Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
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1 Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs Sariel Har-Peled Kent Quanrud University of Illinois at Urbana-Champaign October 19, 2015
2 1. Pregame Gameplan - definition of problems - a hardness result 2. Low-density objects and graphs - basic properties - overview of results 3. Polynomial expansion - basic properties - overview of results halftime! 4. Two proofs - independent set - dominating set
3 f Fat objects
4 f Fat objects
5 Fat objects f b
6 Fat objects f b
7 Fat objects vol(b f) vol(b) = Ω(1) f b
8 Hitting set Input: Set of points P, fat objects F Output: The smallest cardinality subset of P that pierces every object in F.
9 Hitting set Input: Set of points P, fat objects F Output: The smallest cardinality subset of P that pierces every object in F.
10 Set cover Input: Set of points P, fat objects F Output: Find the smallest cardinality subset of F that covers every point in P.
11 Set cover Input: Set of points P, fat objects F Output: Find the smallest cardinality subset of F that covers every point in P.
12 Disks and Pseudo-disks
13 Disks and Pseudo-disks Set Cover NP-Hard PTAS Feder and Greene 1988 Mustafa, Raman and Ray 2014 Hitting Set NP-Hard Feder and Greene 1988 PTAS Mustafa and Ray 2010
14 Fat objects Set cover O(1) approx. for fat triangles of same size Clarkson and Varadarajan 2007
15 Fat objects Set cover O(log OPT) for fat objects in R 2 Aronov, de Berg, Ezra and Sharir 2014
16 Fat objects Hitting set O(log log OPT) for fat triangles of similar size Aronov, Ezra and Sharir 2010
17 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
18 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
19 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
20 Fat, nearly equilateral triangles Set cover APX-Hard 1 Hitting set APX-Hard
21 Fat, nearly equilateral triangles Set cover APX-Hard 1 Hitting set APX-Hard
22 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
23 Fat, nearly equilateral triangles Set cover APX-Hard 1 Hitting set APX-Hard
24 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
25 Fat, nearly equilateral triangles Set cover APX-Hard Hitting set APX-Hard
26 Density F has density ρ if any ball intersects ρ objects with larger diameter. van der Stappen, Overmars, de Berg, Vleugels, 1998
27 Density F has density ρ if any ball intersects ρ objects with larger diameter. van der Stappen, Overmars, de Berg, Vleugels, 1998
28 Density F has density ρ if any ball intersects ρ objects with larger diameter. van der Stappen, Overmars, de Berg, Vleugels, 1998
29 Density F has density ρ if any ball intersects ρ objects with larger diameter. van der Stappen, Overmars, de Berg, Vleugels, 1998
30 Density F has low density if ρ = O(1). van der Stappen, Overmars, de Berg, Vleugels, 1998
31 Intersection graphs F induces an intersection graph G F with objects as vertices and edges representing overlap.
32 Intersection graphs F induces an intersection graph G F with objects as vertices and edges representing overlap.
33 Intersection graphs F induces an intersection graph G F with objects as vertices and edges representing overlap.
34 Intersection graphs A graph has low-density if induced by low-density objects.
35 Examples of low density Interior disjoint disks have O(1) density.
36 Examples of low density Planar graphs have O(1)-density via Circle Packing Theorem. Koebe, Andreev, Thurston
37 Examples of low density Planar graphs have O(1)-density via Circle Packing Theorem. Koebe, Andreev, Thurston
38 Examples of low density Fat convex objects in R d with depth k have density O(k2 d ).
39 Examples of low density r Fat convex objects in R d with depth k have density O(k2 d ).
40 Examples of low density r Fat convex objects in R d with depth k have density O(k2 d ).
41 Examples of low density r Fat convex objects in R d with depth k have density O(k2 d ).
42 Examples of low density r r Fat convex objects in R d with depth k have density O(k2 d ).
43 Examples of low density r r Fat convex objects in R d with depth k have density O(k2 d ).
44 Examples of low density r r r Fat convex objects in R d with depth k have density O(k2 d ).
45 Additivity red density = ρ 1
46 Additivity blue density = ρ 2
47 Additivity red density = ρ 1 blue density = ρ 2
48 Additivity + red density = ρ 1 blue density = ρ 2
49 Additivity + red density = ρ 1 blue density = ρ 2 total density ρ 1 + ρ 2
50 Degeneracy If F has density ρ, then the smallest object intersects at most ρ 1 other objects.
51 Degeneracy If F has density ρ, then the smallest object intersects at most ρ 1 other objects.
52 Degeneracy If F has density ρ, then the smallest object intersects at most ρ 1 other objects.
53 Separators For k F, compute a sphere S that Strictly contains at least k o(k) objects and at most k objects. Intersects O(ρ + ρ 1/d k 1 1/d ) objects. 3r R c r Miller, Teng, Thurston, and Vavasis, 1997; Smith and Wormald, 1998; Chan 2003
54 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
55 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
56 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
57 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
58 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
59 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
60 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
61 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
62 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
63 Graph minors A minor of G is a graph H obtained by contracting edges, deleting edges, and deleting vertices.
64 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
65 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
66 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
67 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
68 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
69 Minors of objects G is a minor of F if it can be obtained by deleting objects and taking unions of overlapping of objects.
70 Large clique minors
71 Large clique minors
72 Large clique minors
73 Large clique minors
74 Large clique minors
75 Large clique minors
76 Large clique minors
77 Large clique minors
78 Large clique minors
79 Large clique minors
80 Large clique minors
81 Large clique minors
82 Large clique minors
83 Shallow minors A vertex in H corresponds to a connected cluster of vertices in G.
84 Shallow minors H is a t-shallow minor if each cluster induces a graph of radius t.
85 Shallow minors H is a 1-shallow minor if each cluster induces a graph of radius 1.
86 Shallow minors H is a 2-shallow minor if each cluster induces a graph of radius 2.
87 Shallow minors of objects Each object in the minor corresponds to a cluster of objects in F.
88 Shallow minors of objects An object minor is a t-shallow minor if the intersection graph of each cluster has radius t.
89 Shallow minors of objects An object minor is a 1-shallow minor if the intersection graph of each cluster has radius 1.
90 Shallow minors of objects An object minor is a 2-shallow minor if the intersection graph of each cluster has radius 2.
91 Shallow minors of objects An object minor is a 3-shallow minor if the intersection graph of each cluster has radius 3.
92 Shallow minors of low-density objects A t-shallow minor of objects with density ρ has density O(tO(d)ρ)
93 Shallow minors of low-density objects A t-shallow minor of objects with density ρ has density O(tO(d)ρ)
94 Shallow minors of low-density objects A t-shallow minor of objects with density ρ has density O(tO(d)ρ)
95 Shallow minors of low-density objects A t-shallow minor of objects with density ρ has density O(tO(d)ρ)
96 Shallow minors of low-density objects A t-shallow minor of objects with density ρ has density O(tO(d)ρ)
97 Recap: low-density graphs are... + red density = ρ 1 blue density = ρ 2 total density ρ 1 + ρ 2 (a) additive 3r (b) degenerate (hence sparse) R c (c) separable r (d) kind of closed under shallow minors
98 Main result: low-density ρ = O(1): PTAS for hitting set, set cover, subset dominating set ρ = polylog(n): QPTAS for same problems. No PTAS under ETH.
99 Main result: low-density density ρ O(1) polylog(n) unbounded hardness NP-Hard No PTAS APX-Hard algo PTAS QPTAS
100 Main result: fat triangles depth density ρ O(1) polylog(n) unbounded hardness NP-Hard No PTAS APX-Hard algo PTAS QPTAS
101
102 Shallow edge density The r-shallow density of a graph is the max edge density over all r-shallow minors. aka greatest reduced average density Nešetřil and Ossona de Mendez, 2008
103 Sparsity is not enough Hide a clique by splitting the edges
104 Sparsity is not enough Hide a clique by splitting the edges
105 Expansion The expansion of a graph is the r-shallow density as a function of r. e.g. constant expansion, polynomial expansion, exponential expansion Nešetřil and Ossona de Mendez, 2008
106 Examples of expansion Planar graphs have constant expansion (Euler s formula) Minor-closed classes have constant expansion
107 Sparsity is not enough Constant degree expanders have exponential expansion Wikipedia
108 Low density polynomial expansion Graphs with density ρ have polynomial expansion f (r) = O(ρrd)
109 Low density polynomial expansion Graphs with density ρ have polynomial expansion f (r) = O(ρrd)
110 expansion constant polynomial exponential examples planar graphs, minor-closed families low-density graphs expander graphs
111 Recap: low-density graphs are... + red density = ρ 1 blue density = ρ 2 total density ρ 1 + ρ 2 (a) additive 3r (b) degenerate (hence sparse) R c (c) separable r (d) kind of closed under shallow minors
112 Lexical product G K h The lexical product G K h blows up each vertex in G with the clique K h.
113 Lexical product G K h V (G K 4 ) = V (G) V (K 4 ) The lexical product G K h blows up each vertex in G with the clique K h.
114 Lexical product G K h V (G K 4 ) = V (G) V (K 4 ) E(G K 4 ) = {((a 1, b 1 ), (a 2, b 2 )) a 1 = a 2 ))} The lexical product G K h blows up each vertex in G with the clique K h.
115 Lexical product G K h V (G K 4 ) = V (G) V (K 4 ) E(G K 4 ) = {((a 1, b 1 ), (a 2, b 2 )) a 1 = a 2 ))} {((a 1, b 1 ), (a 2, b 2 )) (a 1, a 2 ) E(G)} The lexical product G K h blows up each vertex in G with the clique K h.
116 Lexical product G K h V (G K 4 ) = V (G) V (K 4 ) E(G K 4 ) = {((a 1, b 1 ), (a 2, b 2 )) a 1 = a 2 ))} {((a 1, b 1 ), (a 2, b 2 )) (a 1, a 2 ) E(G)} If G has polynomial expansion, then G K h has polynomial expansion.
117 Small separators Graphs with subexponential expansion have sublinear separators. Nešetřil and Ossona de Mendez (2008)
118 Small separators Graphs with subexponential expansion have sublinear separators. Nešetřil and Ossona de Mendez (2008) Why?
119 Small separators Graphs with subexponential expansion have sublinear separators. Nešetřil and Ossona de Mendez (2008) Why? 1. No K h as an l-shallow minor implies a separator of size O(n/l + 4lh 2 log n). Plotkin, Rao, and Smith (1994)
120 Small separators Graphs with subexponential expansion have sublinear separators. Nešetřil and Ossona de Mendez (2008) Why? 1. No K h as an l-shallow minor implies a separator of size O(n/l + 4lh 2 log n). Plotkin, Rao, and Smith (1994) 2. Small expansion => small clique minors as a function of depth
121 Shallow minors of polynomial expansion Shallow minors of graphs with polynomial expansion have polynomial expansion. (If H is an r 1 -shallow minor of G, then an r 2 -shallow minor of H is an (r 1 r 2 )-shallow minor of G, and poly(r 1 r 2 ) = poly(r 2 ).)
122 Recap: polynomial expansion graphs are... V (G K4) = V (G) V (K4) E(G K4) = {((a1, b1), (a2, b2)) a1 = a2))} {((a1, b1), (a2, b2)) (a1, a2) E(G)} (a) closed under lexical product via separator for excluded shallow minors (c) separable FREE (b) degenerate FREE (d) kind of closed under shallow minors
123 Main result: polynomial expansion Graph G with polynomial expansion PTAS for (subset) dominating set Extensions: multiple demands, reach, connected dominating set, vertex cover.
124 PTAS for independent set
125 Recap: low-density graphs are... + red density = ρ 1 blue density = ρ 2 total density ρ 1 + ρ 2 (a) additive 3r (b) degenerate (hence sparse) R c (c) separable r (d) kind of closed under shallow minors
126 Balanced separators Input: G = (V, E) with n vertices. Output: Partition V = X S Y s.t. (a) X, Y.99n. (b) S n.99. (c) No edges run between X and Y.
127 Balanced separators Input: G = (V, E) with n vertices. Output: Partition V = X S Y s.t. (a) X, Y.99n. (b) S n.99. (c) No edges run between X and Y.
128 Balanced separators Input: G = (V, E) with n vertices. Output: Partition V = X S Y s.t. (a) X, Y.99n. (b) S n.99. (c) No edges run between X and Y.
129 Well-separated divisions Input: G = (V, E) w/ n vertices, ɛ (0, 1) Output: Cover V = C 1 C 2 C m s.t. (a) C i = poly(1/ɛ) for all i (b) No edges between C i \ C j and C j \ C i (c) i C i (1 + ɛ)n
130 Well-separated divisions Input: G = (V, E) w/ n vertices, ɛ (0, 1) Output: Cover V = C 1 C 2 C m s.t. (a) C i = poly(1/ɛ) for all i (b) No edges between C i \ C j and C j \ C i (c) i C i (1 + ɛ)n
131 Well-separated divisions Input: G = (V, E) w/ n vertices, ɛ (0, 1) Output: Cover V = C 1 C 2 C m s.t. (a) C i = poly(1/ɛ) for all i (b) No edges between C i \ C j and C j \ C i (c) i C i (1 + ɛ)n
132 Well-separated divisions Input: G = (V, E) w/ n vertices, ɛ (0, 1) Output: Cover V = C 1 C 2 C m s.t. (a) C i = poly(1/ɛ) for all i (b) No edges between C i \ C j and C j \ C i (c) i C i (1 + ɛ)n
133 Separators divisions Frederickson (1987)
134 Local search local-search(g = (V, E), ɛ) L, λ poly(1/ɛ) while there exists S V s.t. (a) S λ (b) L S is an independent set (c) L S > L do L L S end while return L
135 Independent set: proof setup O: Optimal solution L: λ-locally optimal sol n for λ = poly(1/ɛ)
136 Independent set: proof setup O: Optimal solution L: λ-locally optimal sol n for λ = poly(1/ɛ) O L has low density (by additivity)
137 Independent set: proof setup O: Optimal solution L: λ-locally optimal sol n for λ = poly(1/ɛ) O L has low density (by additivity) {C 1,..., C m }: w.s.-division of O L
138 Independent set: proof setup O: Optimal solution L: λ-locally optimal sol n for λ = poly(1/ɛ) O L has low density (by additivity) {C 1,..., C m }: w.s.-division of O L ( ) B i = C i j i C j, b i = B i O i = (O C i ) \ B i, o i = O i L i = (L C i ), l = L i,
139 Independent set: proof setup O: Optimal solution L: λ-locally optimal sol n for λ = poly(1/ɛ) O L has low density (by additivity) {C 1,..., C m }: w.s.-division of O L ( ) B i = C i j i C j, b i = B i O i = (O C i ) \ B i, o i = O i L i = (L C i ), l = L i,
140 PTAS for dominating set
141 Recap: low-density graphs are... + red density = ρ 1 blue density = ρ 2 total density ρ 1 + ρ 2 (a) additive 3r (b) degenerate (hence sparse) R c (c) separable r (d) kind of closed under shallow minors
142 Local search local-search(g = (V, E), ɛ) L V, λ poly(1/ɛ) while there exists S V s.t. (a) S λ (b) L S is a dominating set (c) L S < L do L L S end while return L
143 Flowers Glue an object in the dominating set with the neighboring objects it dominates.
144 Flowers Petal Head Glue an object in the dominating set with the neighboring objects it dominates.
145 Flowers Glue an object in the dominating set with the neighboring objects it dominates.
146 Flowers Glue an object in the dominating set with the neighboring objects it dominates.
147 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L
148 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L Õ L has low density
149 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L Õ L has low density { } C1,..., Cm : w.s.-division of Õ L
150 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L Õ L has low density { } C1,..., Cm : w.s.-division of Õ L {C 1,..., C m }: centers of C i for each i
151 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L Õ L has low density { } C1,..., Cm : w.s.-division of Õ L {C 1,..., C m }: centers of C i for each i ( ) B i = C i j i C j O i = O C i, o i = O i L i = L C i, l i = L i
152 Dominating set: proof setup O = optimal solution, Õ = flowers of O L = locally-optimal sol n, L = flowers of L Õ L has low density { } C1,..., Cm : w.s.-division of Õ L {C 1,..., C m }: centers of C i for each i ( ) B i = C i j i C j O i = O C i, o i = O i L i = L C i, l i = L i
153 thanks!
154 References I G. N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput., 16(6): , J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion ii. algorithmic aspects. European J. Combin., 29(3): , 2008.
155 References II S. Plotkin, S. Rao, and W.D. Smith. Shallow excluded minors and improved graph decompositions. In Proc. 5th ACM-SIAM Sympos. Discrete Algs. (SODA), pages , 1994.
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