10,452 research outputs found
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
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
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