Expected Running Time Research Articles

Special Soundness Revisited

We generalize and abstract the problem of extracting a witness from a prover of a special sound protocol into a combinatorial problem induced by a sequence of matroids and a predicate, and present a parametrized algorithm for solving this problem. The parametrization provides a tight tradeoff between the running time and the extraction error of the algorithm, which allows optimizing the parameters to minimize: the soundness error for interactive proofs, or the extraction time for proofs of knowledge. In contrast to previous work we bound the distribution of the running time and not only the expected running time. Tail bounds give a tighter analysis when applied recursively and a concentrated running time.

Towards Running Time Analysis of Interactive Multi-Objective Evolutionary Algorithms

Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the DM has subjective preferences. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax, OneJumpZeroJump problems are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.

Unbiased Time-Average Estimators for Markov Chains

We consider a time-average estimator fk of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of fk has a limit μ as the number of time steps goes to infinity. We describe a modification of fk that yields an unbiased estimator [Formula: see text] of μ. It is shown that [Formula: see text] is square integrable and has finite expected running time. Under certain conditions, [Formula: see text] can be built without any precomputations and is asymptotically at least as efficient as fk , up to a multiplicative constant arbitrarily close to one. Our approach also provides an unbiased estimator for the bias of fk . We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings.

Run Time Analysis for Random Local Search on Generalized Majority Functions

Run time analysis of evolutionary algorithms recently makes significant progress in linking algorithm performance to algorithm parameters. However, settings that study the impact of problem parameters are rare. The recently proposed W-model provides a good framework for such analyses, generating pseudo-Boolean optimization problems with tunable properties. We initiate theoretical research of the W-model by studying how one of its properties-neutrality-influences the run time of random local search. Neutrality creates plateaus in the search space by first performing a majority vote for subsets of the solution candidate and then evaluating the smaller-dimensional string via a low-level fitness function. We prove upper bounds for the expected run time of random local search on this MAJORITY problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of MAJORITY with its symmetric version HASMAJORITY, where a sufficient majority is needed to optimize the subset. We also introduce a generalized version of classic drift theorems as well as a generalized version of Wald’s equation, both of which we believe to be of independent interest.

Virtual chiral recognition of eugenol derivatives on amylose tris(3-chloro-5-methylphenylcarbamate) chiral stationary phase in unusual normal-phase mode

Chromatographic enantioseparation on polysaccharide-based chiral stationary phases has undergone explosive development over the last three decades as a method for separating the enantiomers of chiral compounds on an analytical and preparative scale. In this context, understanding the nature of the intermolecular interactions involved in retention and recognition processes is an interesting scientific challenge. In the present study, three eugenol derivatives were used as chiral references to elucidate some unexplored aspects of the enantioselective and retention properties of the Chiralpak IG-U chiral stationary phase based on amylose-tris(3-chloro-5-methylphenylcarbamate).The performance of the ultra-high performance liquid chromatography chiral packing material Chiralpak IG-U was evaluated using a two-step approach. First, binary mixtures containing variable proportions of alcohol (ethanol or 2-propanol) in n-hexane were used as mobile phases and the retention factors were recorded at three different temperatures. A rational analysis of this set of chromatographic data shows the leading role played by hydrogen bond between the OH group linked to the stereogenic centre of the analytes and the active sites of the chiral chromatographic material in obtaining a high degree of enantioseparation.The retention factors were then plotted against the percentage of alcohol modifiers to obtain retention maps with a non-linear performance trend with correlation factors >0.9990.The proposed retention map model was used to extrapolate and describe virtual chiral recognition of chiral analytes on the Chiralpak IG-U chiral stationary phase under extreme elution conditions with expected run times of hundreds or thousands of years. The presented virtual chiral recognition approach is based on a generic concept and therefore opens new possibilities for understanding the performance of other polysaccharide-based chiral stationary phases.

Crossover for Cardinality Constrained Optimization

To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length- n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/ n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ ( n 2 ) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n ) if n-B = O (1). However, if n-B = Θ ( n ), we show a bound of Ω ( n 2 ), just as for classic bit mutation. Crossover together with a simple island model gives running times of O ( n 2 / log n ) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O ( n log n ). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.

Automated Configuration of Genetic Algorithms by Tuning for Anytime Performance

Finding the best configuration of algorithms’ hyperparameters for a given optimization problem is an important task in evolutionary computation. We compare in this work the results of four different hyperparameter optimization (HPO) approaches for a family of genetic algorithms (GAs) on 25 diverse pseudo-Boolean optimization (PBO) problems. More precisely, we compare previously obtained results from a grid search with those obtained from three automated configuration techniques: 1) iterated racing; 2) mixed-integer parallel-efficient global optimization (MIP-EGO); and 3) mixed-integer evolutionary strategies. Using two different cost metrics: 1) expected running time (ERT) and 2) the area under the empirical cumulative distribution function (ECDF) curve, we find that in several cases the best configurations with respect to ERT are obtained when using the area under the ECDF curve as the cost metric during the configuration process. Our results suggest that even when interested in ERT performance, it might be preferable to use anytime performance measures for the configuration task. We also observe that tuning for ERT is much more sensitive with respect to the budget that is allocated to the target algorithms.

A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such that AV=US, with S the Smith normal form of A. The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as A. Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of V, the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear system solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.

The smoothed number of Pareto-optimal solutions in bicriteria integer optimization

A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which is exponential in the worst case. We study bicriteria integer optimization problems in the model of smoothed analysis, in which inputs are subject to a small amount of random noise, and we prove an almost tight polynomial bound on the expected number of Pareto-optimal solutions. Our results give rise to tight polynomial bounds for the expected running time of the Nemhauser-Ullmann algorithm for the knapsack problem and they improve known results on the running times of heuristics for the bounded knapsack problem and the bicriteria shortest path problem.

Influence Maximization Revisited: Efficient Sampling with Bound Tightened

Given a social network G with n nodes and m edges, a positive integer k , and a cascade model C , the influence maximization (IM) problem asks for k nodes in G such that the expected number of nodes influenced by the k nodes under cascade model C is maximized. The state-of-the-art approximate solutions run in O(k(n+m) log n/ ε 2 ) expected time while returning a (1 - 1/ e - ε) approximate solution with at least 1 - 1/ n probability. A key phase of these IM algorithms is the random reverse reachable (RR) set generation, and this phase significantly affects the efficiency and scalability of the state-of-the-art IM algorithms. In this article, we present a study on this key phase and propose an efficient random RR set generation algorithm under IC model. With the new algorithm, we show that the expected running time of existing IM algorithms under IC model can be improved to O(k ċ n log n ċ 2 ), when for any node v , the total weight of its incoming edges is no larger than a constant. For the general IC model where the weights are skewed, we present a sampling algorithm SKIP. To the best of our knowledge, it is the first index-free algorithm that achieves the optimal time complexity of the sorted subset sampling problem. Moreover, existing approximate IM algorithms suffer from scalability issues in high influence networks where the size of random RR sets is usually quite large. We tackle this challenging issue by reducing the average size of random RR sets without sacrificing the approximation guarantee. The proposed solution is orders of magnitude faster than states of the art as shown in our experiment. Besides, we investigate the issues of forward propagation and derive its time complexity with our proposed subset sampling techniques. We also present a heuristic condition to indicate when the forward propagation approach should be utilized to estimate the expected influence of a given seed set.

A Generalization of Self-Improving Algorithms

Ailon et al. [SICOMP’11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances x 1 , ... , x n follow some unknown product distribution . That is, x i is drawn independently from a fixed unknown distribution 𝒟 i . After spending O ( n 1+ε ) time in a learning phase, the subsequent expected running time is O (( n + H )/ε), where H ∊ { H S , H DT }, and H S and H DT are the entropies of the distributions of the sorting and DT output, respectively. In this article, we allow dependence among the x i ’s under the group product distribution . There is a hidden partition of [1, n ] into groups; the x i ’s in the k th group are fixed unknown functions of the same hidden variable u k ; and the u k ’s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map u k to x i ’s are well-behaved. After an O (poly( n ))-time training phase, we achieve O ( n + H S ) and O ( n α ( n ) + H DT ) expected running times for sorting and DT, respectively, where α (⋅) is the inverse Ackermann function.

Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs

We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k > 3\Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is \emph{perfectly} uniform over the set of all proper $k$-colorings; the expected running time of the algorithm is $\mathrm{poly}(k,n)=\widetilde{O}(n\Delta^2\cdot \log(k))$. This improves upon a result of Huber~(STOC 1998) who obtained a polynomial time perfect sampling algorithm for $k>\Delta^2+2\Delta$. Prior to our work, no algorithm with expected running time $\mathrm{poly}(k,n)$ was known to guarantee perfectly sampling with sub-quadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Huber's) is based on the Coupling from the Past method. Inspired by the \emph{bounding chain} approach, pioneered independently by Huber~(STOC 1998) and H\aggstr\om \& Nelander~(Scand.{} J.{} Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem.

Rejection- and importance-sampling-based perfect simulation for Gibbs hard-sphere models

AbstractCoupling-from-the-past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes consisting of a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned so that spheres do not overlap with each other. We propose an alternative importance-sampling-based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases to infinity while the (expected) sphere radius decreases to zero at varying rates. We further compare the performance of the proposed method analytically and numerically with that of a naive rejection algorithm and of popular dominated CFTP algorithms. Our analysis relies upon identifying large deviations decay rates of the non-overlapping probability of spheres whose centers are distributed as a homogeneous PPP.

The simultaneous conjugacy problem in the symmetric group

The transitive simultaneous conjugacy problem asks whether there exists a permutation $\tau \in S_n$ such that $b_j = \tau^{-1} a_j \tau$ holds for all $j = 1,2, \ldots, d$, where $a_1, a_2, \ldots, a_d$ and $b_1, b_2, \ldots, b_d$ are given sequences of $d$ permutations in $S_n$, each of which generates a transitive subgroup of $S_n$. As from mid 70' it has been known that the problem can be solved in $O(dn^2)$ time. An algorithm with running time $O(dn \log(dn))$, proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in $O(n^2 \log d / \log n + dn\log n)$ time and $O(n^{3/ 2} + dn)$ space. Experimental evaluation on random instances shows that the expected running time of our algorithm is considerably better, perhaps even nearly linear in $n$ at given $d$.

Analysis of Work Stealing with latency

We study the impact of communication latency on the classical Work Stealing load balancing algorithm. Our paper extends the reference model in which we introduce a latency parameter. By using a theoretical analysis and simulation, we study the overall impact of this latency on the Makespan (maximum completion time). We derive a new expression of the expected running time of a bag of independent tasks scheduled by Work Stealing. This expression enables us to predict under which conditions a given run will yield acceptable performance. For instance, we can easily calibrate the maximal number of processors to use for a given work/platform combination. All our results are validated through simulation on a wide range of parameters.

Fast uniform generation of random graphs with given degree sequences

AbstractIn this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that , where is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power‐law degree sequences in cases where the previous best was O(n4.081), and (ii) O(nd + d4) for d‐regular graphs when , where the previous best was O(nd3).

Coarse Grained Parallel Selection

Several efficient, but non-optimal, solutions to the Selection Problem on coarse grained parallel computers have appeared in the literature. We consider the example of the Saukas-Song algorithm; we analyze it without expressing the running time in terms of communication rounds. This shows that while in the best case the Saukas-Song algorithm runs in asymptotically optimal time, in general it does not. We propose another algorithm for coarse grained selection that has optimal expected running time.

A Stochastic Approach to Eulerian Numbers

We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of n particles perform random walks on the integers, beginning at the origin. Each particle walks until it reaches an unoccupied site, at which point it occupies that site and the next particle begins its walk. After all walks are complete, the set of occupied sites is an interval of length n containing the origin. We show the probability that k of the occupied sites are positive is given by an Eulerian probability distribution. Having made this connection, we use generating function techniques to compute the expected run time of the model.

Running time analysis of the (1+1)-EA for robust linear optimization

Evolutionary algorithms (EAs) have found many successful real-world applications, where the optimization problems are often subject to a wide range of uncertainties. To understand the practical behaviors of EAs theoretically, there are a series of efforts devoted to analyzing the running time of EAs for optimization under uncertainties. Existing studies mainly focus on noisy and dynamic optimization, while another common type of uncertain optimization, i.e., robust optimization, has been rarely touched. In this paper, we analyze the expected running time of the (1+1)-EA solving robust linear optimization problems (i.e., linear problems under robust scenarios) with a cardinality constraint k. Two common robust scenarios, i.e., deletion-robust and worst-case, are considered. Particularly, we derive tight ranges of the robust parameter d or budget k allowing the (1+1)-EA to find an optimal solution in polynomial running time, which disclose the potential of EAs for robust optimization.

Sequential and parallel algorithms for all-pair [formula omitted]-mismatch maximal common substrings

Identifying long pairwise maximal common substrings among a large set of sequences is a frequently used construct in computational biology, with applications in DNA sequence clustering and assembly. Due to errors made by sequencers, algorithms that can accommodate a small number of differences are of particular interest. Formally, let D be a collection of n sequences of total length N, ϕ be a length threshold, and k be a mismatch threshold. The goal is to identify and report all k-mismatch maximal common substrings of length at least ϕ over all pairs of strings in D. Heuristics based on seed-and-extend style filtering techniques are often employed in such applications. However, such methods cannot provide any provably efficient run time guarantees. To this end, we present a sequential algorithm with an expected run time of O(NlogkN+occ), where occ is the output size. We then present a distributed memory parallel algorithm with an expected run time of ONplogN+occlogkN using Ologk+1N expected rounds of global communications, under some realistic assumptions, where p is the number of processors. Finally, we demonstrate the performance and scalability of our algorithms using experiments on large high throughput sequencing data.