54,088 research outputs found
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
Novel Algorithms for LDD Motif Search
Background: Motifs are crucial patterns that have numerous applications including the identification of transcription factors and their binding sites, composite regulatory patterns, similarity between families of proteins, etc. Several motif models have been proposed in the literature. The (l,d)-motif model is one of these that has been studied widely. However, this model will sometimes report too many spurious motifs than expected. We interpret a motif as a biologically significant entity that is evolutionarily preserved within some distance. It may be highly improbable that the motif undergoes the same number of changes in each of the species. To address this issue, in this paper, we introduce a new model which is more general than (l,d)-motif model. This model is called (l,d1,d2)-motif model (LDDMS) and is NP-hard as well. We present three elegant as well as efficient algorithms to solve the LDDMS problem, i.e., LDDMS1, LDDMS2 and LDDMS3. They are all exact algorithms. Results: We did both theoretical analyses and empirical tests on these algorithms. Theoretical analyses demonstrate that our algorithms have less computational cost than the pattern driven approach. Empirical results on both simulated datasets and real datasets show that each of the three algorithms has some advantages on some (l,d1,d2) instances. Conclusions: We proposed LDDMS model which is more practically relevant. We also proposed three exact efficient algorithms to solve the problem. Besides, our algorithms can be nicely parallelized. We believe that the idea in this new model can also be extended to other motif search problems such as Edit-distance-based Motif Search (EMS) and Simple Motif Search (SMS)
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances
The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth
quantum circuit to produce a parameter dependent state. For a given
combinatorial optimization problem instance, the quantum expectation of the
associated cost function is the parameter dependent objective function of the
QAOA. We demonstrate that if the parameters are fixed and the instance comes
from a reasonable distribution then the objective function value is
concentrated in the sense that typical instances have (nearly) the same value
of the objective function. This applies not just for optimal parameters as the
whole landscape is instance independent. We can prove this is true for low
depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our
results generalize beyond this example. We support the arguments with numerical
examples that show remarkable concentration. For higher depth circuits the
numerics also show concentration and we argue for this using the Law of Large
Numbers. We also observe by simulation that if we find parameters which result
in good performance at say 10 bits these same parameters result in good
performance at say 24 bits. These findings suggest ways to run the QAOA that
reduce or eliminate the use of the outer loop optimization and may allow us to
find good solutions with fewer calls to the quantum computer.Comment: 16 pages, 1 figur
Large neighborhood search for the most strings with few bad columns problem
In this work, we consider the following NP-hard combinatorial optimization problem from computational biology. Given a set of input strings of equal length, the goal is to identify a maximum cardinality subset of strings that differ maximally in a pre-defined number of positions. First of all, we introduce an integer linear programming model for this problem. Second, two variants of a rather simple greedy strategy are proposed. Finally, a large neighborhood search algorithm is presented. A comprehensive experimental comparison among the proposed techniques shows, first, that larger neighborhood search generally outperforms both greedy strategies. Second, while large neighborhood search shows to be competitive with the stand-alone application of CPLEX for small- and medium-sized problem instances, it outperforms CPLEX in the context of larger instances.Peer ReviewedPostprint (author's final draft
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