8,466 research outputs found
A constructive proof of the general Lovasz Local Lemma
The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove
the existence of combinatorial objects meeting a prescribed collection of
criteria. In his breakthrough paper [Bec91], Beck demonstrated that a
constructive variant can be given under certain more restrictive conditions.
Simplifications of his procedure and relaxations of its restrictions were
subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06,
Sri08, Mos08]. In [Mos09], a constructive proof was presented that works under
negligible restrictions, formulated in terms of the Bounded Occurrence
Satisfiability problem. In the present paper, we reformulate and improve upon
these findings so as to directly apply to almost all known applications of the
general Local Lemma.Comment: 8 page
Efficient algorithms for three-dimensional axial and planar random assignment problems
Beautiful formulas are known for the expected cost of random two-dimensional
assignment problems, but in higher dimensions even the scaling is not known. In
three dimensions and above, the problem has natural "Axial" and "Planar"
versions, both of which are NP-hard. For 3-dimensional Axial random assignment
instances of size , the cost scales as , and a main result of
the present paper is a linear-time algorithm that, with high probability, finds
a solution of cost . For 3-dimensional Planar assignment, the
lower bound is , and we give a new efficient matching-based
algorithm that with high probability returns a solution with cost
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated
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