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Hypertree decomposition Hypertree decomposition 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 8 9 6 14 15 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h1(1,2,3) h6(7,9,10) 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) Hypertee decomposition of width 3 Hypertee decomposition of width 3 Hypertree decomposition Hypertree decomposition 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) Hypertee decomposition of width 3 Hypertee decomposition of width 3
Hypertree decomposition Hypertree decomposition 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) Hypertee decomposition of width 3 Hypertee decomposition of width 3 Hypertree decomposition Positive Results on Hypertree Decompositions 12 h9 h14 18 11 h8 h1 1 2 h15 h10 h2 h3 13 19 4 h6 h7 h11 17 7 5 h4 16 14 15 8 9 6 h5 h12 h13 10 3 h10(12,13,19), h12(14,15,18), h14(16,17,18) h9(11,12,18), h15(1,17,19) h2(1,4,5,6), h3(3,4,7,8) h4(5,7), h5(6,8,9) h6(7,9,10) h1(1,2,3) Hypertee decomposition of width 3 • For each query Q, hw(Q) qw(Q) • In some cases, hw(Q) < qw(Q) • For fixed k, deciding whether hw(Q) k is in polynomial time (LOGCFL) • Computing hypertree decompositions is feasible in polynomial time (for fixed k). But: FP-intractable wrt k: W[2]-hard.
Page 1 and 2: Three Problems: CQs and Structural
Page 3 and 4: Boolean Conjunctive Queries The pro
Page 5 and 6: Combinatorial Crossword Puzzle All
Page 7 and 8: ExampleConcluded Q1: p(X,Y) r(X,Z)
Page 9 and 10: Example Continued Q1: p(X,Y) r(X,Y)
Page 11 and 12: Cores endomorphism h: {Y b} B = {
Page 13 and 14: CQminimization Ans(X,Y,Z) R(X 2 ,Y
Page 15 and 16: Acyclic CSP Acyclic queries or CSPs
Page 17 and 18: GYO’ algorithm- removingears Algo
Page 19 and 20: d: 3 8 3 7 5 7 6 7 d(Y,P) r(Y,Z,U)
Page 21 and 22: s: 3 8 9 9 3 8 8 3 8 3 8 4 3 8 3 8
Page 23 and 24: … d: 3 8 3 7 5 7 6 7 d(Y,P) r(Y
Page 25 and 26: p(x,y) q q(x,z) … ABCQ is in LO
Page 27 and 28: NearlyAcyclicGraphs y Example: a cy
Page 29 and 30: Playing the game Playing the game q
Page 31 and 32: Beyond treewidth Bounded Degree of
Page 33 and 34: Hypertree Decompositions Grouping a
Page 35 and 36: Hypergraph and the tree decompositi
Page 37: QUESTION: Hypertree Decomposition =
Page 41 and 42: Relationship GHW vs. HW: Observatio

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