A fortification game (FG) is a three-level, two-player Stackelberg game, also known as defender-attacker-defender game, in which at the uppermost level, the defender selects some assets to be protected from potential malicious attacks. At the middle level, the attacker solves an interdiction game by depreciating unprotected assets, i.e., reducing the values of such assets for the defender, while at the innermost level the defender solves a recourse problem over the surviving or partially damaged assets. Fortification games have applications in various important areas, such as military operations, design of survivable networks, protection of facilities, or power grid protection. In this work, we present an exact solution algorithm for FGs, in which the recourse problems correspond to (possibly NP-hard) combinatorial optimization problems. The algorithm is based on a new generic mixed-integer linear programming reformulation in the natural space of fortification variables. Our new model makes use of fortification cuts that measure the contribution of a given fortification strategy to the objective function value. These cuts are generated on-the-fly by solving separation problems, which correspond to (modified) middle-level interdiction games. We design a branch-and-cut-based solution algorithm based on fortification cuts, their lifted versions, and other speed-up techniques. We present a computational study using the knapsack fortification game and the shortest path fortification game. For the latter one, we include a comparison with a state-of-the-art solution method from the literature. Our algorithm outperforms this method and allows us to solve previously unsolved instances to optimality.
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We study the problem of group testing with non-identical, independent priors. So far, the pooling strategies that have been proposed in the literature take the following approach: a hand-crafted test design along with a decoding strategy is proposed, and guarantees are provided on how many tests are sufficient in order to identify all infections in a population. In this paper, we take a different, yet perhaps more practical, approach: we fix the decoder and the number of tests, and we ask, given these, what is the best test design one could use? We explore this question for the Definite Non-Defectives (DND) decoder. We formulate a (non-convex) optimization problem, where the objective function is the expected number of errors for a particular design. We find approximate solutions via gradient descent, which we further optimize with informed initialization. We illustrate through simulations that our method can achieve significant performance improvement over traditional approaches.
In a round-robin tournament, if the final position of a team is already secured independently of the outcomes of the remaining games, it might play with little enthusiasm. This is detrimental to attendance and can inspire collusion and match-fixing. We demonstrate that tie-breaking rules might affect the occurrence of such a situation. Its probability is quantified via simulations for the four groups of the 2022/23 UEFA Nations League A, each of them being a home-away round-robin with four teams and 12 matches. The competitiveness of the last two games can be promoted by preferring goal difference over head-to-head results: the priority of goal difference reduces the chance of a fixed position in the group ranking by at least two and usually five percentage points. Our findings provide important lessons on how to design ranking systems.
The fractional knapsack problem is one of the classical problems in combinatorial optimization, which is well understood in the offline setting. However, the corresponding online setting has been handled only briefly in the theoretical computer science literature so far, although it appears in several applications. Even the previously best known guarantee for the competitive ratio was worse than the best known for the integral problem in the popular random order model. We show that there is an algorithm for the online fractional knapsack problem that admits a competitive ratio of 4.39. Our result significantly improves over the previously best known competitive ratio of 9.37 and surpasses the current best 6.65-competitive algorithm for the integral case. Moreover, our algorithm is deterministic in contrast to the randomized algorithms achieving the results mentioned above.
Unmanned aerial vehicles (UAVs) are envisioned to be extensively employed for assisting wireless communications in Internet of Things (IoT) applications. On the other hand, terahertz (THz) enabled intelligent reflecting surface (IRS) is expected to be one of the core enabling technologies for forthcoming beyond-5G wireless communications that promise a broad range of data-demand applications. In this paper, we propose a UAV-mounted IRS (UIRS) communication system over THz bands for confidential data dissemination from an access point (AP) towards multiple ground user equipments (UEs) in IoT networks. Specifically, the AP intends to send data to the scheduled UE, while unscheduled UEs may pose potential adversaries. To protect information messages and the privacy of the scheduled UE, we aim to devise an energy-efficient multi-UAV covert communication scheme, where the UIRS is for reliable data transmissions, and an extra UAV is utilized as a cooperative jammer generating artificial noise (AN) to degrade unscheduled UEs detection. We then formulate a novel minimum average energy efficiency (mAEE) optimization problem, targetting to improve the covert throughput and reduce UAVs' propulsion energy consumption subject to the covertness requirement, which is determined analytically. Since the optimization problem is non-convex, we tackle it via the block successive convex approximation (BSCA) approach to iteratively solve a sequence of approximated convex sub-problems, designing the binary user scheduling, AP's power allocation, maximum AN jamming power, IRS beamforming, and both UAVs' trajectory planning. Finally, we present a low-complex overall algorithm for system performance enhancement with complexity and convergence analysis. Numerical results are provided to verify our analysis and demonstrate significant outperformance of our design over other existing benchmark schemes.
We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear partial differential equations that characterizes Hopf bifurcation points. The flexibility and robustness of the method allows us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency with respect to a parameter of the system or the shape of the domain on which solutions are defined. Numerical applications are presented in systems arising from biology and fluid dynamics, such as the FitzHugh-Nagumo model, Ginzburg-Landau equation, Rayleigh-B\'enard convection problem, and Navier-Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high interest.
This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O(F (m + n) + mn + M ) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher's well-known method for the determination of the K best integer flows at the first sub-optimal flow results in an algorithm with a running time of O(F m(n log n + m) + M ) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O(Kn3 + M ). Besides, we give lower and upper bounds for the number of all optimal integer and feasible integer solutions. Our bounds are based on the fact that any optimal solution can be obtained by an initial optimal tree solution plus a conical combination of incidence vectors of all induced cycles with bounded coefficients.
Tensor optimization is crucial to massive machine learning and signal processing tasks. In this paper, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a nonconvex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of the best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.
Finding optimal paths in connected graphs requires determining the smallest total cost for traveling along the graph's edges. This problem can be solved by several classical algorithms where, usually, costs are predefined for all edges. Conventional planning methods can, thus, normally not be used when wanting to change costs in an adaptive way following the requirements of some task. Here we show that one can define a neural network representation of path finding problems by transforming cost values into synaptic weights, which allows for online weight adaptation using network learning mechanisms. When starting with an initial activity value of one, activity propagation in this network will lead to solutions, which are identical to those found by the Bellman Ford algorithm. The neural network has the same algorithmic complexity as Bellman Ford and, in addition, we can show that network learning mechanisms (such as Hebbian learning) can adapt the weights in the network augmenting the resulting paths according to some task at hand. We demonstrate this by learning to navigate in an environment with obstacles as well as by learning to follow certain sequences of path nodes. Hence, the here-presented novel algorithm may open up a different regime of applications where path-augmentation (by learning) is directly coupled with path finding in a natural way.
We study the problem of estimating the diagonal of an implicitly given matrix $A$. For such a matrix we have access to an oracle that allows us to evaluate the matrix vector product $Av$. For random variable $v$ drawn from an appropriate distribution, this may be used to return an estimate of the diagonal of the matrix $A$. Whilst results exist for probabilistic guarantees relating to the error of estimates of the trace of $A$, no such results have yet been derived for the diagonal. We analyse the number of queries $s$ required to guarantee that with probability at least $1-\delta$ the estimates of the relative error of the diagonal entries is at most $\varepsilon$. We extend this analysis to the 2-norm of the difference between the estimate and the diagonal of $A$. We prove, discuss and experiment with bounds on the number of queries $s$ required to guarantee a probabilistic bound on the estimates of the diagonal by employing Rademacher and Gaussian random variables. Two sufficient upper bounds on the minimum number of query vectors are proved, extending the work of Avron and Toledo [JACM 58(2)8, 2011], and later work of Roosta-Khorasani and Ascher [FoCM 15, 1187-1212, 2015]. We find that, generally, there is little difference between the two, with convergence going as $O(\log(1/\delta)/\varepsilon^2)$ for individual diagonal elements. However for small $s$, we find that the Rademacher estimator is superior. These results allow us to then extend the ideas of Meyer, Musco, Musco and Woodruff [SOSA, 142-155, 2021], suggesting algorithm Diag++, to speed up the convergence of diagonal estimation from $O(1/\varepsilon^2)$ to $O(1/\varepsilon)$ and make it robust to the spectrum of any positive semi-definite matrix $A$.
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.
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