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The problem of maximizing a normalized monotone non-submodular set function subject to a cardinality constraint arises in the context of extracting information from massive streaming data. In this paper, we present four streaming algorithms for this problem by utilizing the concept of diminishing-return ratio. We analyze these algorithms to obtain the corresponding approximation ratios, which generalize the previous results for the submodular case. The numerical experiments show that our algorithms have better solution quality and competitive running time when compared to an existing algorithm.

Diversity maximization is a fundamental problem with broad applications in data summarization, web search, and recommender systems. Given a set X of n elements, the problem asks for a subset S of k≪n elements with maximum diversity, as quantified by the dissimilarities among the elements in S. In this paper, we study diversity maximization with fairness constraints in streaming and sliding-window models. Specifically, we focus on the max–min diversity maximization problem, which selects a subset S that maximizes the minimum distance (dissimilarity) between any pair of distinct elements within it. Assuming that the set X is partitioned into m disjoint groups by a specific sensitive attribute, e.g., sex or race, ensuring fairness requires that the selected subset S contains ki elements from each group i∈[m]. Although diversity maximization has been extensively studied, existing algorithms for fair max–min diversity maximization are inefficient for data streams. To address the problem, we first design efficient approximation algorithms for this problem in the (insert-only) streaming model, where data arrive one element at a time, and a solution should be computed based on the elements observed in one pass. Furthermore, we propose approximation algorithms for this problem in the sliding-window model, where only the latest w elements in the stream are considered for computation to capture the recency of the data. Experimental results on real-world and synthetic datasets show that our algorithms provide solutions of comparable quality to the state-of-the-art offline algorithms while running several orders of magnitude faster in the streaming and sliding-window settings.

Given an undirected graph G on n nodes and m edges in the form of a data stream we study the problem of finding an Euler tour in G. Our main result is the first one-pass streaming algorithm computing an Euler tour of G in the form of an edge successor function with only O(nlog(n)) RAM, which is optimal for this setting (e.g. Sun and Woodruff (2015)). Since the output size can be much larger, we use a write-only tape to gradually output the solution. The previously best-known result for finding Euler tours in data streams is implicitly given by the W-stream algorithm of Demetrescu et al. (2010) using O(m/n) passes under the same RAM limitation. Our approach is to partition the edges into edge-disjoint cycles and to merge the cycles until a single Euler tour is achieved. In the streaming environment such a merging is far from being obvious as the limited RAM allows the processing of only a constant number of cycles at once. This enforces merging of cycles that partially are no longer present in RAM. We solve this problem with a new edge swapping technique, for which storing two certain edges per node is sufficient to merge tours without having all tour edges in RAM. The mathematical key is to model tours and their merging in an algebraic way, where certain equivalence classes represent subtours. This quite general approach might be of interest also in other routing problems.

An ordering constraint satisfaction problem (OCSP) is defined by a family F of predicates mapping permutations on { 1 , … , k } to { 0 , 1 } . An instance of Max-OCSP( F ) on n variables consists of a list of constraints, each consisting of a predicate from F applied on k distinct variables. The goal is to find an ordering of the n variables that maximizes the number of constraints for which the induced ordering on the k variables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ (MAS) and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every F , Max-OCSP( F ) is approximation-resistant to o( n )-space streaming algorithms, i.e., algorithms using o( n ) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS, our result shows that for every ϵ > 0 , MAS is not ( 1 / 2 + ϵ ) -approximable in o( n ) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in o ( n ) space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet size q ) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of the n variables into small blocks, for most of the constraints, all variables are in distinct blocks.

We demonstrate how to evaluate stepwise hedge automata (Shas) with subhedge projection while completely projecting irrelevant subhedges. Since this requires passing finite state information top-down, we introduce the notion of downward stepwise hedge automata. We use them to define in-memory and streaming evaluators with complete subhedge projection for Shas. We then tune the evaluators so that they can decide on membership at the earliest time point. We apply our algorithms to the problem of answering regular XPath queries on Xml streams. Our experiments show that complete subhedge projection of Shas can indeed speed up earliest query answering on Xml streams so that it becomes competitive with the best existing streaming tools for XPath queries.

In this paper, we investigate the maximization of the differences between a nonnegative monotone diminishing return submodular (DR-submodular) function and a nonnegative linear function on the integer lattice. As it is almost unapproximable for maximizing a submodular function without the condition of nonnegative, we provide weak (bifactor) approximation algorithms for this problem in two online settings, respectively. For the unconstrained online model, we combine the ideas of single-threshold greedy, binary search and function scaling to give an efficient algorithm with a 1/2 weak approximation ratio. For the online streaming model subject to a cardinality constraint, we provide a one-pass ( 3 - 5 ) / 2 weak approximation ratio streaming algorithm. Its memory complexity is ( k log k / ε ) , and the update time for per element is ( log 2 k / ε ) .

We consider online algorithms with respect to the competitive ratio. In this paper, we explore one-way automata as a model for online algorithms. We focus on quantum and classical online algorithms. For a specially constructed online minimization problem, we provide a quantum log-bounded automaton that is more effective (has less competitive ratio) than classical automata, even with advice, in the case of the logarithmic size of memory. We construct an online version of the well-known Disjointness problem as a problem. It was investigated by many researchers from a communication complexity and query complexity point of view.

We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking O(ε− 1) passes: (1) a (1 − e− 1 − ε)-approximation algorithm for the cardinality-constrained problem, (2) a (0.5 − ε)-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run deterministically in O∗(n) time, using O∗(K) space, where n is the size of the ground set and K is the size of the knapsack. Here the term O∗ hides a polynomial of logK and ε− 1. Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes O(nε−1log(ε−1logK)) time, improving on the algorithm of Badanidiyuru and Vondrák that takes O(nε−1log(ε−1K)) time.

Submodular optimization problem has been concerned in recent years. The problem of maximizing submodular and non-submodular functions on the integer lattice has received a lot of recent attention. In this paper, we study streaming algorithms for the problem of maximizing a monotone non-submodular functions with cardinality constraint on the integer lattice. For a monotone non-submodular function f : Z + n → R + defined on the integer lattice with diminishing-return (DR) ratio γ , we present a one pass streaming algorithm that gives a ( 1 - 1 2 γ - ϵ ) -approximation, requires at most O ( k ϵ - 1 log k / γ ) space and O ( ϵ - 1 log k / γ · log ‖ B ‖ ∞ ) update time per element. We then modify the algorithm and improve the memory complexity to O ( k γ ϵ ) . To the best of our knowledge, this is the first streaming algorithm on the integer lattice for this constrained maximization problem.