MOTIVATION: Multiplex PCR-based enrichment is widely used in viral genome sequencing and pathogen surveillance. However, designing large sets of primers that maximize genome coverage while minimizing primer-primer interactions remains a major computational challenge. Existing methods such as SADDLE and Olivar use heuristics to optimize a Badness score for primer dimers but lack theoretical guarantees on solution quality. RESULTS: We introduce PRISM, a new framework that formulates multiplex primer design as a constrained submodular maximization problem. Our method defines an objective that balances genome coverage and dimer risk, and applies a local search algorithm with a constant-factor approximation guarantee. Evaluations on viral genome datasets demonstrate that PRISM consistently achieves lower Badness scores compared to PrimalScheme, Olivar, and primerJinn. These results highlight the scalability and theoretical rigor of submodular optimization in primer design. AVAILABILITY: PRISM is open-source and available at https://github.com/yhhan19/PRISM-new. The experimental data, scripts, and results used in this paper are archived on Figshare at https://doi.org/10.6084/m9.figshare.32806499.

With the rapid growth of data in modern applications, parallel combinatorial algorithms for maximizing non-monotone submodular functions have gained significant attention. In the parallel computation setting, the state-of-the-art approximation ratio of $1/e$ is achieved by a continuous algorithm (Ene & Nguyen, 2020) with adaptivity $\mathcal O (log(n))$. In this work, we focus on size constraints and present the first combinatorial algorithm matching this bound – a randomized parallel approach achieving $1/e - \epsilon$ approximation ratio. This result bridges the gap between continuous and combinatorial approaches for this problem. As a byproduct, we also develop a simpler $(1/4 - \epsilon)$-approximation algorithm with high probability $(\ge 1 - 1/n)$. Both algorithms achieve $\mathcal O (log(n) log(k))$ adaptivity and $\mathcal O (n log(n) log(k)) query complexity. Empirical results show our algorithms achieve competitive objective values, with the $(1/4 - $\epsilon$)$-approximation algorithm particularly efficient in queries.

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

Distributed maximization of a submodular function in the MapReduce (MR) model has received much attention, culminating in two frameworks that allow a centralized algorithm to be run in the MR setting without loss of approximation, as long as the centralized algorithm satisfies a certain consistency property – which had previously only been known to be satisfied by the standard greedy and continous greedy algorithms. A separate line of work has studied parallelizability of submodular maximization in the adaptive complexity model, where each thread may have access to the entire ground set. For the size-constrained maximization of a monotone and submodular function, we show that several sublinearly adaptive (highly parallelizable) algorithms satisfy the consistency property required to work in the MR setting, which yields practical, parallelizable and distributed algorithms. Separately, we develop the first distributed algorithm with linear query complexity for this problem. Finally, we provide a method to increase the maximum cardinality constraint for MR algorithms at the cost of additional MR rounds.

We present combinatorial and parallelizable algorithms for the maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to 1/6 − ε, and even further to 0.193 − ε by increasing the adaptivity by a factor of O(log(k)). The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in O(log(n)) adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio 1/6 − ε, adaptivity O(log(n)), and query complexity O(n log(k)), while the second has approximation ratio 0.193 − ε, adaptivity O(log(n) log(k)), and query complexity O(n log(k)). Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

MapReduce (MR) algorithms for maximizing monotone, submodular functions subject to a cardinality constraint (SMCC) are currently restricted to the use of the linear-adaptive (non-parallelizable) algorithm GREEDY. Low-adaptive algorithms do not satisfy the requirements of these distributed MR frameworks, thereby limiting their performance. We study the SMCC problem in a distributed setting and propose the first MR algorithms with sublinear adaptive complexity. Our algorithms, R-DASH, T-DASH and G-DASH provide 0.316 - ε, 3/8 - ε , and (1 - 1/e - ε) approximation ratios, respectively, with nearly optimal adaptive complexity and nearly linear time complexity. Additionally, we provide a framework to increase, under some mild assumptions, the maximum permissible cardinality constraint from O( n / ℓ^2) of prior MR algorithms to O( n / ℓ ), where n is the data size and ℓ is the number of machines; under a stronger condition on the objective function, we increase the maximum constraint value to n. Finally, we provide empirical evidence to demonstrate that our sublinear-adaptive, distributed algorithms provide orders of magnitude faster runtime compared to current state-of-the-art distributed algorithms.

In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and nonlinear regression. We provide the first deterministic, linear-time approximation algorithms for this problem that do not assume the objective is monotone. We present three deterministic, linear-time algorithms: a single-pass streaming algorithm with a ratio of 23.313 + ε, which is the first linear-time streaming algorithm; a simpler deterministic linear-time algorithm with a ratio of 11.657; and a (4 + O(ε))-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of e + ε in O_ε (n log(n)) time, close to the best known expected ratio of e - 0.121 in polynomial time.

For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its theoretical properties, in terms of adaptive complexity, query complexity, and approximation ratio; that is, it obtains, with high probability, query complexity of $O(n)$ in expectation, adaptivity of $O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is assembled from two components which may be of independent interest. The first component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm to improve the query complexity of many algorithms. Moreover, a variant of LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is smaller than that of any previous algorithm in the literature. The second component is a parallelizable thresholding procedure THRESHOLDSEQ for adding elements with gain above a constant threshold. Finally, we demonstrate that our main algorithm empirically outperforms, in terms of runtime, adaptive rounds, total queries, and objective values, the previous state-of-the-art algorithm FAST in a comprehensive evaluation with six submodular objective functions.