Proof Of Optimality Research Articles

Revisiting simultaneous consensus with crash failures

This paper addresses the “consensus with simultaneous decision” problem in a synchronous system prone to t process crashes. This problem requires that all the processes that do not crash decide on the same value (consensus), and that all decisions are made during the very same round (simultaneity). So, there is a double agreement, one on the decided value (data agreement) and one on the decision round (time agreement). This problem was first defined by Dwork and Moses who analyzed it and solved it using an analysis of the evolution of states of knowledge in a system with crash failures. The current paper presents a simple algorithm that optimally solves simultaneous consensus. Optimality means in this case that the simultaneous decision is taken in each and every run as soon as any protocol decides, given the same failure pattern and initial value. The design principle of this algorithm is simplicity, a first-class criterion. A new optimality proof is given that is stated in purely combinatorial terms.

Lower bounds for strictly fundamental cycle bases in grid graphs

AbstractConsider the following problem: compute a spanning tree such that the sum of the lengths of its induced fundamental circuits is as small as possible. We motivate why planar square grid graphs are very relevant instances for this problem. In particular, other contributions already showed that the identification of strong lower bounds is highly challenging. Asymptotically, for a graph on n vertices, Alon et al. [SIAM J Comput 24(1995), 78–100] obtained a lower bound of Ω(n log n). We raise the n log n coefficient by a factor of 325. Concerning optimality proofs, the largest grid for which provably optimum solutions were known is 6 × 6, and it was obtained by massive MIP computing power. Here, we present a combinatorial optimality proof even for the 8 × 8 grid. These two results are complemented by new combinatorial lower bounds for the dimensions in which earlier empirical computations were performed, i.e., for up to 10,000 vertices. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Limitations of restricted branching in clause learning

The techniques for making decisions, that is, branching, play a central role in complete methods for solving structured instances of constraint satisfaction problems (CSPs). In this work we consider branching heuristics in the context of propositional satisfiability (SAT), where CSPs are expressed as propositional formulas. In practice, there are cases when SAT solvers based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) benefit from limiting the set of variables the solver is allowed to branch on to so called input variables which provide a strong unit propagation backdoor set to any SAT instance. Theoretically, however, restricting branching to input variables implies a super-polynomial increase in the length of the optimal proofs for DPLL (without clause learning), and thus input-restricted DPLL cannot polynomially simulate DPLL. In this paper we settle the case of DPLL with clause learning. Surprisingly, even with unlimited restarts, input-restricted clause learning DPLL cannot simulate DPLL (even without clause learning). The opposite also holds, and hence DPLL and input-restricted clause learning DPLL are polynomially incomparable. Additionally, we analyze the effect of input-restricted branching on clause learning solvers in practice with various structured real-world benchmarks.

Size-Space Tradeoffs for Resolution

We investigate tradeoffs of various basic complexity measures such as size, space, and width. We show examples of formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results have implications to the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof relies on a novel connection of the variable space of a proof to the black-white pebbling measure of an underlying graph.

On the expected diameter of an L2-bounded martingale

Dedicated to the memory of Gideon Schwarz (1933-2007) It is shown that the ratio between the expected diameter of an L2-bounded martingale and the standard deviation of its last term cannot exceed p 3. Moreover, a one-parameter family of stopping times on standard Brownian Motion is exhibited, for which the p 3 upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost ct. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be bounded by p 2 times the standard deviation of the last term of the martingale. These results complement the Dubins & Schwarz respective bounds 1 and p 2 for the ratios between the expected maximum and maximal absolute value of the martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.

A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies

This paper reviews the most relevant literature on efficient models and methods for packing circular objects/items into Euclidean plane regions where the objects/items and regions are either two- or three-dimensional. These packing problems are NP hard optimization problems with a wide variety of applications. They have been tackled using various approaches-based algorithms ranging from computer-aided optimality proofs, to branch-and-bound procedures, to constructive approaches, to multi-start nonconvex minimization, to billiard simulation, to multiphase heuristics, and metaheuristics.

Distributed Uplink Power Control for Optimal SIR Assignment in Cellular Data Networks

This paper solves the joint power control and SIR assignment problem through distributed algorithms in the uplink of multi-cellular wireless networks. The 1993 Foschini-Miljanic distributed power control can attain a given fixed and feasible SIR target. In data networks, however, SIR needs to be jointly optimized with transmit powers in wireless data networks. In the vast research literature since the mid-1990s, solutions to this joint optimization problem are either distributed but suboptimal, or optimal but centralized. For convex formulations of this problem, we report a distributed and optimal algorithm. The main issue that has been the research bottleneck is the complicated, coupled constraint set, and we resolve it through a re-parametrization via the left Perron Frobenius eigenvectors, followed by development of a locally computable ascent direction. A key step is a new characterization of the feasible SIR region in terms of the loads on the base stations, and an indication of the potential interference from mobile stations, which we term spillage. Based on this load-spillage characterization, we first develop a distributed algorithm that can achieve any Pareto-optimal SIR assignment, then a distributed algorithm that picks out a particular Pareto-optimal SIR assignment and the associated powers through utility maximization. Extensions to power-constrained and interference-constrained cases are carried out. The algorithms are theoretically sound and practically implementable: we present convergence and optimality proofs as well as simulations using 3GPP network and path loss models.

Soft car sequencing with colors: Lower bounds and optimality proofs

This paper is a study of the car sequencing problem, when feature spacing constraints are soft and colors of vehicles are taken into account. Both pseudo-polynomial algorithms and lower bounds are presented for parts of the problem or family of instances. With this set of lower bounds, we establish the optimality (up to the first non-trivial criteria) of 54% of best known solutions for the benchmark used for the Roadef Challenge 2005. We also prove that the optimal penalty for a single ratio constraint N/ P can be computed in O( P) and that determining the feasibility of a car sequencing instance limited to a pair of simple ratio constraints can be achieved by dynamic programming. Finally, we propose a solving algorithm exploiting these results within a local search approach. To achieve this goal, a new meta-heuristic ( star relinking) is introduced, designed for the optimization of an aggregation of criteria, when the optimization of each single criterion is a polynomial problem.

An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “ no more than l c cars are allowed to require a component c in any subsequence of m c consecutive cars”. The first method is an exact one based on integer linear programming (ILP). The second approach is hybrid: it uses ILP techniques within a general variable neighborhood search (VNS) framework for examining large neighborhoods. We tested the two methods on benchmark instances provided by CSP LIB and the automobile manufacturer RENAULT for the ROADEF Challenge 2005. These tests reveal that our approaches are competitive to previous reported algorithms. For the CSP LIB instances we were able to shorten the required computation time for reaching and proving optimality. Furthermore, we were able to obtain tight bounds on some of the ROADEF instances. For two of these instances the proposed ILP-method could provide new optimality proofs for already known solutions. For the VNS, the individual contributions of the used neighborhoods are also experimentally analyzed. Results highlight the significant impact of each structure. In particular the large ones examined using ILP techniques enhance the overall performance significantly, so that the hybrid approach clearly outperforms variants including only commonly defined neighborhoods.

Asymptotic optimality of maximum pressure policies in stochastic processing networks

We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks.

Another brief recursion excursion to Hanoi

We propose another simple Towers of Hanoi variant, a hybrid between classical Hanoi and linear Hanoi, in which the rules governing movement depend on ring color. An optimal algorithm is presented. The problem and its heavily recursive solution are not difficult; perhaps one of its more interesting facets is that the optimality proof uses simultaneous induction on four statements. This paper can be viewed as similar in purpose and spirit to the author's previous work [1]; the goal here is again to present a fun example of potential usefulness in teaching discrete mathematics and data structures courses.

On an infinite family of solvable Hanoi graphs

The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed graph G with two distinguished vertices, S and D , and allowing moves only along arcs of this graph. An optimal solution for such a graph G is an algorithm that completes the task of moving a tower of any given number of disks from S to D in a minimal number of disk moves. In this article we present an algorithm which solves the problem for two infinite families of graphs, and prove its optimality. To the best of our knowledge, this is the first optimality proof for an infinite family of graphs. Furthermore, we present a unified algorithm that solves the problem for a wider family of graphs and conjecture its optimality.

A continuous-time search model with job switch and jumps

We study a new search problem in continuous time. In the traditional approach, the basic formulation is to maximize the expected (discounted) return obtained by taking a job, net of search cost incurred until the job is taken. Implicitly assumed in the traditional modeling is that the agent has no job at all during the search period or her decision on a new job is independent of the job situation she is currently engaged in. In contrast, we incorporate the fact that the agent has a job currently and starts searching a new job. Hence we can handle more realistic situation of the search problem. We provide optimal decision rules as to both quitting the current job and taking a new job as well as explicit solutions and proofs of optimality. Further, we extend to a situation where the agent’s current job satisfaction may be affected by sudden downward jumps (e.g., de-motivating events), where we also find an explicit solution; it is rather a rare case that one finds explicit solutions in control problems using a jump diffusion.

LIMITATIONS OF SOME SIMPLE ADIABATIC QUANTUM ALGORITHMS

Let H(t) = (1 - t/T)H0 + (t/T)H1, t ∈ [0,T], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.

Optimality of Feedback Control Strategies for Qubit Purification

Recently two papers [K. Jacobs, Phys. Rev. A 67, 030301(R) (2003); H. M. Wiseman and J. F. Ralph, New J. Physics 8, 90 (2006)] have derived a number of control strategies for rapid purification of qubits, optimized with respect to various goals. In the former paper the proof of optimality was not mathematically rigorous, while the latter gave only heuristic arguments for optimality. In this paper we provide rigorous proofs of optimality in all cases, by applying simple concepts from optimal control theory, including Bellman equations and verification theorems.

Adaptive clustering for scalable key management in dynamic group communications

Group key management has received a particular attention, these last years, in both academic and industry research communities. This is due to the economical relevance of group-based applications, such as video on demand, video-conferencing, collaborative work, and so forth. The dynamic aspect of group applications due to free membership joins and leaves in addition to members' mobility, makes security further challenging in the context of group communications. In this paper, we propose a novel key management approach based on adaptive clustering of encryption areas in order to tackle both scalability and dynamism issues in secure group communications. We demonstrate the pertinence of our solution through a proof of optimality with respect to the stated optimisation problem, and simulation results that illustrate when and how our solution outperforms other approaches proposed in the literature.

Provably Secure Multisignatures in Formal Security Model and Their Optimality

We first model the formal security model of multisignature scheme following that of group signature scheme. Second, we prove that the following three probabilistic multisignature schemes based on a trapdoor permutation have tight security; PFDH (probabilistic full domain hash) based multisignature scheme (PFDH-MSS), PSS (probabilistic signature scheme) based multisignature scheme (PSS-MSS), and short signature PSS based multisignature scheme (S-PSS-MSS). Third, we give an optimal proof (general result) for multisignature schemes, which derives the lower bound for the length of random salt. We also estimate the upper bound for the length in each scheme and derive the optimal length of a random salt. Two of the schemes are promising in terms of security tightness and optimal signature length. In appendix, we describe a multisignature scheme using the claw-free permutation and discuss its security.

Another short proof of optimality for the MIN cache replacement algorithm

When a new item is written into a cache, an old item must be evicted; the latter can be chosen by the offline MIN algorithm, which was first proven to be optimal by Mattson et al. [R.L. Mattson, J. Gecsei, D.R. Slutz, I.L. Traiger, Evaluation techniques for storage hierarchies, IBM Systems J. 9 (2) (1970) 78–117]. Recently a new, short proof was given by van Roy [B. van Roy, A short proof of optimality for the MIN cache replacement algorithm, Inform. Process. Lett. 102 (2007) 72–73]; here we give another short proof using what we call an amortized simulation.

Editor's puzzle

Solutions should be sent to the editor at conitzer@cs.duke.edu with subject header SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to ...

MANAGEMENT OF VIRTUAL SOCIETIES IN MAS ENVIRONMENT

Environmental engineering is emerging in multi-agent system research. There are numerous recent state-of-the-art publications defining short and long term challenges. One of these challenges addressed by this paper is incorporating knowledge system into multi-agent system environment. Experience management of agents' virtual society as application knowledge system is introduced by this paper. This application enhances managerial properties of the environment. The paper also presents several application dependent problems in experience management, such as experience item description, ontology specification and similarity measurement with proof of optimality.