Box-Inequalities for Quadratic Assignment Polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. They have turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started by Rijal (1995), Padberg and Rijal (1996), and Jünger and Kaibel (1996, 1997). They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope

Relativistic center-vortex dynamics of a confining area law

We offer a physicists' proof that center-vortex theory requires the area in the Wilson-loop area law to involve an extremal area. Area-law dynamics is determined by integrating over Wilson loops only, not over surface fluctuations for a fixed loop. Fluctuations leading to to perimeter-law corrections come from loop fluctuations as well as integration over finite -thickness center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD in many respects, except that there are both quartic and quadratic terms in the action. One major result is that at large angular momentum \ell in d=3+1 the center-vortex extremal-area picture yields a linear Regge trajectory with Regge slope--string tension product \alpha'(0)K_F of 1/(2\pi), which is the canonical Veneziano/string value. In a curious effect traceable to retardation, the quark kinetic terms in the action vanish relative to area-law terms in the large-\ell limit, in which light-quark masses \sim K_F^{1/2} are negligible. This corresponds to string-theoretic expectations, even though we emphasize that the extremal-area law is not a string theory quantum-mechanically. We show how some quantum trajectory fluctuations as well as non-leading classical terms for finite mass yield corrections scaling with \ell^{-1/2}. We compare to old semiclassical calculations of relativistic q\bar{q} bound states at large \ell, which also yield asymptotically-linear Regge trajectories, finding agreement with a naive string picture (classically, not quantum-mechanically) and disagreement with an effective-propagator model. We show that contour forms of the area law can be expressed in terms of Abelian gauge potentials, and relate this to old work of Comtet.Comment: 20 pages RevTeX4 with 3 .eps figure

Reconstruction of the early Universe as a convex optimization problem

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour clarifications in the text, additional material on the history of the Monge-Ampere equation, improved description of the auction algorithm, updated bibliography. Version 3: several misprints correcte