Statistical Mechanics of maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation

Trajectory and Policy Aware Sender Anonymity in Location Based Services

We consider Location-based Service (LBS) settings, where a LBS provider logs the requests sent by mobile device users over a period of time and later wants to publish/share these logs. Log sharing can be extremely valuable for advertising, data mining research and network management, but it poses a serious threat to the privacy of LBS users. Sender anonymity solutions prevent a malicious attacker from inferring the interests of LBS users by associating them with their service requests after gaining access to the anonymized logs. With the fast-increasing adoption of smartphones and the concern that historic user trajectories are becoming more accessible, it becomes necessary for any sender anonymity solution to protect against attackers that are trajectory-aware (i.e. have access to historic user trajectories) as well as policy-aware (i.e they know the log anonymization policy). We call such attackers TP-aware. This paper introduces a first privacy guarantee against TP-aware attackers, called TP-aware sender k-anonymity. It turns out that there are many possible TP-aware anonymizations for the same LBS log, each with a different utility to the consumer of the anonymized log. The problem of finding the optimal TP-aware anonymization is investigated. We show that trajectory-awareness renders the problem computationally harder than the trajectory-unaware variants found in the literature (NP-complete in the size of the log, versus PTIME). We describe a PTIME l-approximation algorithm for trajectories of length l and empirically show that it scales to large LBS logs (up to 2 million users)

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

2 CSPs All Are Approximable Within a Constant Differential Factor

International audienceOnly a few facts are known regarding the approximability of optimization CSPs with respect to the differential approximation measure, which compares the gain of a given solution over the worst solution value to the instance diameter. Notably, the question whether is approximable within any constant factor is open in case when or. Using a family of combinatorial designs we introduce for our purpose, we show that, given any three constant integers , and , reduces to with an expansion of on the approximation guarantee. When , this implies together with the result of Nesterov as regards [1] that for all constant integers , is approximable within factor

2 CSPs All Are Approximable Within a Constant Differential Factor

International audienceOnly a few facts are known regarding the approximability of optimization CSPs with respect to the differential approximation measure , which compares the gain of a given solution over the worst solution value to the instance diameter. Notably, the question whether k CSP−q is approximable within any constant factor is open in case when q ≥ 3 or k ≥ 4. Given three integers k ≥ 2, p ≥ k and q > p, we analyse the expansion of a precise reduction from k CSP−q to k CSP−p. We introduce a family of combinatorial designs from which we deduce a lower bound of 1/(q − p + k/2) k for this expansion. When p = k = 2, this implies together with the result of Nesterov as regards 2 CSP−2 [?] that for all constant integers q ≥ 2, 2 CSP−q is approximable within factor (2 − π/2)/(q − 1) 2

Quantum Algorithms for Scientific Computing and Approximate Optimization

Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations