Differentially Private Linear Programming: Reduced Sub-Optimality and Guaranteed Constraint Satisfaction

In location-based services(LBSs), it is promising for users to crowdsource and share their Point-of-Interest(PoI) information with each other in a common cache to reduce query frequency and preserve location privacy. Yet most studies on multi-user privacy preservation overlook the opportunity of leveraging their service flexibility. This paper is the first to study multiple users' strategic cooperation against an adversary's optimal inference attack, by leveraging mutual service flexibility. We formulate the multi-user privacy cooperation against the adversary as a max-min adversarial game and solve it in a linear program. Unlike the vast literature, even if a user finds the cached information useful, we prove it beneficial to still query the platform to further confuse the adversary. As the linear program's computational complexity still increases superlinearly with the number of users' possible locations, we propose a binary obfuscation scheme in two opposite spatial directions to achieve guaranteed performance with only constant complexity. Perhaps surprisingly, a user with a greater service flexibility should query with a less obfuscated location to add confusion. Finally, we provide guidance on the optimal query sequence among LBS users. Simulation results show that our crowdsourced privacy protection scheme greatly improves users' privacy as compared with existing approaches.

Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3/2-epsilon inapproximability for Vertex Cover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1/2+e for bounded degree IndependentSet, and we establish inapproximability of MaxMULTICUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

Distributed control and optimization form the backbone of many critical control systems, with typical examples including sensor networks, power grids, and intelligent transportation systems. As these systems grow in scale and complexity, ensuring the privacy of individual agents becomes increasingly essential. Privacy protection is no longer an optional add-on; it is a crucial element in the design of these systems. Sensitive data, such as states, measurements, and decisions, must be shielded from unauthorized access or malicious attacks, while still allowing the system to function effectively. This special issue aims to explore novel techniques and strategies that protect privacy in distributed control and optimization, showcasing how these approaches can be seamlessly integrated into real-world applications. The diverse contributions in this issue demonstrate the growing importance of privacy in the evolving field of distributed control systems. RNC7747 introduces a privacy-preserving method for coverage control in multi-agent systems with limited communication ranges, ensuring efficient operation without compromising agents' privacy. RNC7798 proposes a privacy-preserving consensus mechanism based on pulse-coupled oscillators, helping maintain privacy while agents coordinate in distributed networks. RNC7778 presents privacy preservation techniques for cloud-based cooperative LQG control systems, enabling privacy protection even when leveraging cloud resources. RNC8051 discusses achieving privacy-preserving average consensus in unreliable network environments, ensuring privacy despite communication challenges. RNC7929 introduces a novel approach to open privacy-preserving consensus via state decomposition, allowing agents to reach consensus while maintaining privacy. RNC7966 investigates leader-follower consensus in multi-agent systems with distributed event-triggered observations, ensuring privacy during agent coordination in real-time systems. RNC7789 discusses the tradeoff between privacy protection strength and control performance in control systems. RNC8029 proposes a new privacy mechanism for distributed average consensus on time-varying directed graphs. RNC7776 investigates event-triggered distributed optimization with differential privacy, reducing communication costs while safeguarding sensitive data. RNC7885 develops an event-triggered privacy-preserving optimization method for directed communication networks, emphasizing privacy during optimization tasks. RNC7791 focuses on robust optimization for virtual power plant scheduling under uncertainty, incorporating privacy preservation into the optimization process. RNC7730 explores prescribed-time distributed optimization with set constraints, offering time guarantees while maintaining privacy in dynamic optimization scenarios. RNC7926 provides a privacy-preserving algorithm for constrained resource allocation with communication delays, crucial for real-time decision-making in distributed systems. RNC7848 presents an adaptive event-triggered control strategy for cyber-physical systems under DoS and deception attacks, ensuring privacy and system security. RNC8068 addresses initial state privacy in nonlinear systems on Riemannian manifolds, offering methods to protect privacy in complex, high-dimensional control systems. RNC7906 focuses on dynamic consensus-based formation control for multi-robot systems, preserving privacy through output masking techniques while achieving coordination. RNC7688 introduces a periodic dynamic encoding mechanism for stealthy attack detection in distributed state estimation, focusing on security and privacy under attack scenarios. RNC70033 tackles robust consensus Kalman filtering for distributed state-saturated systems, incorporating dynamic-disturbed saturation levels and censored measurements to ensure privacy under uncertain conditions. The contributions in this special issue span a broad spectrum of topics related to privacy in distributed control and optimization. As distributed systems continue to evolve, these papers contribute to the ongoing conversation about balancing privacy protection with efficient system operation, highlighting the importance of privacy as a key consideration in the design and optimization of future distributed control systems. The authors declare no conflicts of interest.

Playing a fundamental role in mathematic optimization, linear programming (LP) problems have been widely encountered in various scientific disciplines and industrial applications. Although static LP problems have been investigated extensively and applied to abundant scientific fields through the last decades, researches concerning time-varying linear programming (TVLP) problem solving are in relatively small amount. In this paper, the TVLP problems are solved by a linear-variational-inequality based primal-dual neural network (LVI-PDNN), which is originally designed for static LP problem solving. Numerical examples and computer simulations further reveal that LVI-PDNN could approach the theoretical solution when solving TVLP problems subject to equality, inequality and bound constraints simultaneously.

Secure multiparty computation is a basic concept of growing interest in modern cryptography. It allows a set of mutually distrusting parties to perform a computation on their private information in such a way that as little as possible is revealed about each private input. The early results of multiparty computation have only theoretical signi??cance since they are not able to solve computationally complex functions in a reasonable amount of time. Nowadays, e??ciency of secure multiparty computation is an important topic of cryptographic research. As a case study we apply multiparty computation to solve the problem of secure linear programming. The results enable, for example in the context of the EU-FP7 project SecureSCM, collaborative supply chain management. Collaborative supply chain management is about the optimization of the supply and demand con??guration of a supply chain. In order to optimize the total bene??t of the entire chain, parties should collaborate by pooling their sensitive data. With the focus on e??ciency we design protocols that securely solve any linear program using the simplex algorithm. The simplex algorithm is well studied and there are many variants of the simplex algorithm providing a simple and e??cient solution to solving linear programs in practice. However, the cryptographic layer on top of any variant of the simplex algorithm imposes restrictions and new complexity measures. For example, hiding the number of iterations of the simplex algorithm has the consequence that the secure implementations have a worst case number of iterations. Then, since the simplex algorithm has exponentially many iterations in the worst case, the secure implementations have exponentially many iterations in all cases. To give a basis for understanding the restrictions, we review the basic theory behind the simplex algorithm and we provide a set of cryptographic building blocks used to implement secure protocols evaluating basic variants of the simplex algorithm. We show how to balance between privacy and e??ciency; some protocols reveal data about the internal state of the simplex algorithm, such as the number of iterations, in order to improve the expected running times. For the sake of simplicity and e??ciency, the protocols are based on Shamir's secret sharing scheme. We combine and use the results from the literature on secure random number generation, secure circuit evaluation, secure comparison, and secret indexing to construct e??cient building blocks for secure simplex. The solutions for secure linear programming in this thesis can be split into two categories. On the one hand, some protocols evaluate the classical variants of the simplex algorithm in which numbers are truncated, while the other protocols evaluate the variants of the simplex algorithms in which truncation is avoided. On the other hand, the protocols can be separated by the size of the tableaus. Theoretically there is no clear winner that has both the best security properties and the best performance.

It is a long-standing open question in combinatorial optimization whether the integrality gap of the subtour linear program relaxation (subtour LP) for the asymmetric traveling salesman problem (ATSP) is a constant. The study on the structure of this linear program is important and extensive. In this paper, we give a new and simpler LP relaxation for the ATSP. Our linear program consists of a single type of constraints that combine both the subtour elimination and the degree constraints in the traditional subtour LP. As a result, we obtain a much simpler relaxation. In particular, it is shown that the extreme solutions of our program have at most 2n − 2 non-zero variables, improving the bound 3n − 2, proved by Vempala and Yannakakis, for the ones obtained by the subtour LP. Nevertheless, the integrality gap of the new linear program is larger than that of the traditional subtour LP by at most a constant factor.

Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Differential privacy has emerged as a leading theoretical framework for privacy-preserving data gathering and analysis. It allows meaningful statistics to be collected for a population without revealing ``too much'' information about any individual member of the population. For software profiling, this machinery allows profiling data from many users of a deployed software system to be collected and analyzed in a privacy-preserving manner. Such a solution is appealing to many stakeholders, including software users, software developers, infrastructure providers, and government agencies.We propose an approach for differentially-private collection of frequency vectors from software executions. Frequency information is reported with the addition of random noise drawn from the Laplace distribution. A key observation behind the design of our scheme is that event frequencies are closely correlated due to the static code structure. Differential privacy protections must account for such relationships; otherwise, a seemingly-strong privacy guarantee is actually weaker than it appears. Motivated by this observation, we propose a novel and general differentially-private profiling scheme when correlations between frequencies can be expressed through linear inequalities. Using a linear programming formulation, we show how to determine the magnitude of random noise that should be added to achieve meaningful privacy protections under such linear constraints. Next, we develop an efficient instance of this general machinery for an important subclass of constraints. Instead of LP, our solution uses a reachability analysis of a constraint graph. As an exemplar, we employ this approach to implement differentially-private method frequency profiling for Android apps.Any differentially-private scheme has to balance two competing aspects: privacy and accuracy. Through an experimental study to characterize these trade-offs, we (1) show that our proposed randomization achieves much higher accuracy compared to related prior work, (2) demonstrate that high accuracy and high privacy protection can be achieved simultaneously, and (3) highlight the importance of linear constraints in the design of the randomization. These promising results provide evidence that our approach is a good candidate for privacy-preserving frequency profiling of deployed software.

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max cx, Ax = b, x ≥ 0, A g m × n, Vavasis and Ye developed a primal-dual interior point method using a g€layered least squares' (LLS) step, and showed that O(n3.5 log(χA+n)) iterations suffice to solve (LP) exactly, where χA is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure χA∗, defined as the minimum χAD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an O(m2 n2 + n3) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying χAD ≤ n(χ∗)3. This algorithm also allows us to approximate the value of χA up to a factor n (χ∗)2. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NP-hardness for approximating χA to within 2poly(rank(A)). The key insight for our algorithm is to work with ratios gi/gj of circuits of A - i.e., minimal linear dependencies Ag=0 - which allow us to approximate the value of χA∗ by a maximum geometric mean cycle computation in what we call the g€circuit ratio digraph' of A. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n2.5 lognlog(χA∗+n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/logn improvement on the iteration complexity bound of the original Vavasis-Ye algorithm.

We are witnessing the era of Internet of Things (IoT), where its applications such as smart cities and smart homes catch sensitive data gathered mostly by IoT surveillance cameras among other sensors or devices. Therefore, security and privacy protection is a key concern during transmitting such sensitive data across the IoT network to be processed and stored on Cloud. In this paper, we proposed a lightweight encryption model that complies with the limited resources of IoT devices in terms of process and memory. Also, the encryption model also provides a high level of security for the transmitted data through a constant change of the key used for encrypting of transmitted IoT data. In addition, the key size used to encrypt transmitted data in the proposed model is large enough which makes it hard to break by the attackers. The experimental results show outstanding results with an average of 150.5 ms of encryption time for a key size 80 bits where the key size is relatively large and with an average PSNR of 7.13 compared to other algorithms.

In multi-stakeholder systems such as healthcare, the Internet of Things, and supply chain management, data is often generated and shared. As a result, data owners aim to maintain control and privacy over their data, while data consumers need reliable methods to verify the data’s origins and creators. These conflicting interests require data governance systems to ensure data provenance, privacy protection, consent management, and selective disclosure. To address these challenges, this research proposed a decentralized data governance system that utilizes blockchain technology, proxy re-encryption (PRE), and Boneh, Boyen, and Shacham (BBS) signatures. The proposed system enables data owners to control, selectively share, and track their data through privacy protection, consent management, and selective disclosure mechanisms. It also allows data consumers to understand the data’s lineage through a blockchain-based provenance mechanism. The research focused on analyzing electronic prescription use cases that contain sensitive data and involve multiple stakeholders, such as patients, doctors, and pharmacists. The research was structured as a collection of published articles organized in the following sequence: problem formulation and developing smart contracts, implementing privacy and consent management through PRE, and applying BBS signatures for selective data sharing. The proof-of-concept implementation and evaluations were conducted using CosmWasm, Hyperledger Besu, Ethereum, pyUmbral PRE, and BBS signatures. The evaluations show that the proposed decentralized system is platform-agnostic, scalable, and capable of providing higher transparency, privacy, and trust, all with minimal overhead.

Linear programming remains a central subject of Mathematical Programming although already in 1947, Dantzig developed the Simplex method for solving Linear Programming problems. That method has been implemented in quite powerful codes which are very efficient for solving real world problems. With the growing interest in the complexity of algorithms, however, it turned out that the Simplex method is exponential in the worst case (Klee-Minty [1972]). Until now the existence of pivoting rules resulting in a polynomial variant of the Simplex Method is an open question. For most known rules conterexamples to polynomiality have been constructed. The worst case approach itself is due to much criticizm. The analysis of the average case is usually much harder, but seems to explain the typically efficient performance of the Simplex method much better. For a broad discussion of average case analysis of the Simplex method we refer the reader to Borgwardt [1984].

Collaborative filtering recommendation systems generate personalized recommendation results by analyzing and collaboratively processing a large numerous of user ratings or behavior data. The widespread use of recommendation systems in daily decision-making also brings potential risks of privacy leakage. Recent literature predominantly employs differential privacy to achieve privacy protection, however, many schemes struggle to balance user privacy and recommendation performance effectively. In this work, we present a practical privacy-preserving scheme for user-based collaborative filtering recommendation that utilizes fuzzy C-means clustering and Shapley value, FSPPCFs, aiming to enhance the recommendation performance while ensuring privacy protection. Specifically, (i) we have modified the traditional recommendation scheme by introducing a similarity balance factor integrated into the Pearson similarity algorithm, enhancing recommendation system performance; (ii) FSPPCFs first clusters the dataset through fuzzy C-means clustering and Shapley value, grouping users with similar interests and attributes into the same cluster, thereby providing more accurate data support for recommendations. Then, differential privacy is used to achieve the user’s personal privacy protection when selecting the neighbor set from the target cluster. Finally, it is theoretically proved that our scheme satisfies differential privacy. Experimental results illustrate that our scheme significantly outperforms existing methods.

This paper proposes an implementation of linear programming in tsunami horizontal evacuation routes decision making. A tsunami is a huge wave that travels with high speed towards a coastal area. Padang is the capital city of West Sumatera Province that is located in the coastal area as well as have a history of the tsunami disaster. The main problem is only a few minutes of the interval for the tsunami waves arrive at the coast after tsunami hazard warning. In the evacuation concept, the evacuees from the high-risk zone are moving to the shelter through a safe evacuation route. This research aims to determine the optimal evacuation routes with the constraints of a few minutes of the interval of evacuation time, the capacity of each shelter, and the number of evacuees. The evacuation time is an objective function that depends on the number of population, people density, road width, distance from the high-risk zone to the shelter, and velocity. The numerical simulation is done using linear programming with Matlab. The results show that all evacuees in each high-risk region are proportionally evacuated to available shelter areas within allocated evacuation time. The results of this research can be used for supporting tsunami evacuation plans and disaster resilience management by the authorities in Padang.