arXiv subjects

Lisa Hellerstein

Publications and source records attributed to Lisa Hellerstein.

At least 19 recordsLinked to original sources

arXiv2026

Approximating Matroid Basis Testing for Partition Matroids using Budget-In-Expectation

We consider the following Stochastic Boolean Function Evaluation problem, which is closely related to several problems from the literature. A matroid $\mathcal{M}$ (in compact representation) on ground set $E$ is given, and each element $i\in E$ is active independently with known probability $p_i\in(0,1)$. The elements can be queried, upon which it is revealed whether the respective element is active or not. The goal is to find an adaptive querying strategy for determining whether there is a basis of $\mathcal{M}$ in which all elements are active, with the objective of minimizing the expected number of queries. When $\mathcal{M}$ is a uniform matroid, this is the problem of evaluating a $k$-of-$n$ function, first studied in the 1970s. This problem is well-understood, and has an optimal adaptive strategy that can be computed in polynomial time. Taking $\mathcal{M}$ to instead be a partition matroid, we show that previous approaches fail to give a constant-factor approximation. Our main result is a polynomial-time constant-factor approximation algorithm producing a randomized strategy for this partition matroid problem. We obtain this result by combining a new technique with several well-established techniques. Our algorithm adaptively interleaves solutions to several instances of a novel type of stochastic querying problem, with a constraint on the $\textit{expected}$ cost. We believe that this type of problem is of independent interest, will spark follow-up work, and has the potential for additional applications.

cs.DS↗

arXiv2025

Optimal Verification of a Minimum-Weight Basis in an Uncertainty Matroid

Research in explorable uncertainty addresses combinatorial optimization problems where there is partial information about the values of numeric input parameters, and exact values of these parameters can be determined by performing costly queries. The goal is to design an adaptive query strategy that minimizes the query cost incurred in computing an optimal solution. Solving such problems generally requires that we be able to solve the associated verification problem: given the answers to all queries in advance, find a minimum-cost set of queries that certifies an optimal solution to the combinatorial optimization problem. We present a polynomial-time algorithm for verifying a minimum-weight basis of a matroid, where each weight lies in a given uncertainty area. These areas may be finite sets, real intervals, or unions of open and closed intervals, strictly generalizing previous work by Erlebach and Hoffman which only handled the special case of open intervals. Our algorithm introduces new techniques to address the resulting challenges. Verification problems are of particular importance in the area of explorable uncertainty, as the structural insights and techniques used to solve the verification problem often heavily influence work on the corresponding online problem and its stochastic variant. In our case, we use structural results from the verification problem to give a best-possible algorithm for a promise variant of the corresponding adaptive online problem. Finally, we show that our algorithms can be applied to two learning-augmented variants of the minimum-weight basis problem under explorable uncertainty.

cs.DS↗

arXiv2025

An Exact Algorithm for the Unanimous Vote Problem

Consider $n$ independent, biased coins, each with a known probability of heads. Presented with an ordering of these coins, flip (i.e., toss) each coin once, in that order, until we have observed both a *head* and a *tail*, or flipped all coins. The Unanimous Vote problem asks us to find the ordering that minimizes the expected number of flips. Gkenosis et al. [arXiv:1806.10660] gave a polynomial-time $\phi$-approximation algorithm for this problem, where $\phi \approx 1.618$ is the golden ratio. They left open whether the problem was NP-hard. We answer this question by giving an exact algorithm that runs in time $O(n \log n)$. The Unanimous Vote problem is an instance of the more general Stochastic Boolean Function Evaluation problem: it thus becomes one of the only such problems known to be solvable in polynomial time. Our proof uses simple interchange arguments to show that the optimal ordering must be close to the ordering produced by a natural greedy algorithm. Beyond our main result, we compare the optimal ordering with the best adaptive strategy, proving a tight adaptivity gap of $1.2\pm o(1)$ for the Unanimous Vote problem.

cs.DS↗

arXiv2024

Minimizing Cost Rather Than Maximizing Reward in Restless Multi-Armed Bandits

Restless Multi-Armed Bandits (RMABs) offer a powerful framework for solving resource constrained maximization problems. However, the formulation can be inappropriate for settings where the limiting constraint is a reward threshold rather than a budget. We introduce a constrained minimization problem for RMABs that balances the goal of achieving a reward threshold while minimizing total cost. We show that even a bi-criteria approximate version of the problem is PSPACE-hard. Motivated by the hardness result, we define a decoupled problem, indexability and a Whittle index for the minimization problem, mirroring the corresponding concepts for the maximization problem. Further, we show that the Whittle index for the minimization problem can easily be computed from the Whittle index for the maximization problem. Consequently, Whittle index results on RMAB instances for the maximization problem give Whittle index results for the minimization problem. Despite the similarities between the minimization and maximization problems, solving the minimization problem is not as simple as taking direct analogs of the heuristics for the maximization problem. We give an example of an RMAB for which the greedy Whittle index heuristic achieves the optimal solution for the maximization problem, while the analogous heuristic yields the worst possible solution for the minimization problem. In light of this, we present and compare several heuristics for solving the minimization problem on real and synthetic data. Our work suggests the importance of continued investigation into the minimization problem.

cs.DS↗

arXiv2024

Quickly Determining Who Won an Election

This paper considers elections in which voters choose one candidate each, independently according to known probability distributions. A candidate receiving a strict majority (absolute or relative, depending on the version) wins. After the voters have made their choices, each vote can be inspected to determine which candidate received that vote. The time (or cost) to inspect each of the votes is known in advance. The task is to (possibly adaptively) determine the order in which to inspect the votes, so as to minimize the expected time to determine which candidate has won the election. We design polynomial-time constant-factor approximation algorithms for both the absolute-majority and the relative-majority version. Both algorithms are based on a two-phase approach. In the first phase, the algorithms reduce the number of relevant candidates to $O(1)$, and in the second phase they utilize techniques from the literature on stochastic function evaluation to handle the remaining candidates. In the case of absolute majority, we show that the same can be achieved with only two rounds of adaptivity.

cs.DS↗

arXiv2022

A Local Search Algorithm for the Min-Sum Submodular Cover Problem

We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a $(4+\epsilon)$-approximate solution in time $O(n^3\log(n/\epsilon))$, provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets.

cs.DS↗

arXiv2022

Adaptivity Gaps for the Stochastic Boolean Function Evaluation Problem

We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the variables of $x$, each of which is associated with a cost of testing and an independent probability of being true. If a strategy for solving the problem is adaptive in the sense that its next test can depend on the outcomes of previous tests, it has lower expected cost but may take up to exponential space to store. In contrast, a non-adaptive strategy may have higher expected cost but can be stored in linear space and benefit from parallel resources. The adaptivity gap, the ratio between the expected cost of the optimal non-adaptive and adaptive strategies, is a measure of the benefit of adaptivity. We present lower bounds on the adaptivity gap for the SBFE problem for popular classes of Boolean functions, including read-once DNF formulas, read-once formulas, and general DNFs. Our bounds range from $Ω(\log n)$ to $Ω(n/\log n)$, contrasting with recent $O(1)$ gaps shown for symmetric functions and linear threshold functions.

cs.DS↗

arXiv2022

The Stochastic Boolean Function Evaluation Problem for Symmetric Boolean Functions

We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of Deshpande, Hellerstein and Kletenik. Our second algorithm, which is simple, is based on the algorithm solving the SBFE problem for $k$-of-$n$ functions, due to Salloum, Breuer, and Ben-Dov. It achieves a $(B-1)$ approximation factor, where $B$ is the number of blocks of 0's and 1's in the standard vector representation of the symmetric Boolean function. As part of the design of the first algorithm, we prove that the goal value of any symmetric Boolean function is less than $n(n+1)/2$. Finally, we give an example showing that for symmetric Boolean functions, minimum expected verification cost and minimum expected evaluation cost are not necessarily equal. This contrasts with a previous result, given by Das, Jafarpour, Orlitsky, Pan and Suresh, which showed that equality holds in the unit-cost case.

cs.DS↗

arXiv2021

A Game Theoretic Approach to a Problem in Polymatroid Maximization

We consider the problem of maximizing the minimum (weighted) value of all components of a vector over a polymatroid. This is a special case of the lexicographically optimal base problem introduced and solved by Fujishige. We give an alternative formulation of the problem as a zero-sum game between a maximizing player whose mixed strategy set is the base of the polymatroid and a minimizing player whose mixed strategy set is a simplex. We show that this game and three variations of it unify several problems in search, sequential testing and queuing. We give a new, short derivation of optimal strategies for both players and an expression for the value of the game. Furthermore, we give a characterization of the set of optimal strategies for the minimizing player and we consider special cases for which optimal strategies can be found particularly easily.

math.OC↗

arXiv2021

A Tight Bound for Stochastic Submodular Cover

We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achieves an approximation bound of $(\ln (Q/η)+1)$ for Stochastic Submodular Cover: here $Q$ is the "goal value" and $η$ is the smallest non-zero marginal increase in utility deliverable by an item. (For integer-valued utility functions, we show a bound of $H(Q)$, where $H(Q)$ is the $Q^{th}$ Harmonic number.) Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan and Saligrama (2017). The subsequent corrected proof of Golovin and Krause (2017) gives a quadratic bound of $(\ln(Q/η) + 1)^2$. Other previous bounds for the problem are $56(\ln(Q/η) + 1)$, implied by work of Im et al. (2016) on a related problem, and $k(\ln (Q/η)+1)$, due to Deshpande et al. (2016) and Hellerstein and Kletenik (2018), where $k$ is the number of states. Our bound generalizes the well-known $(\ln~m + 1)$ approximation bound on the greedy algorithm for the classical Set Cover problem, where $m$ is the size of the ground set.

cs.DS↗

arXiv2021

A General Framework for Approximating Min Sum Ordering Problems

We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the Min Sum Ordering Problem (MSOP) which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time $α$-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time $4 α$-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results.

cs.DS↗

arXiv2018

The Stochastic Score Classification Problem

Consider the following Stochastic Score Classification Problem. A doctor is assessing a patient's risk of developing a certain disease, and can perform $n$ tests on the patient. Each test has a binary outcome, positive or negative. A positive test result is an indication of risk, and a patient's score is the total number of positive test results. The doctor needs to classify the patient into one of $B$ risk classes, depending on the score (e.g., LOW, MEDIUM, and HIGH risk). Each of these classes corresponds to a contiguous range of scores. Test $i$ has probability $p_i$ of being positive, and it costs $c_i$ to perform the test. To reduce costs, instead of performing all tests, the doctor will perform them sequentially and stop testing when it is possible to determine the risk category for the patient. The problem is to determine the order in which the doctor should perform the tests, so as to minimize the expected testing cost. We provide approximation algorithms for adaptive and non-adaptive versions of this problem, and pose a number of open questions.

cs.DS↗

arXiv2018

Solving Zero-sum Games using Best Response Oracles with Applications to Search Games

We present efficient algorithms for computing optimal or approximately optimal strategies in a zero-sum game for which Player I has n pure strategies and Player II has an arbitrary number of pure strategies. We assume that for any given mixed strategy of Player I, a best response or "approximate" best response of Player II can be found by an oracle in time polynomial in n. We then show how our algorithms may be applied to several search games with applications to security and counter-terrorism. We evaluate our main algorithm experimentally on a prototypical search game. Our results show it performs well compared to an existing, well-known algorithm for solving zero-sum games that can also be used to solve search games, given a best response oracle.

math.OC↗

arXiv2017

Submodular Goal Value of Boolean Functions

Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of $f$ is defined in terms of a monotone, submodular utility function associated with $f$. As shown by Deshpande et al., proving that a Boolean function $f$ has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for $f$. Also, if $f$ has small goal value, it indicates a close relationship between two other measures of the complexity of $f$, its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions provoked by our work.

cs.DM↗

arXiv2016

Scenario Submodular Cover

Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause. In Scenario Submodular Cover, the goal is to produce a cover with minimum expected cost, where the expectation is with respect to an empirical joint distribution, given as input by a weighted sample of realizations. In contrast, in Stochastic Submodular Cover, the variables of the input distribution are assumed to be independent, and the distribution of each variable is given as input. Building on algorithms developed by Cicalese et al. and Golovin and Krause for related problems, we give two approximation algorithms for Scenario Submodular Cover over discrete distributions. The first achieves an approximation factor of O(log Qm), where m is the size of the sample and Q is the goal utility. The second, simpler algorithm achieves an approximation bound of O(log QW), where Q is the goal utility and W is the sum of the integer weights. (Both bounds assume an integer-valued utility function.) Our results yield approximation bounds for other problems involving non-independent distributions that are explicitly specified by their support.

cs.DS↗

arXiv2015

Discrete Stochastic Submodular Maximization: Adaptive vs. Non-Adaptive vs. Offline

We consider the problem of stochastic monotone submodular function maximization, subject to constraints. We give results on adaptivity gaps, and on the gap between the optimal offline and online solutions. We present a procedure that transforms a decision tree (adaptive algorithm) into a non-adaptive chain. We prove that this chain achieves at least $τ$ times the utility of the decision tree, over a product distribution and binary state space, where $τ = \min_{i,j} \Pr[x_i=j]$. This proves an adaptivity gap of $1/τ$ (which is $2$ in the case of a uniform distribution) for the problem of stochastic monotone submodular maximization subject to state-independent constraints. For a cardinality constraint, we prove that a simple adaptive greedy algorithm achieves an approximation factor of $(1-1/e^τ)$ with respect to the optimal offline solution; previously, it has been proven that the algorithm achieves an approximation factor of $(1-1/e)$ with respect to the optimal adaptive online solution. Finally, we show that there exists a non-adaptive solution for the stochastic max coverage problem that is within a factor $(1-1/e)$ of the optimal adaptive solution and within a factor of $τ(1-1/e)$ of the optimal offline solution.

cs.DS↗

arXiv2014

Evaluation of DNF Formulas

Stochastic Boolean Function Evaluation (SBFE) is the problem of determining the value of a given Boolean function $f$ on an unknown input $x$, when each bit of $x_i$ of $x$ can only be determined by paying a given associated cost $c_i$. Further, $x$ is drawn from a given product distribution: for each $x_i$, $Prob[x_i=1] = p_i$, and the bits are independent. The goal is to minimize the expected cost of evaluation. Stochastic Boolean Function Evaluation (SBFE) is the problem of determining the value of a given Boolean function $f$ on an unknown input $x$, when each bit of $x_i$ of $x$ can only be determined by paying a given associated cost $c_i$. Further, $x$ is drawn from a given product distribution: for each $x_i$, $Prob[x_i=1] = p_i$, and the bits are independent. The goal is to minimize the expected cost of evaluation. In this paper, we study the complexity of the SBFE problem for classes of DNF formulas. We consider both exact and approximate versions of the problem for subclasses of DNF, for arbitrary costs and product distributions, and for unit costs and/or the uniform distribution.

cs.CC↗

arXiv2013

Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause.

cs.DS↗