algorithms for $2$-CatSP withoutand with triangle inequalities in terms of a new positivesemidefinite matrix reflecting more information on unbalancedproperties. The performance ratios of our algorithm are muchbetter than the current best ones of Xu et al in (Optim. Method. Softw., 18(6):705-719, 2003) for general cases underunbalanced parameters $\sigma=t/n\geq 0.50$ by our rigorous analysis. In particular,we get 0.6700 and 0.7250 without and with triangle inequalities for the most important subcase $\sigma=0.50$,improving previously best ratios 0.6430 and 0.6736 in corresponding environment. Indeedrecently Wu et al (J. Ind. Manag. Optim. 8: 117-126, 2012) and Xu et alin (J. Comb. Optim., 27: 315-327, 2014)also considered approximate strategies for its disjoint subcase $A_1\cap A_2=\emptyset$ as $\sigma=0.50$with ratios 0.7317 and 0.7469, which can be reduced to max bisection problems of graphs.

A Randomized Approach to Volume Constrained Polyhedronization Problem

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

Deterministic approximation algorithms for the maximum traveling salesman and maximum triangle packing problems

We give derandomizations of known randomized approximation algorithms for the maximum traveling salesman problem and the maximum triangle packing problem: we show how to define pessimistic estimators for certain probabilities, based on the analysis of the randomized algorithms, and show that we can multiply the estimators to obtain pessimistic estimators for the expected weight of the solution. The method of pessimistic estimators (Raghavan (1988) [14]) then immediately implies that the randomized algorithms can be derandomized. For the maximum triangle packing problem, this gives deterministic algorithms with better approximation guarantees than what was previously known.The key idea in our analysis is the specification of conditions on pessimistic estimators of two expectations E[Y] and E[Z], under which the product of the pessimistic estimators is a pessimistic estimator of E[YZ], where Y and Z are two random variables. This approach can be useful when derandomizing algorithms for which one needs to bound the probability of some event that can be expressed as an intersection of multiple events; using our method, one can define pessimistic estimators for the probabilities of the individual events, and then multiply them to obtain a pessimistic estimator for the probability of the intersection of the events.

Approximation Algorithms for Discrete Polynomial Optimization

In this paper, we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete (typically binary) variables. Such models have natural applications in graph theory, neural networks, error-correcting codes, among many others. In particular, we focus on three types of optimization models: (1) maximizing a homogeneous polynomial function in binary variables; (2) maximizing a homogeneous polynomial function in binary variables, mixed with variables under spherical constraints; (3) maximizing an inhomogeneous polynomial function in binary variables. We propose polynomial-time randomized approximation algorithms for such polynomial optimization models, and establish the approximation ratios (or relative approximation ratios whenever appropriate) for the proposed algorithms. Some examples of applications for these models and algorithms are discussed as well.

Improved inapproximability results for counting independent sets in the hard‐core model

AbstractWe study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph and an activity , we are interested in the quantity defined as the sum over independent sets I weighted as . In statistical physics, is the partition function for the hard‐core model, which is an idealized model of a gas where the particles have non‐negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and . The quantity is the critical point for the so‐called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for , unless , for some function . We remove the upper bound in the assumptions of Sly's result for , that is, we show that there does not exist efficient randomized approximation algorithms for all for and . Sly's inapproximability result uses a clever reduction, combined with a second‐moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014

On approximating multicriteria TSP

We present approximation algorithms for almost all variants of the multicriteria traveling salesman problem (TSP). First, we devise randomized approximation algorithms for multicriteria maximum traveling salesman problems (Max-TSP). For multicriteria Max-STSP where the edge weights have to be symmetric, we devise an algorithm with an approximation ratio of 2/3 - ε. For multicriteria Max-ATSP where the edge weights may be asymmetric, we present an algorithm with a ratio of 1/2 - ε. Our algorithms work for any fixed number k of objectives. Furthermore, we present a deterministic algorithm for bicriteria Max-STSP that achieves an approximation ratio of 7/27. Finally, we present a randomized approximation algorithm for the asymmetric multicriteria minimum TSP with triangle inequality (Min-ATSP). This algorithm achieves a ratio of log n + ε.

Technical Note—Approximation Algorithms for VRP with Stochastic Demands

We consider the vehicle routing problem with stochastic demands (VRPSD). We give randomized approximation algorithms achieving approximation guarantees of 1 + α for split-delivery VRPSD, and 2 + α for unsplit-delivery VRPSD; here α is the best approximation guarantee for the traveling salesman problem. These bounds match the best known for even the respective deterministic problems [Altinkemer, K., B. Gavish. 1987. Heuristics for unequal weight delivery problems with a fixed error guarantee. Oper. Res. Lett. 6(4) 149–158; Altinkemer, K., B. Gavish. 1990. Heuristics for delivery problems with constant error guarantees. Transportation Res. 24(4) 294–297]. We also show that the “cyclic heuristic” for split-delivery VRPSD achieves a constant approximation ratio, as conjectured in Bertsimas [Bertsimas, D. J. 1992. A vehicle routing problem with stochastic demand. Oper. Res. 40(3) 574–585].

Robust Multicast Beamforming for Spectrum Sharing-Based Cognitive Radios

We consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) network. The minimization of transmit power is formulated subject to both quality-of-service (QoS) constraints on the secondary receivers and interference temperature constraints on the primary users, under the assumption of imperfect channel state information (CSI). The problem is a nonconvex quadratically constrained quadratic program (QCQP), and in general it is hard to achieve the global optimality. As a compromise, we present two randomized approximation algorithms for the problem via convex optimization techniques. Apart from the general setting of the robust beamforming problem, we identify one interesting special case, the robust problem of which can be solved efficiently. Simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design.

Multi-criteria TSP: Min and max combined

We present randomized approximation algorithms for multi-criteria traveling salesman problems (TSP), where some objective functions should be minimized while others should be maximized. For the symmetric multi-criteria TSP (STSP), we present an algorithm that computes ( 2 / 3 , 3 + ε ) -approximate Pareto curves. Here, the first parameter is the approximation ratio for the objectives that should be maximized, and the second parameter is the ratio for the objectives that should be minimized. For the asymmetric multi-criteria TSP (ATSP), we obtain an approximation performance of ( 1 / 2 , log 2 n + ε ) .

On distributed scheduling in wireless networks exploiting broadcast and network coding

In this paper, we consider cross-layer optimization in wireless networks with wireless broadcast advantage, focusing on the problem of distributed scheduling of broadcast links. The wireless broadcast advantage is most useful in multicast scenarios. As such, we include network coding in our design to exploit the throughput gain brought in by network coding for multicasting. We derive a subgradient algorithm for joint rate control, network coding and scheduling, which however requires centralized link scheduling. Under the primary interference model, link scheduling problem is equivalent to a maximum weighted hypergraph matching problem that is NP-complete. To solve the scheduling problem distributedly, locally greedy and randomized approximation algorithms are proposed and shown to have bounded worst-case performance. With random network coding, we obtain a fully distributed cross-layer design. Numerical results show promising throughput gain using the proposed algorithms, and surprisingly, in some cases even with less complexity than cross-layer design without broadcast advantage.

Mechanism design for fractional scheduling on unrelated machines

Scheduling on unrelated machines is one of the most general and classical variants of the task scheduling problem. Fractional scheduling is the LP-relaxation of the problem, which is polynomially solvable in the nonstrategic setting, and is a useful tool to design deterministic and randomized approximation algorithms. The mechanism design version of the scheduling problem was introduced by Nisan and Ronen. In this article, we consider the mechanism design version of the fractional variant of this problem. We give lower bounds for any fractional truthful mechanism. Our lower bounds also hold for any (randomized) mechanism for the integral case. In the positive direction, we propose a truthful mechanism that achieves approximation 3/2 for 2 machines, matching the lower bound. This is the first new tight bound on the approximation ratio of this problem, after the tight bound of 2, for 2 machines, obtained by Nisan and Ronen. For n machines, our mechanism achieves an approximation ratio of n +1/2. Motivated by the fact that all the known deterministic and randomized mechanisms for the problem assign each task independently from the others, we focus on an interesting subclass of allocation algorithms, the task-independent algorithms. We give a lower bound of n +1/2, that holds for every (not only monotone) allocation algorithm that takes independent decisions. Under this consideration, our truthful independent mechanism is the best that we can hope from this family of algorithms.

Computational experience with a SDP‐based algorithm for maximum cut with limited unbalance

AbstractIn the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B, so that the sum of the weights of the crossing edges is maximum. This problem has been introduced in (Galbiati and Maffioli, Theor Comput Sci 385 (2007), 78–87) where polynomial time randomized approximation algorithms are proposed and their performance guarantees are analyzed in the case of non‐negative integer weights. In this article, we present extensive computational experience with these algorithms on a large number of different graphs. We then extend the analysis of these algorithms to integer weights not restricted in sign, and continue the computational testing. It turns out that the approximation ratios obtained are always substantially better than those guaranteed by the theoretical analysis. © 2010 Wiley Periodicals, Inc. NETWORKS 2010