Statistical Physics of Hard Optimization Problems

Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the NP-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this thesis is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.Comment: PhD thesi

Path integrals on ultrametric spaces

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994.Includes bibliographical references (p. 51-52).by Alan Blair.Ph.D

'Vivre et mourir en la religion ancienne romaine et Catholique'. Catholic activism in South-west France, 1560-1570

This thesis determines the impact of Catholic activism in south-west France during the 1560s. Using manuscript and archival material, it examines the nascence, infrastructure and aims of the various ligues and associations of the period, and assesses their ability to secure hegemony by usurping local administrative and military apparatus. The thesis is divided into four parts: Part I studies Catholic activism at Bordeaux and the Bordelais. Chapters 1 and 2 establish the demography and structure of government there and evaluate antecedents to the ligues of the 1560s: namely the activities of the confraternity of Saint- Yves, the basoche and the syndicat of 1561. Chapter 3 examines the role of the Catholic nobility in the internecine struggles of the period, and Chapter 4 explores the post-1568 domination of military and bureaucratic offices at Bordeaux by a coalition of leading Catholic activists. Part II focuses on events at Agen and the Agenais. Chapter 5 details the establishment of an executive council of Catholic notables within the town, while Chapter 6 assesses the role of the local nobility in supporting this coalition government after 1563. Part III concentrates on affairs at Toulouse. Chapter 7 examines the birth of formal Catholic activism within the town, focussing on the enterprises of the confraternity, basochiens and militant court officials. Chapter 8 analyses the reinvigoration of Catholic endeavours at Toulouse after 1567, gauging the success of a policy to finance the Catholic war effort by sequestering Protestant goods, and appraising the somewhat anachronistic phenomenon of the croisade of 1568. Part IV considers the extent to which national and European dimensions influenced the form and nature of Catholic activism during the 1560s. Chapter 9 examines the impact of Catholic grandee patronage on the ligues of the south-west, and assesses the resurgence in confraternal activism after 1567. Chapter 10 examines the influence of Spain and the papacy on the evolution of the associations, and explores the level of reciprocity between ligueurs and European potentates. Chapter 11 details the ambitions of Catholic powers to remove the Calvinist queen of Navarre, Jeanne d' Albret, from power in Béarn and Navarre, and assesses the role played by the ligueurs in this enterprise

'Vivre et mourir en la religion ancienne romaine et Catholique' : Catholic activism in South-west France, 1560-1570

This thesis determines the impact of Catholic activism in south-west France during the 1560s. Using manuscript and archival material, it examines the nascence, infrastructure and aims of the various ligues and associations of the period, and assesses their ability to secure hegemony by usurping local administrative and military apparatus. The thesis is divided into four parts: Part I studies Catholic activism at Bordeaux and the Bordelais. Chapters 1 and 2 establish the demography and structure of government there and evaluate antecedents to the ligues of the 1560s: namely the activities of the confraternity of Saint- Yves, the basoche and the syndicat of 1561. Chapter 3 examines the role of the Catholic nobility in the internecine struggles of the period, and Chapter 4 explores the post-1568 domination of military and bureaucratic offices at Bordeaux by a coalition of leading Catholic activists. Part II focuses on events at Agen and the Agenais. Chapter 5 details the establishment of an executive council of Catholic notables within the town, while Chapter 6 assesses the role of the local nobility in supporting this coalition government after 1563. Part III concentrates on affairs at Toulouse. Chapter 7 examines the birth of formal Catholic activism within the town, focussing on the enterprises of the confraternity, basochiens and militant court officials. Chapter 8 analyses the reinvigoration of Catholic endeavours at Toulouse after 1567, gauging the success of a policy to finance the Catholic war effort by sequestering Protestant goods, and appraising the somewhat anachronistic phenomenon of the croisade of 1568. Part IV considers the extent to which national and European dimensions influenced the form and nature of Catholic activism during the 1560s. Chapter 9 examines the impact of Catholic grandee patronage on the ligues of the south-west, and assesses the resurgence in confraternal activism after 1567. Chapter 10 examines the influence of Spain and the papacy on the evolution of the associations, and explores the level of reciprocity between ligueurs and European potentates. Chapter 11 details the ambitions of Catholic powers to remove the Calvinist queen of Navarre, Jeanne d' Albret, from power in Béarn and Navarre, and assesses the role played by the ligueurs in this enterprise.EThOS - Electronic Theses Online ServiceArtsHumanities Research Council (Great Britain)GBUnited Kingdo

Case studies in quantum adiabatic optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 139-143).Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization problem is encoded in the ground state of a "problem Hamiltonian" Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian [mathematical formula] ... where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We present I A random ensemble of 3SAT instances which this algorithm does not solve efficiently. For these instances H(s) has a small eigenvalue gap at a value s* which approaches 1 as n - oc. II Theorems concerning the interpolating Hamiltonian when Hp is "scrambled" by conjugating with a random permutation matrix. III Results pertaining to phase transitions that occur as a function of the transverse field. IV A new quantum monte carlo method which can be used to compute ground state properties of such quantum systems. We discuss the implications of our results for the performance of quantum adiabatic optimization algorithms.by David Gosset.Ph.D