The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [X. Chen et al., 2025; Chen et al., 2025; Chen et al., 2025]. In this paper we consider the problem of relative-error testing an unknown and arbitrary f: {0,1}ⁿ → {0,1} for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the n input variables. Our first result is a one-sided non-adaptive algorithm for this problem that makes Õ(log(N)/ε) samples and queries, where N = |f^{-1}(1)| is the number of satisfying assignments of the function that is being tested and the value of N is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of N and with high probability makes Õ(log(N)/ε) samples and queries. We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of N up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.

Differential privacy (DP) is the de facto notion of privacy both in theory and in practice. However, despite its popularity, DP imposes strict requirements which guard against strong worst-case scenarios. For example, it guards against seemingly unrealistic scenarios where an attacker has full information about all but one point in the data set, and still nothing can be learned about the remaining point. While preventing such a strong attack is desirable, many works have explored whether average-case relaxations of DP are easier to satisfy [Hall et al., 2013; Wang et al., 2016; Bassily and Freund, 2016; Liu et al., 2023]. In this work, we are motivated by the question of whether alternate, weaker notions of privacy are possible: can a weakened privacy notion still guarantee some basic level of privacy, and on the other hand, achieve privacy more efficiently and/or for a substantially broader set of tasks? Our main result shows the answer is no: even in the statistical setting, any reasonable measure of privacy satisfying nontrivial composition is equivalent to DP. To prove this, we identify a core set of four axioms or desiderata: pre-processing invariance, prohibition of blatant non-privacy, strong composition, and linear scalability. Our main theorem shows that any privacy measure satisfying our axioms is equivalent to DP, up to polynomial factors in sample complexity. We complement this result by showing our axioms are minimal: removing any one of our axioms enables ill-behaved measures of privacy.

This paper considers the junta testing problem in a recently introduced “relative error” variant of the standard Boolean function property testing model. In relative-error testing we measure the distance from $f$ to $g$, where $f,g: \{0,1\}^n \to \{0,1\}$, by the ratio of $|f^{-1}(1) \triangle g^{-1}(1)|$ (the number of inputs on which $f$ and $g$ disagree) to $|f^{-1}(1)|$ (the number of satisfying assignments of $f$), and we give the testing algorithm both black-box access to $f$ and also access to independent uniform samples from $f^{-1}(1)$. Chen et al. (SODA 2025) observed that the class of $k$-juntas is poly$(2^k,1/\epsilon)$-query testable in the relative-error model, and asked whether poly$(k,1/\epsilon)$ queries is achievable. We answer this question affirmatively by giving a $\tilde{O}(k/\epsilon)$-query algorithm, matching the optimal complexity achieved in the less challenging standard model. Moreover, as our main result, we show that any subclass of $k$-juntas that is closed under permuting variables is relative-error testable with a similar complexity. This gives highly efficient relative-error testing algorithms for a number of well-studied function classes, including size-$k$ decision trees, size-$k$ branching programs, and size-$k$ Boolean formulas.

We study the relative-error property testing model for Boolean functions that was recently introduced in the work of [X. Chen et al., 2025]. In relative-error testing, the testing algorithm gets uniform random satisfying assignments as well as black-box queries to f, and it must accept f with high probability whenever f has the property that is being tested and reject any f that is relative-error far from having the property. Here the relative-error distance from f to a function g is measured with respect to |f^{-1}(1)| rather than with respect to the entire domain size 2ⁿ as in the Hamming distance measure that is used in the standard model; thus, unlike the standard model, relative-error testing allows us to study the testability of sparse Boolean functions that have few satisfying assignments. It was shown in [X. Chen et al., 2025] that relative-error testing is at least as difficult as standard-model property testing, but for many natural and important Boolean function classes the precise relationship between the two notions is unknown. In this paper we consider the well-studied and fundamental properties of being a conjunction and being a decision list. In the relative-error setting, we give an efficient one-sided error tester for conjunctions with running time and query complexity O(1/ε). Secondly, we give a two-sided relative-error Õ(1/ε) tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model [Nader H. Bshouty, 2020; I. Diakonikolas et al., 2007].

It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, \({\text{ PLS}} \not\subseteq {\text{ PPP}}\) , \({\text{ SOPL}} \not\subseteq {\text{ PPA}}\) , and \({\text{ EOPL}} \not\subseteq {\text{ UEOPL}}\) . In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s.

Abstract. We show [Formula: see text]. Here the class [Formula: see text] consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse [Formula: see text] by Fearnley et al. (STOC 2021). We also prove a companion result [Formula: see text], where [Formula: see text] is the class associated with the Sink-of-Potential-Line problem.

In this work, we investigate how to minimize the number of gateways deployed in Unmanned Aerial Vehicles (UAVs) needed to meet the demand for non-3GPP Internet of Things (IoT) devices, seeking to improve the Quality of Service (QoS), keeping a balance between delay and data rate. Gateways deployed in UAVs add the mobility flexibility of UAVs, which paves the way for meeting emergency demand increments. The 5thGeneration Networks (5G) and Beyond 5thGeneration Networks (B5G) systems incorporated access to IoT devices via non-3GPP access, opening up new integration possibilities. Furthermore, Low Power Wide Area Network (LPWAN) networks, especially Long Range Wide Area Network (LoRaWAN), allow access over long distances with reduced energy consumption. In this scenario, our work proposes a Mixed Integer Linear Programming (MILP) optimization model to minimize the number of UAVs that meet the increment of emergency demand, comply with limits of QoS, and maintain the compromise between delay and data rate. Simulation results show that the proposed model significantly reduces the number of gateways, maintains optimal levels of QoS, and maintains the compromise between delay and data rate compared to the presented baselines.

We show $\textsf{EOPL}=\textsf{PLS}\cap\textsf{PPAD}$. Here the class $\textsf{EOPL}$ consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubacek and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse $\textsf{CLS}=\textsf{PLS}\cap\textsf{PPAD}$ by Fearnley et al. (STOC 2021). We also prove a companion result $\textsf{SOPL}=\textsf{PLS}\cap\textsf{PPADS}$, where $\textsf{SOPL}$ is the class associated with the Sink-of-Potential-Line problem.

It is well-known that Resolution proofs can be efficiently simulated by Sherali-Adams (SA) proofs. We show1, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, PLS $\nsubseteq$ PPP, SOPL $\nsubseteq$ PPA, and EOPL $\nsubseteq$ UEOPL. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s.1This is an extended abstract. For the full version of this article, please refer to [GHJ+22b].

While energy efficiency is an important issue in virtualized RAN due to its impact on OPEX, the centralization level of virtualized RAN functions is another relevant concern that can conflict with the former. In this paper, we introduce a bi-objective problem formulation representing these two objectives and solution strategy based on the ϵ-constraint approach to generate the minimal complete set of Pareto-optimal solutions. We investigate the trade-off between energy efficiency and centralization level in traditional and next-generation RAN topologies. We show scenarios allowing noticeable improvement in the centralization level (e.g., from near 10% to 30%) without impacting energy consumption. However, after a certain value of centralization level, the impact in the energy consumption may become high and hard to justify.