Planted Clique Conjectures Are Equivalent Shuichi Hirahara (National Institute of Informatics) Nobutaka Shimizu (Tokyo Institute of Technology) STOC2024

•Theme: Success probability of Planted Clique Problem •Our Results: how to increase the success probability of algorithms If one can solve Planted Clique Problem with success prob , then one can solve the same problem (slight loss in parameter) with success prob . •Consequences ‣ Decision-Recovery gap ‣ Equivalence of many variants of Planted Clique Conjecture e.g., improved search-to-decision reduction 1 poly(n) 1 − exp(−nc) Message 2 [Alon, Andoni, Kaufman, Matulef, Rubinfeld, Xie, 07] [Hirahara, S., 23]

Planted Clique Random Graph 3 •Planted Clique Random Graph ‣ Sample an ER graph (each edge occurs with prob. independently) ‣ Choose a random -vertex subset ‣ Make a -clique by adding edges and output PC(n, k) ER(n) 1/2 k C ⊆ V C k

Planted Clique Random Graph 4 •Planted Clique Random Graph ‣ Sample an ER graph (each edge occurs with prob. independently) ‣ Choose a random -vertex subset ‣ Make a -clique by adding edges and output PC(n, k) ER(n) 1/2 k C ⊆ V C k

Search Planted Clique 5 Input : Output : any -clique PC(n, k) k Def (Search Planted Clique) •A randomized algo has success prob if A γ Pr G ∼ PC(n, k) A [A succeeds on G] ≥ γ [Jerrum, 92][Kučera, 95]

Search Planted Clique 6 Input : Output : any -clique PC(n, k) k Def (Search Planted Clique) •A randomized algo has success prob if A γ Pr G ∼ PC(n, k) A [A succeeds on G] ≥ γ [Jerrum, 92][Kučera, 95] •Uniqueness ‣ We always assume ‣ Maximum clique of k ≫ log n ER(n) ≈ 2 log2 n many -cliques O(log n) unique -clique k

Decision Planted Clique 7 Input : or Question: Which distribution? PC(n, k) ER(n) Def (Decision Planted Clique) •Algo decides and with advantage if A PC(n, k) ER(n) γ Pr G ∼ ER(n) A [A(G) = 1] − Pr G ∼ PC(n, k) A [A(G) = 1] ≥ γ ER ? PC ?

Decision Planted Clique 8 Input : or Question: Which distribution? PC(n, k) ER(n) Def (Decision Planted Clique) •Algo decides and with advantage if •search with success prob decision with adv •decide with adv seach with success prob (low error) •decide with adv search with success prob prob A PC(n, k) ER(n) γ Pr G ∼ ER(n) A [A(G) = 1] − Pr G ∼ PC(n, k) A [A(G) = 1] ≥ γ γ ⇒ Ω(γ) 1 − δ ⇒ 1 − nδ 0.01 ⇒ 0.99 [Alon, Andoni, Kaufman, Matulef, Rubinfeld, Xie, 07] ER ? PC ? [Hirahara, S., 23]

•Can we recover the -clique hidden in a random graph? k Computational-Statistical Gap 9

•Can we recover the -clique hidden in a random graph? •Information-theoretically tractable if ‣ unique -clique threshold k k ≥ (2 + ε)log2 n k Computational-Statistical Gap 10

•Can we recover the -clique hidden in a random graph? •Information-theoretically tractable if ‣ unique -clique threshold •Computationally tractable if ‣ success prob can be k k ≥ (2 + ε)log2 n k k ≥ n ≈ 1 − exp(−nc) Computational-Statistical Gap 11 [Alon, Krivelevich, Sudakov, 98] [Dekel, Gurel-Gurevich, Peres, 14]

•Can we recover the -clique hidden in a random graph? •Information-theoretically tractable if ‣ unique -clique threshold •Computationally tractable if ‣ success prob can be •Is computationally tractable? ‣ major open problem for decades ‣ believed to be computationally intractable (Planted Clique Conjecture) - computational-statistical gap k k ≥ (2 + ε)log2 n k k ≥ n ≈ 1 − exp(−nc) log n ≪ k ≪ n Computational-Statistical Gap 12 [Alon, Krivelevich, Sudakov, 98] [Dekel, Gurel-Gurevich, Peres, 14]

•True for restricted class of algorithms ‣ Metropolis process ‣ Lovász-Schrijver semidefinite hierarchy ‣ Statistical query model ‣ low-degree polynomials •The constant can be arbitrary constant 1/2 Planted Clique Conjecture 13 For any small , any poly-time algorithm has success prob for search PC of . α > 0 ≤ 1/2 k = n1/2−α Conjecture (Planted Clique Conjecture) [Jerrum, 92] [Feige, Krauthgamer, 03] [Feldman, Grigorescu, Reyzin, Vempala, Xiao, 17] [Barak, Hopkins, Kelner, Kothari, Moitra, Potechin, 19] [Hirahara, S., 23]

14 Hardness of PC Community Detection [Hajek, Wu, Xu, 15] Compressed Sensing [Koiran, Zouzias, 14] Nash Equilibrium [Hazan, Krauthgamer, 11] [Austin, Braverman, Chlamtac, 13] Densest Subgraph [Manurangsi, Rubinstein, Schramm, 21] Principal Components [Brennan, Bresler, Huleihel, 18] [Berthet, Rigollet, 13] Cryptography [Juels, Peeinado, 00] [Applebaum, Barak, Wigderson, 09] Testing k-wise Independence [Alon, Andoni, Kaufman, Matulef, Rubinfeld, Xie, 07] computational lower bounds

Planted Clique Conjecture”s” 15 •Variants of PC conjectures ‣ decision version ‣ partial recovery - Find a subset of of size ‣ strong hardness - any poly-time algo has success prob •Question: Are they equivalent? C β log2 n ≤ 1/poly(n) β log2 n These variants are equivalent to Planted Clique Conjecture. Theorem (this work)

Core Result 16 Theorem 1 (this work) If we can solve Search PC on with success prob , then we can solve Search PC on with success prob . PC(n, n1/2−α) 1/nc PC(n, n1/2−Θ(α)) 1 − exp(−nc′  ) •Success prob: from to •Under Planted Clique Conjecture, any poly-time algo has negligible (i.e., ) success prob 1/poly(n) 1 − exp(−nc) n−ω(1)

Core Result 17 Theorem 2 (this work) If we can decide and for with advantage for some , then we can solve Search PC on with success prob ˜ PC(n, k) ER(n) k2 n ⋅ nγ k PC(n, n1/2−α) 1 − exp(−nc) • ‣ choose PC(n, k) C ∼ ( [n] k ) • ‣ Each in in with prob ‣ clique size ˜ PC(n, k) v C k/n = Bin(n, k/n) ≈ k

Core Result 18 Theorem 2 (this work) If we can decide and for with advantage for some , then we can solve Search PC on with success prob ˜ PC(n, k) ER(n) k2 n ⋅ nγ k PC(n, n1/2−α) 1 − exp(−nc) •Folklore algo has advantage ‣ Under Planted Clique Conjecture, this is almost optimal! Ω(k2/n) • ‣ choose PC(n, k) C ∼ ( [n] k ) • ‣ Each in in with prob ‣ clique size ˜ PC(n, k) v C k/n = Bin(n, k/n) ≈ k

Detection-Recovery Gap 19 Search k success prob n1/2 2 log2 n 1 − exp(nc) Under Planted Clique Conjecture… n1/2−α n−ω(1) k advantage n 2 log2 n Ω(k2/n) 1 − exp(nc) o( n) Decision this is optimal for k ≤ n1/2−α negligible

Proof Outline

Outline 21 Decide with adv k2 n ⋅ nγ Decide with adv 1 − exp(−nγ/6) Search with prob 1 − exp(−nc′  ) Shrinking Reduction [Alon et al. ’07]

Outline 22 Decide with adv k2 n ⋅ nγ Decide with adv 1 − exp(−nγ/6) Search with prob 1 − exp(−nc′  ) Search with prob 1 nc Search with prob 1/2 Shrinking Reduction Embedding Reduction Shrinking Reduction [Alon et al. ’07] [Hirahara, S., 23]

Overall Reduction 23 input graph G PC(n, n1/2−α)

Overall Reduction 24 input graph G PC(n, n1/2−α) Choose random vertices n′  = n1−c

Overall Reduction 25 input graph G PC(n, n1/2−α) Choose random vertices n′  = n1−c Induced subgraph Shrinking intermediate graph for H PC(n′  , ℓ) ℓ ≈ n1/2−α−c

Overall Reduction 26 input graph G PC(n, n1/2−α) Choose random vertices n′  = n1−c Prepare for ER(N) N = n′  log n′  ER(N) Induced subgraph Shrinking intermediate graph for H PC(n′  , ℓ) ℓ ≈ n1/2−α−c

Overall Reduction 27 input graph G PC(n, n1/2−α) Choose random vertices n′  = n1−c intermediate graph for H PC(n′  , ℓ) ℓ ≈ n1/2−α−c Embed into ER(N) Induced subgraph Embedding Shrinking query G′  PC(N, ℓ)

Analysis: Reduction Graph 28 Set of all possible inputs Set of all possible queries Set of all possible intermediate graphs

Analysis: Reduction Graph 29 Set of all possible inputs Set of all possible intermediate graphs Set of all possible queries An input An intermediate graph A query Embedding

Analysis: Reduction Graph 30 Theorem (informal) The bipartite graph from blue/red edges are expanders (samplers). Expansion hardness amplification ⇒ [Impagliazzo, Jaiswal, Kabanets, Wigderson, 10] [Hirahara, S., 23]

•Our Results: boosting success probability of algorithms If one can solve Planted Clique Problem with success prob , then one can solve the same problem (slight loss in parameter) with success prob . •Consequences ‣ Decision-Recovery gap ‣ Equivalence of many variants of Planted Clique Conjecture e.g., search-to-decision reduction •Future Direction ‣ other problem (e.g., planted dense subgraph) 1 poly(n) 1 − exp(−nc) Conclusion 31