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# On NP-Hardness of the Paired de Bruijn Sound Cycle Problem

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Talk at WABI-2013

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### On NP-Hardness of the Paired de Bruijn Sound Cycle Problem

1. 1. On NP-Hardness of the Paired de Bruijn Sound Cycle Problem Evgeny Kapun Fedor Tsarev Genome Assembly Algorithms Laboratory University ITMO, St. Petersburg, Russia September 2, 2013
2. 2. Genome assembly models Shortest common superstring – NP-hard. Shortest common superwalk in a de Bruijn graph – NP-hard. Superwalk in a de Bruijn graph with known edge multiplicities – NP-hard. Path in a paired de Bruijn graph?
3. 3. Paired de Bruijn graph TT CT TA TT AG TC GC CT CA TT AT TC TTA CTT TAG TTC AGC TCT GCA CTT CAT TTC ATT TCT TAT TTC CAG TTC
4. 4. Sound path Sound path: strings match with shift d = 6. TAGCTCACCCGTTGGT ACCCGTTGGTAATTGC Sound cycle: cyclic strings match with shift d = 6. TGATAAGTAGGCTAAG GTAGGCTAAGTGATAA
5. 5. Paired de Bruijn Sound Cycle Problem 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.
6. 6. Paired de Bruijn Covering Sound Cycle 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.
7. 7. Parameters |Σ|: size of the alphabet. k: length of vertex labels. |V |, |E|: size of the graph (bounded in terms of |Σ| and k). d: shift distance.
8. 8. Simple cases |Σ| = 1: at most one vertex, at most one edge. k = 0: at most one vertex, at most |Σ|2 edges, reduces to the problem of computing strongly connected components. d is ﬁxed: ﬁnd a cycle in a graph of |V ||Σ|d states.
9. 9. 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.
10. 10. The intermediate problem Given an undirected graph, if it contains a hamiltonian cycle, output 1. if it doesn’t contain hamiltonian paths, output 0. otherwise, invoke undeﬁned behavior.
11. 11. Solution, step 1 G1 G2 a1 a2 a3 b1 b2 b3
12. 12. Properties of a graph with a hamiltonian cycle Such graph contains hamiltonian paths ending at any vertex. passing through any edge. for any edge {i, j} and vertex k = i, j, passing through {i, j} such that j is between i and k on the path.
13. 13. Solution, step 2 V2n+2 s1 s2 V1 s2 s3 V2 . . . . . . s1. . . . . . . . . . . . s2. . . . . . . . . . . . s2. . . . . . . . . . . . s3. . . . . .
14. 14. Solution, step 2 In V1, V2n+2: i j −→ i j
15. 15. Solution, step 2 In V3. . . V2n+1: i
16. 16. How to make a covering cycle Pass along the loop multiple times, covering more and more edges with each iteration.
17. 17. 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.
18. 18. 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.
19. 19. Results Paired de Bruijn (Covering) Sound Cycle Problem is NP-hard for any ﬁxed k ≥ 1 (can be reduced from k = 1). NP-hard for any ﬁxed |Σ| ≥ 2 (trivially reduced from |Σ| = 2). NP-hard in the general case.
20. 20. Results Paired de Bruijn (Covering) Sound Cycle Problem is Not NP-hard if both k and |Σ| are ﬁxed, unless P=NP. Solvable in polynomial time if k = 0, |Σ| = 1, or d is ﬁxed.
21. 21. Thank you!