On NP-Hardness of the Paired de Bruijn Sound Cycle Problem
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On NP-Hardness of thePaired de Bruijn Sound Cycle
Problem
Evgeny Kapun Fedor Tsarev
Genome Assembly Algorithms Laboratory
University ITMO, St. Petersburg, Russia
September 2, 2013
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Genome assembly models
I Shortest common superstring –
NP-hard.
I Shortest common superwalk in a de
Bruijn graph – NP-hard.
I Superwalk in a de Bruijn graph with
known edge multiplicities – NP-hard.
I Path in a paired de Bruijn graph?
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Paired de Bruijn graph
TT
CT
TA
TT
AG
TC
GC
CT
CA
TT
AT
TC
TTACTT
TAGTTC
AGCTCT
GCACTT
CATTTC
ATTTCT
TATTTC
CAGTTC
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Sound path
Sound path: strings match with shift d = 6.
TAGCTCACCCGTTGGT
ACCCGTTGGTAATTGCSound cycle: cyclic strings match with shift
d = 6.
TGATAAGTAGGCTAAG
GTAGGCTAAGTGATAA
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Paired de Bruijn Sound CycleProblem
Given a paired de Bruijn graph G and an
integer d (represented in unary coding), find
if G has a sound cycle with respect to shift d .
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Paired de Bruijn Covering SoundCycle Problem
Given a paired de Bruijn graph G and an
integer d (represented in unary coding), find
if G has a covering sound cycle with respect
to shift d .
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Parameters
I |Σ|: size of the alphabet.
I k : length of vertex labels.
I |V |, |E |: size of the graph (bounded in
terms of |Σ| and k).
I d : shift distance.
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Simple cases
I |Σ| = 1: at most one vertex, at most
one edge.
I k = 0: at most one vertex, at most |Σ|2edges, reduces to the problem of
computing strongly connected
components.
I d is fixed: find a cycle in a graph of
|V ||Σ|d states.
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Interesting case: k = 1
With fixed k = 1, the problem is NP-hard.
Proof outline:
1. Reduce Hamiltonian Cycle Problem,
which is NP-hard, to an intermediate
problem.
2. Reduce that problem to Paired de Bruijn
(Covering) Sound Cycle Problem.
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The intermediate problem
Given an undirected graph,
I if it contains a hamiltonian cycle, output
1.
I if it doesn’t contain hamiltonian paths,
output 0.
I otherwise, invoke undefined behavior.
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Solution, step 1
G1 G2
a1
a2
a3
b1
b2
b3
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Properties of a graph with ahamiltonian cycle
Such graph contains hamiltonian paths
I ending at any vertex.
I passing through any edge.
I for any edge {i , j} and vertex k 6= i , j ,
passing through {i , j} such that j is
between i and k on the path.
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Solution, step 2
V2n+2s1s2 V1
s2s3 V2
. . . . . . s1. . . . . . . . . . . . s2. . . . . .
. . . . . . s2. . . . . . . . . . . . s3. . . . . .
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Solution, step 2
In V1, V2n+2:
i
j
−→
i
j
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Solution, step 2
In V3. . .V2n+1:
i
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How to make a covering cycle
Pass along
the loop
multiple
times,
covering
more and
more edges
with each
iteration.
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Interesting case: |Σ| = 2
With fixed |Σ| = 2, the problem is NP-hard.
The proof is done by reduction from the case
k = 1. The characters are replaced with
binary sequences, and a transformation is
done to avoid undesired overlaps.
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Fix both k and |Σ|If both k and |Σ| are fixed, the number of
different de Bruijn graphs is finite. The only
argument which can take infinitely many
values is d , and it is an integer represented in
unary coding. As a result, the number of
valid problem instances having any fixed
length is bounded. So, the language defined
by the problem is sparse. Therefore, the
problem is not NP-hard unless P=NP.
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Results
Paired de Bruijn (Covering) Sound Cycle
Problem is
I NP-hard for any fixed k ≥ 1 (can be
reduced from k = 1).
I NP-hard for any fixed |Σ| ≥ 2 (trivially
reduced from |Σ| = 2).
I NP-hard in the general case.
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Results
Paired de Bruijn (Covering) Sound Cycle
Problem is
I Not NP-hard if both k and |Σ| are fixed,
unless P=NP.
I Solvable in polynomial time if k = 0,
|Σ| = 1, or d is fixed.
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Thank you!