Algorithmic Regularity Lemmas and Applications

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1 Algorithmic Regularity Lemmas and Applications László Miklós Lovász Massachusetts Institute of Technology Proving and Using Pseudorandomness Simons Institute for the Theory of Computing Joint work with Jacob Fox and Yufei Zhao March 8, 2017

2 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

3 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

4 Szemerédi s Regularity Lemma Szemerédi s regularity lemma Roughly speaking, in any graph, the vertices can be partitioned into a bounded number of parts, such that the graph is random-like between almost all pairs of parts.

5 Szemerédi s Regularity Lemma Szemerédi s regularity lemma Roughly speaking, in any graph, the vertices can be partitioned into a bounded number of parts, such that the graph is random-like between almost all pairs of parts.

6 Szemerédi s Regularity Lemma Szemerédi s regularity lemma Roughly speaking, in any graph, the vertices can be partitioned into a bounded number of parts, such that the graph is random-like between almost all pairs of parts. Very important tool in graph theory

7 Szemerédi s Regularity Lemma Szemerédi s regularity lemma Roughly speaking, in any graph, the vertices can be partitioned into a bounded number of parts, such that the graph is random-like between almost all pairs of parts. Very important tool in graph theory Gives a rough structural result for all graphs

8 Regularity of Sets Let X and Y be two sets of vertices in a graph G.

9 Regularity of Sets Let X and Y be two sets of vertices in a graph G. e(x, Y ): number of pairs of vertices in X Y that have an edge between them.

10 Regularity of Sets Let X and Y be two sets of vertices in a graph G. e(x, Y ): number of pairs of vertices in X Y that have an edge between them. d(x, Y ) = e(x,y ) X Y.

11 Regularity of Sets Let X and Y be two sets of vertices in a graph G. e(x, Y ): number of pairs of vertices in X Y that have an edge between them. d(x, Y ) = e(x,y ) X Y. Definition Given a graph G and two sets of vertices X and Y, we say the pair (X, Y ) is ɛ-regular if for any X X with X ɛ X, Y Y with Y ɛ Y, we have d(x, Y ) d(x, Y ) ɛ.

12 Regularity of Sets Let X and Y be two sets of vertices in a graph G. e(x, Y ): number of pairs of vertices in X Y that have an edge between them. d(x, Y ) = e(x,y ) X Y. Definition Given a graph G and two sets of vertices X and Y, we say the pair (X, Y ) is ɛ-regular if for any X X with X ɛ X, Y Y with Y ɛ Y, we have d(x, Y ) d(x, Y ) ɛ. Roughly says graph between X and Y is random-like.

13 Szemerédi s Regularity Lemma Definition Given a partition P of the set of vertices V, we say it is equitable if the size of any two parts differs by at most one.

14 Szemerédi s Regularity Lemma Definition Given a partition P of the set of vertices V, we say it is equitable if the size of any two parts differs by at most one. Definition Given an equitable partition P of the set of vertices V, it is ɛ-regular if all but ɛ P 2 pairs are ɛ-regular.

15 Szemerédi s Regularity Lemma Definition Given a partition P of the set of vertices V, we say it is equitable if the size of any two parts differs by at most one. Definition Given an equitable partition P of the set of vertices V, it is ɛ-regular if all but ɛ P 2 pairs are ɛ-regular. Szemerédi s regularity lemma For every ɛ > 0, there is an M(ɛ) such that for any graph G = (V, E), there is an equitable, ɛ-regular partition of the vertices into at most M(ɛ) parts.

16 Regularity Lemma Proof Sketch Definition For a vertex partition P : V = V 1 V 2... V k, define the mean square density: q(p) = i,j p i p j d(v i,v j ) 2, where p i = V i V.

17 Regularity Lemma Proof Sketch Definition For a vertex partition P : V = V 1 V 2... V k, define the mean square density: q(p) = i,j p i p j d(v i,v j ) 2, where p i = V i V. Between 0 and 1.

18 Regularity Lemma Proof Sketch Definition For a vertex partition P : V = V 1 V 2... V k, define the mean square density: q(p) = i,j p i p j d(v i,v j ) 2, where p i = V i V. Between 0 and 1. If we refine the partition, it cannot decrease.

19 Regularity Lemma Proof Sketch Definition For a vertex partition P : V = V 1 V 2... V k, define the mean square density: q(p) = i,j p i p j d(v i,v j ) 2, where p i = V i V. Between 0 and 1. If we refine the partition, it cannot decrease. If a partition into k parts is not ɛ-regular, can divide each piece into at most 2 k+1 parts, according to worst case sets, to get an increase of ɛ 5 (then make equitable).

20 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

21 Algorithmic Regularity Alon-Duke-Lefmann-Rödl-Yuster (1994) If a pair (X, Y ) is not ɛ-regular, find a pair of subsets that show they are not ɛ 4 /16-regular, in time O ɛ (n ω+o(1) ). Implies tower height at most T (ɛ 20 ). (ω < 2.373)

22 Algorithmic Regularity Alon-Duke-Lefmann-Rödl-Yuster (1994) If a pair (X, Y ) is not ɛ-regular, find a pair of subsets that show they are not ɛ 4 /16-regular, in time O ɛ (n ω+o(1) ). Implies tower height at most T (ɛ 20 ). (ω < 2.373) Frieze-Kannan (1999) Regularity lemma algorithmically, through a spectral approach.

23 Algorithmic Regularity Alon-Duke-Lefmann-Rödl-Yuster (1994) If a pair (X, Y ) is not ɛ-regular, find a pair of subsets that show they are not ɛ 4 /16-regular, in time O ɛ (n ω+o(1) ). Implies tower height at most T (ɛ 20 ). (ω < 2.373) Frieze-Kannan (1999) Regularity lemma algorithmically, through a spectral approach. Kohayakawa-Rödl-Thoma (2003) Faster algorithmic lemma, running time O ɛ (n 2 ).

24 Algorithmic Regularity Alon-Duke-Lefmann-Rödl-Yuster (1994) If a pair (X, Y ) is not ɛ-regular, find a pair of subsets that show they are not ɛ 4 /16-regular, in time O ɛ (n ω+o(1) ). Implies tower height at most T (ɛ 20 ). (ω < 2.373) Frieze-Kannan (1999) Regularity lemma algorithmically, through a spectral approach. Kohayakawa-Rödl-Thoma (2003) Faster algorithmic lemma, running time O ɛ (n 2 ). Alon-Naor (2006) Polynomial-time algorithm, at most T (O(ɛ 7 )) parts.

25 Algorithmic Regularity Even though only a tower-type number is guaranteed, most graphs have a much smaller regularity partition. Previous algorithms may not find it.

26 Algorithmic Regularity Even though only a tower-type number is guaranteed, most graphs have a much smaller regularity partition. Previous algorithms may not find it. Fischer-Matsliah-Shapira (2010) Randomized algorithm which runs in time O ɛ,k (1), if there is an ɛ-regular partition with k parts, finds 2ɛ-regular partition with at most k parts.

27 Algorithmic Regularity Even though only a tower-type number is guaranteed, most graphs have a much smaller regularity partition. Previous algorithms may not find it. Fischer-Matsliah-Shapira (2010) Randomized algorithm which runs in time O ɛ,k (1), if there is an ɛ-regular partition with k parts, finds 2ɛ-regular partition with at most k parts. Folklore/Tao blog post (2010) Randomized algorithm in time O ɛ (1), ɛ-regular partition.

28 Finding a regular partition Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts.

29 Finding a regular partition Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts. An intermediate result is testing regularity.

30 Finding a regular partition Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts. An intermediate result is testing regularity. Fox-L.-Zhao An O ɛ,α,k (n 2 )-time deterministic algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular, i.e. U (1 α)ɛ X, W (1 α)ɛ Y, and d(x, Y ) d(u, W ) > (1 α)ɛ.

31 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

32 Frieze-Kannan (weak) regularity lemma Definition Given a partition P = {V 1, V 2,..., V k } of the set of vertices V, it is Frieze-Kannan ɛ-regular (FK-ɛ-regular) if for any pair of sets S, T V, we have k e(s, T ) d(v i, V j ) S V i T V j ɛ V 2 i,j=1

33 Frieze-Kannan (weak) regularity lemma Definition Given a partition P = {V 1, V 2,..., V k } of the set of vertices V, it is Frieze-Kannan ɛ-regular (FK-ɛ-regular) if for any pair of sets S, T V, we have k e(s, T ) d(v i, V j ) S V i T V j ɛ V 2 i,j=1 Frieze-Kannan regularity lemma Let ɛ > 0. Every graph has a Frieze-Kannan ɛ-regular partition with at most 2 2/ɛ2 parts.

34 Frieze-Kannan (weak) regularity lemma Definition Given a partition P = {V 1, V 2,..., V k } of the set of vertices V, it is Frieze-Kannan ɛ-regular (FK-ɛ-regular) if for any pair of sets S, T V, we have k e(s, T ) d(v i, V j ) S V i T V j ɛ V 2 i,j=1 Frieze-Kannan regularity lemma Let ɛ > 0. Every graph has a Frieze-Kannan ɛ-regular partition with at most 2 2/ɛ2 parts. Proof similar: refine by worst case sets, mean square density increases by ɛ 2.

35 Counting Lemma Definition Given two (possibly weighted) graphs G 1 and G 2 on the same vertex set V, we define their cut distance d (G 1, G 2 ) = 1 V 2 max S,T V e G 1 (S, T ) e G2 (S, T ).

36 Counting Lemma Definition Given two (possibly weighted) graphs G 1 and G 2 on the same vertex set V, we define their cut distance d (G 1, G 2 ) = 1 V 2 max S,T V e G 1 (S, T ) e G2 (S, T ). Partition P is FK-ɛ-regular if and only if d (G, G P ) ɛ.

37 Counting Lemma Definition Given two (possibly weighted) graphs G 1 and G 2 on the same vertex set V, we define their cut distance d (G 1, G 2 ) = 1 V 2 max S,T V e G 1 (S, T ) e G2 (S, T ). Partition P is FK-ɛ-regular if and only if d (G, G P ) ɛ. Counting lemma Given two graphs G 1 and G 2 on the same vertex set, for any graph H on k vertices, we have hom(h, G 1 ) hom(h, G 2 ) e(h)d (G 1, G 2 )n k.

38 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

39 Algorithmic Frieze-Kannan Dellamonica-Kalyanasundaram-Martin-Rödl-Shapira Give a deterministic algorithm which finds a Frieze-Kannan ɛ-regular partition in time ɛ 6 n ω+o(1) into at most 2 O(ɛ 7) parts (2012) in time O(2 2ɛ O(1) n 2 ) into at most 2 ɛ O(1) parts (2015)

40 Algorithmic Frieze-Kannan Dellamonica-Kalyanasundaram-Martin-Rödl-Shapira Give a deterministic algorithm which finds a Frieze-Kannan ɛ-regular partition in time ɛ 6 n ω+o(1) into at most 2 O(ɛ 7) parts (2012) in time O(2 2ɛ O(1) n 2 ) into at most 2 ɛ O(1) parts (2015) Dellamonica-Kalyanasundaram-Martin-Rödl-Shapira There is an n ω+o(1) -time algorithm which, given ɛ > 0, an n-vertex graph G and a partition P of V (G), either: 1 Correctly states that P is FK-ɛ-regular; 2 Finds sets S, T which witness the fact that P is not FK-ɛ 3 /1000-regular.

41 Algorithmic Frieze-Kannan Corollary There is an ɛ O(1) n ω+o(1) -time algorithm which, given ɛ > 0, an n-vertex graph G, outputs t ɛ O(1), subsets S 1, S 2,..., S t, T 1, T 2,..., T t V (G) and real numbers c 1, c 2,..., c t such that d (G, d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t ) ɛ.

42 Algorithmic Frieze-Kannan Corollary There is an ɛ O(1) n ω+o(1) -time algorithm which, given ɛ > 0, an n-vertex graph G, outputs t ɛ O(1), subsets S 1, S 2,..., S t, T 1, T 2,..., T t V (G) and real numbers c 1, c 2,..., c t such that d (G, d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t ) ɛ. Can also do in time 2 2ɛ O(1) n 2.

43 Counting subgraphs Algorithmic problem Count the number of copies of a graph H in a graph G on n vertices.

44 Counting subgraphs Algorithmic problem Count the number of copies of a graph H in a graph G on n vertices. Special case: is there a single copy?

45 Counting subgraphs Algorithmic problem Count the number of copies of a graph H in a graph G on n vertices. Special case: is there a single copy? Even for K k, Zuckerman showed NP-hard to approximate the size of the largest clique within a factor n 1 ɛ, building on an earlier result of Hastad.

46 Counting subgraphs Algorithmic problem Count the number of copies of a graph H in a graph G on n vertices. Special case: is there a single copy? Even for K k, Zuckerman showed NP-hard to approximate the size of the largest clique within a factor n 1 ɛ, building on an earlier result of Hastad. How fast can we approximate the count within an additive ɛn V (H)?

47 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph on n vertices, up to an error of at most ɛn k.

48 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph on n vertices, up to an error of at most ɛn k. A simple randomized algorithm gives 99% certainty:

49 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph on n vertices, up to an error of at most ɛn k. A simple randomized algorithm gives 99% certainty: Sample 10/ɛ 2 random k-sets of vertices.

50 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph on n vertices, up to an error of at most ɛn k. A simple randomized algorithm gives 99% certainty: Sample 10/ɛ 2 random k-sets of vertices. What about deterministic algorithms?

51 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph on n vertices, up to an error of at most ɛn k. A simple randomized algorithm gives 99% certainty: Sample 10/ɛ 2 random k-sets of vertices. What about deterministic algorithms? Can use algorithmic regularity lemma.

52 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k.

53 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k. Duke-Lefmann-Rödl (1996) Can be done in time 2 (k/ɛ)o(1) n ω+o(1).

54 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k. Duke-Lefmann-Rödl (1996) Can be done in time 2 (k/ɛ)o(1) n ω+o(1). Fox-L.-Zhao (2017) Can be done in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ).

55 Counting subgraphs Algorithmic problem Count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k. Duke-Lefmann-Rödl (1996) Can be done in time 2 (k/ɛ)o(1) n ω+o(1). Fox-L.-Zhao (2017) Can be done in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ). Corollary We can approximate the count of K 1000 in a graph on n vertices within an additive n in time O(n 2.4 ).

56 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

57 Counting subgraphs proof sketch Fox-L.-Zhao (2017) Can count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ).

58 Counting subgraphs proof sketch Fox-L.-Zhao (2017) Can count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ). Apply algorithmic Frieze-Kannan: In time ɛ O(1) n ω+o(1), get G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t and d (G, G ) ɛ/e(h), t ɛ O(1).

59 Counting subgraphs proof sketch Fox-L.-Zhao (2017) Can count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ). Apply algorithmic Frieze-Kannan: In time ɛ O(1) n ω+o(1), get G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t and d (G, G ) ɛ/e(h), t ɛ O(1). This means that the count is off by at most ɛn k in G.

60 Counting subgraphs proof sketch Fox-L.-Zhao (2017) Can count the number of copies of a graph H on k vertices in a graph G on n vertices, up to an error of at most ɛn k in time O H (ɛ O(e(H)) n + ɛ O(1) n ω+o(1) ). Apply algorithmic Frieze-Kannan: In time ɛ O(1) n ω+o(1), get G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t and d (G, G ) ɛ/e(h), t ɛ O(1). This means that the count is off by at most ɛn k in G. We can compute hom(h, G ) by computing a sum of (t + 1) e(h) terms.

61 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular.

62 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular. Algorithmic Frieze-Kannan: t (αɛ) O(1), G with d (G, G ) αɛ 3 /4, G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t.

63 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular. Algorithmic Frieze-Kannan: t (αɛ) O(1), G with d (G, G ) αɛ 3 /4, G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t. Can check a bounded number of cases based on the sizes of the intersection of U, W with X, Y and each S i, T i. Check feasibility and whether the density is off.

64 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular.

65 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time algorithm which, given ɛ, α and a graph G between sets X, Y of size n, outputs either that (X, Y ) are ɛ-regular. a pair of subsets U X, W Y that show that (X, Y ) are not (1 α)ɛ-regular. Corollary An O ɛ,α,k (n 2 )-time algorithm which, given ɛ, α, k > 0, graph G on n vertices, and a k-part partition P of the vertices, either: correctly states that P is (1 + α)ɛ-regular. correctly states that P is not ɛ-regular.

66 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts.

67 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts. Apply algorithmic Frieze-Kannan to obtain t (αɛ/k) O(1), G such that d (G, G ) αɛ/(10k 2 ), and G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t.

68 Algorithmic regularity proof sketch Fox-L.-Zhao An O ɛ,α (n 2 )-time deterministic algorithm which, given ɛ, α, k and a graph G on n vertices that has an ɛ-regular partition with k parts, gives a (1 + α)ɛ-regular partition into k parts. Apply algorithmic Frieze-Kannan to obtain t (αɛ/k) O(1), G such that d (G, G ) αɛ/(10k 2 ), and G = d(g)k V (G) + c 1 K S1,T 1 + c 2 K S2,T c t K St,T t. Can work with G. Need to check 2 2(k/αɛ)O(1) possible partitions. For each one, either get not (1 + α/2)ɛ-regular, or (1 + 3α/4)ɛ-regular. Second case must happen for a partition.

69 1 Regularity 2 Algorithmic Regularity 3 Frieze-Kannan Regularity 4 Algorithmic Frieze-Kannan Regularity 5 Proof sketches 6 Conclusion

70 Conclusion Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira developed an algorithmic Frieze-Kannan regularity lemma.

71 Conclusion Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira developed an algorithmic Frieze-Kannan regularity lemma. It actually gives a bit more than just a partition: it gives a finite sum structure.

72 Conclusion Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira developed an algorithmic Frieze-Kannan regularity lemma. It actually gives a bit more than just a partition: it gives a finite sum structure. We can use this to count the number of copies of a small graph H in a graph G efficiently.

73 Conclusion Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira developed an algorithmic Frieze-Kannan regularity lemma. It actually gives a bit more than just a partition: it gives a finite sum structure. We can use this to count the number of copies of a small graph H in a graph G efficiently. We can also use this to more efficiently find and test regularity of sets and of partitions.

74 Conclusion Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira developed an algorithmic Frieze-Kannan regularity lemma. It actually gives a bit more than just a partition: it gives a finite sum structure. We can use this to count the number of copies of a small graph H in a graph G efficiently. We can also use this to more efficiently find and test regularity of sets and of partitions. Questions Faster algorithmic regularity lemmas? With what additive error can we count subgraphs?

Edges and Triangles. Po-Shen Loh. Carnegie Mellon University. Joint work with Jacob Fox

Edges and Triangles. Po-Shen Loh. Carnegie Mellon University. Joint work with Jacob Fox Edges and Triangles Po-Shen Loh Carnegie Mellon University Joint work with Jacob Fox Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no