2,013 research outputs found
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
Solution space structure of random constraint satisfaction problems with growing domains
In this paper we study the solution space structure of model RB, a standard
prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using
rigorous the first and the second moment method, we show that in the solvable
phase close to the satisfiability transition, solutions are clustered into
exponential number of well-separated clusters, with each cluster contains
sub-exponential number of solutions. As a consequence, the system has a
clustering (dynamical) transition but no condensation transition. This picture
of phase diagram is different from other classic random CSPs with fixed domain
size, such as random K-Satisfiability (K-SAT) and graph coloring problems,
where condensation transition exists and is distinct from satisfiability
transition. Our result verifies the non-rigorous results obtained using cavity
method from spin glass theory, and sheds light on the structures of solution
spaces of problems with a large number of states.Comment: 8 pages, 1 figure
Satisfiability, sequence niches, and molecular codes in cellular signaling
Biological information processing as implemented by regulatory and signaling
networks in living cells requires sufficient specificity of molecular
interaction to distinguish signals from one another, but much of regulation and
signaling involves somewhat fuzzy and promiscuous recognition of molecular
sequences and structures, which can leave systems vulnerable to crosstalk. This
paper examines a simple computational model of protein-protein interactions
which reveals both a sharp onset of crosstalk and a fragmentation of the
neutral network of viable solutions as more proteins compete for regions of
sequence space, revealing intrinsic limits to reliable signaling in the face of
promiscuity. These results suggest connections to both phase transitions in
constraint satisfaction problems and coding theory bounds on the size of
communication codes
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
This paper first analyzes the resolution complexity of two random CSP models
(i.e. Model RB/RD) for which we can establish the existence of phase
transitions and identify the threshold points exactly. By encoding CSPs into
CNF formulas, it is proved that almost all instances of Model RB/RD have no
tree-like resolution proofs of less than exponential size. Thus, we not only
introduce new families of CNF formulas hard for resolution, which is a central
task of Proof-Complexity theory, but also propose models with both many hard
instances and exact phase transitions. Then, the implications of such models
are addressed. It is shown both theoretically and experimentally that an
application of Model RB/RD might be in the generation of hard satisfiable
instances, which is not only of practical importance but also related to some
open problems in cryptography such as generating one-way functions.
Subsequently, a further theoretical support for the generation method is shown
by establishing exponential lower bounds on the complexity of solving random
satisfiable and forced satisfiable instances of RB/RD near the threshold.
Finally, conclusions are presented, as well as a detailed comparison of Model
RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively,
exhibit three different kinds of phase transition behavior in NP-complete
problems.Comment: 19 pages, corrected mistakes in Theorems 5 and
Exhaustive enumeration unveils clustering and freezing in random 3-SAT
We study geometrical properties of the complete set of solutions of the
random 3-satisfiability problem. We show that even for moderate system sizes
the number of clusters corresponds surprisingly well with the theoretic
asymptotic prediction. We locate the freezing transition in the space of
solutions which has been conjectured to be relevant in explaining the onset of
computational hardness in random constraint satisfaction problems.Comment: 4 pages, 3 figure
- …