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Tree Decompositions and Fixed Parameter Tractability of k -Path

Tree Decompositions and Fixed Parameter Tractability of k -Path. Tree Decomposition. Every edge is in a red ellipse (the bags). The bags are connected in a tree . The bags an original vertex is part of are connected . Example : Independent Set. The root. The k-Path Problem.

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Tree Decompositions and Fixed Parameter Tractability of k -Path

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  1. TreeDecompositions and FixedParameter Tractability ofk-Path

  2. TreeDecomposition • Everyedge is in a red ellipse (the bags). • The bags are connected in a tree. • The bags an original vertex is part of are connected.

  3. Example: Independent Set The root

  4. The k-Path Problem

  5. The k-Path Problem Directed Weighted w1 w3 w2 w4 w5 w6 w7 w9 w8 w10 w11

  6. Complexity • Recursivelybuildpaths: O(nk) • Can there be algorithms with runtimes on the form f(k)nO(1)? … and if so, how small canf(k) be?

  7. Intuition for FPT Considerregulargraphs of degree d: Eitherd>k: There must be k-path. Or d<=k: Wecan list all potential k-paths in ndk<=nkktime.

  8. Bodlaender 2kk!nO(1) Depth First Search B6 <k? B1 B2 B4 B3 B5

  9. Bodlaender 2kk!nO(1) B1 B2 B3 B4 B6 B5

  10. Color codingAlon, Yuster, and Zwick 1 3 5 2 4 6 Prob[rainbowk-path] >= k!/kk ~ e-k

  11. Color codingAlon, Yuster, and Zwick • Dynamicprogramming over color subsets. • LetD(X,v,k) be Trueiffthere is a path of length k ending in vertex v whosevertices are colored as X.

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