1,160 research outputs found
Syntactic Separation of Subset Satisfiability Problems
Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems
A nonmonotone GRASP
A greedy randomized adaptive search procedure (GRASP) is an itera-
tive multistart metaheuristic for difficult combinatorial optimization problems. Each
GRASP iteration consists of two phases: a construction phase, in which a feasible
solution is produced, and a local search phase, in which a local optimum in the
neighborhood of the constructed solution is sought. Repeated applications of the con-
struction procedure yields different starting solutions for the local search and the
best overall solution is kept as the result. The GRASP local search applies iterative
improvement until a locally optimal solution is found. During this phase, starting from
the current solution an improving neighbor solution is accepted and considered as the
new current solution. In this paper, we propose a variant of the GRASP framework that
uses a new “nonmonotone” strategy to explore the neighborhood of the current solu-
tion. We formally state the convergence of the nonmonotone local search to a locally
optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP
on three classical hard combinatorial optimization problems: the maximum cut prob-
lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and
the quadratic assignment problem (QAP)
Using synchronous Boolean networks to model several phenomena of collective behavior
In this paper, we propose an approach for modeling and analysis of a number
of phenomena of collective behavior. By collectives we mean multi-agent systems
that transition from one state to another at discrete moments of time. The
behavior of a member of a collective (agent) is called conforming if the
opinion of this agent at current time moment conforms to the opinion of some
other agents at the previous time moment. We presume that at each moment of
time every agent makes a decision by choosing from the set {0,1} (where
1-decision corresponds to action and 0-decision corresponds to inaction). In
our approach we model collective behavior with synchronous Boolean networks. We
presume that in a network there can be agents that act at every moment of time.
Such agents are called instigators. Also there can be agents that never act.
Such agents are called loyalists. Agents that are neither instigators nor
loyalists are called simple agents. We study two combinatorial problems. The
first problem is to find a disposition of instigators that in several time
moments transforms a network from a state where a majority of simple agents are
inactive to a state with a majority of active agents. The second problem is to
find a disposition of loyalists that returns the network to a state with a
majority of inactive agents. Similar problems are studied for networks in which
simple agents demonstrate the contrary to conforming behavior that we call
anticonforming. We obtained several theoretical results regarding the behavior
of collectives of agents with conforming or anticonforming behavior. In
computational experiments we solved the described problems for randomly
generated networks with several hundred vertices. We reduced corresponding
combinatorial problems to the Boolean satisfiability problem (SAT) and used
modern SAT solvers to solve the instances obtained
Comparing Beliefs, Surveys and Random Walks
Survey propagation is a powerful technique from statistical physics that has
been applied to solve the 3-SAT problem both in principle and in practice. We
give, using only probability arguments, a common derivation of survey
propagation, belief propagation and several interesting hybrid methods. We then
present numerical experiments which use WSAT (a widely used random-walk based
SAT solver) to quantify the complexity of the 3-SAT formulae as a function of
their parameters, both as randomly generated and after simplification, guided
by survey propagation. Some properties of WSAT which have not previously been
reported make it an ideal tool for this purpose -- its mean cost is
proportional to the number of variables in the formula (at a fixed ratio of
clauses to variables) in the easy-SAT regime and slightly beyond, and its
behavior in the hard-SAT regime appears to reflect the underlying structure of
the solution space that has been predicted by replica symmetry-breaking
arguments. An analysis of the tradeoffs between the various methods of search
for satisfying assignments shows WSAT to be far more powerful that has been
appreciated, and suggests some interesting new directions for practical
algorithm development.Comment: 8 pages, 5 figure
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