Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Influence of zeolite nanofillers on properties of polymeric materials

The present work deals with the preparation and study of modified polymeric materials with the replacement of carbon black by nanofillers on the basis of zeolite that is environmentally friendly. Natural zeolites from a group of aluminosilicate nanoporous materials have wide range of possibilities for applications that are environmentally friendly. Zeolites can be used in the role of fillers into the polymer materials too [1]. The given work deals with the preparation and study of modified polymeric materials with the substitution of carbon black by nanofillers on the basis of monoionic form - Ni(II) zeolite. The prepared monoinic forms – Ni(II) zeolite were characterized by the method of infrared spectroscopy. The vulcanization performance of prepared modified polymeric compounds and physical-mechanical properties of vulcanizates were measured and the efficiency of zeolite filler and carbon black filler was evaluated. The obtained values were compared with the values of commercially used polymer materials with the original composition

Linear-time limited automata

The time complexity of 1-limited automata is investigated from a descriptional complexity view point. Though the model recognizes regular languages only, it may use quadratic time in the input length. We show that, with a polynomial increase in size and preserving determinism, each 1-limited automaton can be transformed into an halting linear-time equivalent one. We also obtain polynomial transformations into related models, including weight-reducing Hennie machines, and we show exponential gaps for converse transformations in the deterministic case