622,002 research outputs found
Heuristic Algorithms for the Maximum Colorful Subtree Problem
In metabolomics, small molecules are structurally elucidated using tandem mass spectrometry (MS/MS); this computational task can be formulated as the Maximum Colorful Subtree problem, which is NP-hard. Unfortunately, data from a single metabolite requires us to solve hundreds or thousands of instances of this problem - and in a single Liquid Chromatography MS/MS run, hundreds or thousands of metabolites are measured.
Here, we comprehensively evaluate the performance of several heuristic algorithms for the problem. Unfortunately, as is often the case in bioinformatics, the structure of the (chemically) true solution is not known to us; therefore we can only evaluate against the optimal solution of an instance. Evaluating the quality of a heuristic based on scores can be misleading: Even a slightly suboptimal solution can be structurally very different from the optimal solution, but it is the structure of a solution and not its score that is relevant for the downstream analysis. To this end, we propose a different evaluation setup: Given a set of candidate instances of which exactly one is known to be correct, the heuristic in question solves each instance to the best of its ability, producing a score for each instance, which is then used to rank the instances. We then evaluate whether the correct instance is ranked highly by the heuristic.
We find that one particular heuristic consistently ranks the correct instance in a top position. We also find that the scores of the best heuristic solutions are very close to the optimal score; in contrast, the structure of the solutions can deviate significantly from the optimal structures. Integrating the heuristic allowed us to speed up computations in practice by a factor of 100-fold
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
Construction of near-optimal vertex clique covering for real-world networks
We propose a method based on combining a constructive and a bounding heuristic to solve the vertex clique covering problem (CCP), where the aim is to partition the vertices of a graph into the smallest number of classes, which induce cliques. Searching for the solution to CCP is highly motivated by analysis of social and other real-world networks, applications in graph mining, as well as by the fact that CCP is one of the classical NP-hard problems. Combining the construction and the bounding heuristic helped us not only to find high-quality clique coverings but also to determine that in the domain of real-world networks, many of the obtained solutions are optimal, while the rest of them are near-optimal. In addition, the method has a polynomial time complexity and shows much promise for its practical use. Experimental results are presented for a fairly representative benchmark of real-world data. Our test graphs include extracts of web-based social networks, including some very large ones, several well-known graphs from network science, as well as coappearance networks of literary works' characters from the DIMACS graph coloring benchmark. We also present results for synthetic pseudorandom graphs structured according to the Erdös-Renyi model and Leighton's model
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Phase transition and landscape statistics of the number partitioning problem
The phase transition in the number partitioning problem (NPP), i.e., the
transition from a region in the space of control parameters in which almost all
instances have many solutions to a region in which almost all instances have no
solution, is investigated by examining the energy landscape of this classic
optimization problem. This is achieved by coding the information about the
minimum energy paths connecting pairs of minima into a tree structure, termed a
barrier tree, the leaves and internal nodes of which represent, respectively,
the minima and the lowest energy saddles connecting those minima. Here we apply
several measures of shape (balance and symmetry) as well as of branch lengths
(barrier heights) to the barrier trees that result from the landscape of the
NPP, aiming at identifying traces of the easy/hard transition. We find that it
is not possible to tell the easy regime from the hard one by visual inspection
of the trees or by measuring the barrier heights. Only the {\it difficulty}
measure, given by the maximum value of the ratio between the barrier height and
the energy surplus of local minima, succeeded in detecting traces of the phase
transition in the tree. In adddition, we show that the barrier trees associated
with the NPP are very similar to random trees, contrasting dramatically with
trees associated with the spin-glass and random energy models. We also
examine critically a recent conjecture on the equivalence between the NPP and a
truncated random energy model
The computational difficulty of finding MPS ground states
We determine the computational difficulty of finding ground states of
one-dimensional (1D) Hamiltonians which are known to be Matrix Product States
(MPS). To this end, we construct a class of 1D frustration free Hamiltonians
with unique MPS ground states and a polynomial gap above, for which finding the
ground state is at least as hard as factoring. By lifting the requirement of a
unique ground state, we obtain a class for which finding the ground state
solves an NP-complete problem. Therefore, for these Hamiltonians it is not even
possible to certify that the ground state has been found. Our results thus
imply that in order to prove convergence of variational methods over MPS, as
the Density Matrix Renormalization Group, one has to put more requirements than
just MPS ground states and a polynomial spectral gap.Comment: 5 pages. v2: accepted version, Journal-Ref adde
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