Complexity Theory of Polynomial-Time Problems

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1 Complexity Theory of Polynomial-Time Problems Lecture 8: (Boolean) Matrix Multiplication Karl Bringmann

2 Recall: Boolean Matrix Multiplication given n n matrices A, B with entries in {0,1} compute matrix C with C +,, = 1 /23 A +,/ B /,, what we already know about BMM: BMM is in time O(n 6 / logn) (four Russians) BMM is equivalent to computing the Transitive Closure of a given graph BMM can be reduced to APSP O(n 6 /2?@A 1 ) n n n

3 Exponent of Matrix Multiplication define ω as the infimum over all c such that MM has an O(n F ) algorithm note: MM is in time O(n GHI ) for any ε > 0 we will be sloppy and write: MM is in time O(n G ) note: MM is not in time O(n GNI ) for any ε > 0 ω 2 Thm: ω < 3 this is very fast in theory all these algorithms have impractically large constant factors (maybe except Strassen 69) ω Strassen Pan Bini et al Schönhage Romani Coppersmith,Winograd Strassen Coppersmith,Winograd Stothers Vassilevska-Williams Le Gall

4 Boolean Matrix Multiplication Thm: BMM is in time O(n G ) given n n matrices A, B with entries in {0,1} Q compute standard matrix product C with C +,, = A +,/ B /,, Q define matrix C with C +,, = [C +,, > 0] then C is the Boolean matrix product of A and B Hypothesis: BMM is not in time O(n GNI )

5 Combinatorial Algorithms fast matrix multiplication uses algebraic techniques which are impractical combinatorial algorithms : do not use algebraic techniques not well defined! Arlazarov,Dinic,Kronrod, Faradzhev 70 (four russians) Bansal,Williams 09 Chan 15 Yu 15 O(n 6 /log V n) O(n 6 log log n V /log W/X n) O(n 6 log log n 6 /log 6 n) O(n 6 poly log log n /log X n) Hypothesis: BMM has no combinatorial algorithm in time O(n 6NI )

6 In this lecture you learn that... (B)MM is useful for designing theoretically fast algorithms - Exercise: k-clique in O(n G//6 ) - Exercise: MaxCut in O(2 G1/6 poly(n)) - Node-Weighted Negative in O(n G ) BMM is an obstacle for practically fast / theoretically very fast algorithms no combinatorial O(n 6NI ) not faster than O(n V.6\6 ) - Transitive Closure has no O(n GNI ) / combinatorial O(n 6NI ) algorithm - Exercise: pattern matching with 2 patterns - Sliding Window Hamming Distance - context-free grammar problems

7 Outline 1) Relations to Subcubic Equivalences 2) Strassen s Algorithm 3) Sliding Window Hamming Distance 4) Node-Weighted Negative 5) Context-Free Grammars

8 Corollaries from Subcubic Equivalences APSP BMM All-Pairs- given an unweighted graph G vertices V = I J K i, j: are they in a triangle with some k? given an unweighted graph G does it contain a triangle? Min-Plus Product All-Pairs- Negative- Negative [Vassilevska-Williams,Williams 10]

9 Corollaries from Subcubic Equivalences APSP BMM All-Pairs- n I K J n A B n Min-Plus Product All-Pairs- Negative- Negative

10 Corollaries from Subcubic Equivalences APSP BMM All-Pairs- Given an unweighted undirected graph G Adjacency matrix A, entries in {0,1} 1. Compute Boolean Product C A A: C +,, = k A +,/ A /,, / 2. Compute A +,, C +,, +,, Min-Plus Product All-Pairs- Negative- this equals k A +,, A +,/ A /,, +,,,/ thus we solved triangle detection Negative

11 Corollaries from Subcubic Equivalences APSP BMM All-Pairs- Min-Plus Product All-Pairs- Negative- Negative

12 All-Pairs- to Given graph G Decide whether there are vertices i, j, k such that i, j, k form a triangle All-Pairs- All-Pairs- Given graph G with vertex set V = I J K Decide for every i I, j J whether there is a vertex k K such that i, j, k form a triangle K Split I, J, K into n/s parts of size s: I 3,, I 1/r, J 3,, J 1/r, K 3,, K 1/r I K p J For each of the (n/s) 6 triples (I n, J o, K p ): consider graph G[I n J o K p ] I n J o

] Set C i, j 1 Delete edge (i, j) (i, j) is in no more triangles K All-Pairs- guaranteed

13 All-Pairs- to Initialize C as n n all-zeroes matrix For each of the (n/s) 6 triples of parts (I n, J o, K p ): While G[I n J o K p ] contains a triangle: Find a triangle (i, j, k) in G[I n J o K p ] Set C i, j 1 Delete edge (i, j) (i, j) is in no more triangles K All-Pairs- guaranteed termination: can delete n V edges I K p J correctness: if (i, j) is in a triangle, we will find one I n J o

14 All-Pairs- to Find a triangle (i, j, k) in G[I n J o K p ] How to find a triangle if we can only decide whether one exists? All-Pairs- Partition I n into I n (3), I n (V), J o into J o (3), J o (V), K p into K p (3),K p (V) Since G[I n J o K p ] contains a triangle, at least one of the 2 6 subgraphs G[I (s) n J (t) o K (F) p ] contains a triangle K p Decide for each such subgraph whether it contains a triangle I n J o Recursively find a triangle in one subgraph

15 All-Pairs- to Find a triangle (i, j, k) in G[I n J o K p ] How to find a triangle if we can only decide whether one exists? All-Pairs- Partition I n into I n (3), I n (V), J o into J o (3), J o (V), K p into K p (3),K p (V) Since G[I n J o K p ] contains a triangle, at least one of the 2 6 subgraphs G[I n (s) J o (t) K p (F) ] contains a triangle Decide for each such subgraph whether it contains a triangle Recursively find a triangle in one subgraph Running Time: T vwxyz{w xa?} n 2 6 T ~}wy}z{w xa?} (n) + T vwxyz{w xa?} n/2 = O(T ~}wy}z{w xa?} n )

16 All-Pairs- to Initialize C as n n all-zeroes matrix For each of the (n/s) 6 triples of parts (I n, J o, K p ): While G[I n J o K p ] contains a triangle: Find a triangle (i, j, k) in G[I n J o K p ] Set C i, j 1 Delete edge (i, j) ( ) All-Pairs- Running Time: ( ) = O(T vwxyz{w xa?} (s)) = O(T ~}wy}z{w xa?} (s)) Total time: #triples + #triangles found ( ) n/s 6 + n V T ~}wy}z{w xa?} (s) Set s = n 3/6 and assume T ~}wy}z{w xa?} n = O(n 6NI ) Total time: O n V n 3NI/6 = O(n 6NI/6 )

17 Corollaries from Subcubic Equivalences APSP BMM All-Pairs- If BMM has (combinatorial) O(n 6NI ) algorithm then has (combinatorial) O n 6NI algorithm If has (combinatorial) O(n 6NI ) algorithm then BMM has (combinatorial) O n 6NI/6 algorithm subcubic equivalent, but this mainly makes sense for combinatorial algorithms Min-Plus Product All-Pairs- Negative- Negative

18 Outline 1) Relations to Subcubic Equivalences 2) Strassen s Algorithm 3) Sliding Window Hamming Distance 4) Node-Weighted Negative 5) Context-Free Grammars

19 Strassen s Algorithm shows ω 2.81 A B C A 3,3 A 3,V A V,3 A V,V B 3,3 B 3,V = B V,3 B V,V C 3,3 C 3,V C V,3 C V,V C 3,3 = A 3,3 B 3,3 + A 3,V B V,3 C 3,V = A 3,3 B 3,V + A 3,V B V,V C V,3 = A V,3 B 3,3 + A V,V B V,3 T n 8 T n/2 + O(n V ) T n O(n 6 ) C V,V = A V,3 B 3,V + A V,V B V,V

20 Strassen s Algorithm shows ω 2.81 A B C A 3,3 A 3,V A V,3 A V,V B 3,3 B 3,V = B V,3 B V,V C 3,3 C 3,V C V,3 C V,V M 3 = A 3,3 + A V,V B 3,3 + B V,V M V = A V,3 + A V,V B 3,3 M 6 = A 3,3 B 3,V B V,V M X = A V,V B V,3 B 3,3 M = A 3,3 + A 3,V B V,V M = A V,3 A 3,3 B 3,3 + B 3,V M \ = A 3,V A V,V B V,3 + B V,V C 3,3 = M 3 + M X M + M \ C 3,V = M 6 + M C V,3 = M V + M X C V,V = M 3 M V + M 6 + M T n 7 T n/2 + O(n V ) T n O n?@a \ = O(n V. \X )

21 Faster Matrix Multiplication tensor = 3-dimensional matrix matrix multiplication tensor: n V rows/columns/... entries in {0,1} (i Q, k Q ) entry T +,,, +,/, /,, = 1 iff i = i and j = j QQ and k Q = k QQ i.e. A +,/ B /,, appears in C +,, (i, j) (k QQ, j QQ ) matrix of rank 1: outer product of two vectors matrix of rank r: sum of r rank-1-matrices tensor of rank 1: outer product of three vectors tensor of rank r: sum of r rank-1-tensors matrix rank is in P tensor rank is not known to be in P

22 Faster Matrix Multiplication tensor = 3-dimensional matrix matrix multiplication tensor: n V rows/columns/... entries in {0,1} (i Q, k Q ) entry T +,,, +,/, /,, = 1 iff i = i and j = j QQ and k Q = k QQ i.e. A +,/ B /,, appears in C +,, (i, j) (k QQ, j QQ ) Strassen: rank of MM-tensor for n = 2 is at most 7 any bound on rank of MM-tensor can be transformed into a MM-algorithm thus search for faster MM-algorithms is a mathematical question this is complete: one can find ω by analyzing tensor rank!

23 Outline 1) Relations to Subcubic Equivalences 2) Strassen s Algorithm 3) Sliding Window Hamming Distance 4) Node-Weighted Negative 5) Context-Free Grammars

Sliding Window Hamming Distance given two strings: text T of length n and pattern P of length m < n compute for each i the Hamming distance of P and T[i.

24 Sliding Window Hamming Distance given two strings: text T of length n and pattern P of length m < n compute for each i the Hamming distance of P and T[i.. i + m 1] best known algorithm: O(n m polylog n) O(n 3. 3 ) [Indyk,Porat,Clifford 09] a b c b b c a a b b c a b b c a b b c a b b c a b b c a Thm: Sliding Window Hamming Distance has no O(n G/VNI ) algorithm or combinatorial O(n 3. NI ) algorithm unless the BMM-Hypothesis fails O(n 3.3 ) Open Problem: get rid of combinatorial or design improved algorithm using MM

Sliding Window Hamming Distance given two strings: text T of length n and pattern P of length m < n compute for each i the Hamming distance of P and T[i.. i + m 1] A 1 0 0 1 1 1 0 1 1 B 1 1 0 1 0 1 =?

25 Sliding Window Hamming Distance given two strings: text T of length n and pattern P of length m < n compute for each i the Hamming distance of P and T[i.. i + m 1] A B =? x x x y 2 y 2 y 3 3 alphabet: {1,2,...,n,x,y,$} pattern = concat rows: 1 x x x 2 3 text = concat columns + padding: $ $ $ $ $ $ 1 2 y $ 1 y 3 $ y 2 3 $ $ $ $ $ $ 1 x x x 2 3

26 Sliding Window Hamming Distance $ $ $ $ $ $ 1 2 y $ 1 y 3 $ y 2 3 $ $ $ $ $ $ 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x 2 3 put a 1 if there is at least one match =

27 Sliding Window Hamming Distance given Boolean n n-matrices A, B we construct text+pattern of length O(n V ) (in time O n V ) thus, an O(n G/VNI ) algorithm for Sliding Window Hamming Distance would yield an O(n GNVI ) algorithm for BMM, contradicting BMM-Hypothesis and an O(n 3. NI ) combinatorial algorithm for Sliding Window Hamming Dist. would yield an O(n 6NVI ) combinatorial algorithm for BMM Thm: Sliding Window Hamming Distance has no O(n G/VNI ) algorithm or combinatorial O(n 3. NI ) algorithm unless the BMM-Hypothesis fails

28 Outline 1) Relations to Subcubic Equivalences 2) Strassen s Algorithm 3) Sliding Window Hamming Distance 4) Node-Weighted Negative 5) Context-Free Grammars

29 Node-Weighted Negative (Edge- Weighted) Negative O(n 6 ) given a directed graph with weights w +,, on edges, is there a triangle i, j, k: w,,+ + w +,/ + w /,, < 0? Node- Weighted Negative given a directed graph with weights w + on nodes, is there a triangle i, j, k: w + + w, + w / < 0? w + 1 O(n G ) given an unweighted undirected graph, is there a triangle?

30 Node-Weighted Negative (Edge- Weighted) Negative Node- Weighted Negative O(n 6 ) find appropriate edge weights that simulate the given node weights: set w +,, (w + + w, )/2 then for a triangle i, j, k: w,,+ + w +,/ + w /,, = w + + w, + w / O(n G )

31 Node-Weighted Negative (Edge- Weighted) Negative O(n 6 ) Node- Weighted Negative O(n G ) [Czumaj,Lingas 07] actually for Node-Weighted Minimum Weight O(n G )

32 Node-Weighted Minimum Weight we can assume that the graph is tripartite: G: G Q : u V u u 3 v V u 6 v V V V v 3 V 3 v 6 V 6 triangle i, j, k triangle i 3,j V, k 6

33 Node-Weighted Minimum Weight given graph G = (V, E) with V = I J K and node weights w Ÿ, compute minimum weight q s.t. there are i I, j J, k K with i, j, j, k, k, i E and w + + w, + w / = q - assume that I, J, K are sorted by weight - parameter p (=sufficiently large constant) - assume n I = J = K = p l for some l N (add isolated dummy vertices)

34 Node-Weighted Minimum Weight given graph G = (V, E) with V = I J K and node weights w Ÿ, compute minimum weight q s.t. there are i I, j J, k K with i, j, j, k, k, i E and w + + w, + w / = q K - assume that I, J, K are sorted by weight - parameter p (=sufficiently large constant) - assume n I = J = K = p l for some l N (add isolated dummy vertices) I K p J ALG(G): 0) if n = O(1) then solve in constant time I n J o 1) split I = I 3 I, J = J 3 J, K = K 3 K (in sorted order: max w(i n ) min w(i nh3 ) and so on) 2) R x, y, z 1,, p 6 G I n J o K p contains a triangle} 3) for each x, y, z R s.t. there is no x Q, y Q, z Q R with x Q < x, y Q < y, z Q < z run ALG(G I n J o K p )

35 Node-Weighted Minimum Weight 3) for each x, y, z R s.t. there is no x Q, y Q, z Q R with x Q < x, y Q < y, z Q < z run ALG(G I n J o K p ) Correctness: if there is no triangle in G I n J o K p then we can ignore it if x, y, z R is dominated by x Q, y Q, z Q R: let i, j, k be a triangle in G I n J o K p, and i, j, k a triangle in G I n J o K p then w + + w, + w / min w(i n ) + min w(j o ) + min w(k p ) and w + + w, + w / max w(i n ) + max w(j o ) + max w(k p ) so we can safely ignore G I n J o K p

36 Node-Weighted Minimum Weight given graph G = (V, E) with V = I J K and node weights w Ÿ, compute minimum weight q s.t. there are i I, j J, k K with i, j, j, k, k, i E and w + + w, + w / = q K - assume that I, J, K are sorted by weight - parameter p (=sufficiently large constant) - assume n I = J = K = p l for some l N (add isolated dummy vertices) I K p J ALG(G): 0) if n = O(1) then solve in constant time I n J o 1) split I = I 3 I, J = J 3 J, K = K 3 K (in sorted order: max I n min I nh3 aso.) 2) R x, y, z 1,, p 6 G I n J o K p contains a triangle} O(p 6 n G ) 3) for each x, y, z R s.t. there is no x Q, y Q, z Q R with x Q < x, y Q < y, z Q < z run ALG(G I n J o K p ) size n/p how many iterations?

37 Node-Weighted Minimum Weight 3) for each x, y, z R s.t. there is no x Q, y Q, z Q R with x Q < x, y Q < y, z Q < z How many iterations? define Δ,r x, y, z 1,, p 6 x y = r and x z = s} = 1,1 r, 1 s, 2,2 r, 2 s,, p, p r, p s 1,, p 6 for r, s { p,, p} the sets Δ,r cover 1,, p 6 line 3) applies to at most one (x, y, z) in Δ,r for any r, s! hence, there are at most (2p) V = 4p V recursive calls to ALG recursion: T n 4p V T n/p + O(p 6 n G )

38 Node-Weighted Minimum Weight given graph G = (V, E) with V = I J K and node weights w Ÿ, compute minimum weight q s.t. there are i I, j J, k K with i, j, j, k, k, i E and w + + w, + w / = q K - assume that I, J, K are sorted by weight - parameter p (=sufficiently large constant) - assume n I = J = K = p l for some l N (add isolated dummy vertices) I K p J ALG(G): 0) if n = O(1) then solve in constant time I n J o 1) split I = I 3 I, J = J 3 J, K = K 3 K (in sorted order: max I n min I nh3 aso.) 2) R x, y, z 1,, p 6 G I n J o K p contains a triangle} O(p 6 n G ) 3) for each x, y, z R s.t. there is no x Q, y Q, z Q R with x Q < x, y Q < y, z Q < z run ALG(G I n J o K p ) size n/p recursion: T n 4p V T n/p + O(p 6 n G )

39 Node-Weighted Minimum Weight recursion: T n 4p V T n/p + O(p 6 n G ) T n 4p V T n/p + α p 6 n G for some constant α want to show: T n 2α p 6 n G plug in: T n 4p V 2α p 6 (n/p) G +α p 6 n G = α p 6 n G (1 + 8p VNG ) assume ω > 2: 2α p 6 n G for a sufficiently large constant p so we have: T n O(n G ) (if ω = 2: show that T n 2α p 6 n GHI ) in total: T n O(n G + n VHI ) for any ε > 0

40 In this lecture you learned that... (B)MM is useful for designing theoretically fast algorithms - Exercise: k-clique in O(n G//6 ) - Exercise: MaxCut in O(2 G1/6 poly(n)) - Node-Weighted Negative in O(n G ) BMM is an obstacle for practically fast / theoretically very fast algorithms no combinatorial O(n 6NI ) not faster than O(n V.6\6 ) - Transitive Closure has no O(n GNI ) / combinatorial O(n 6NI ) algorithm - Exercise: pattern matching with 2 patterns - Sliding Window Hamming Distance - context-free grammar problems

Complexity Theory of Polynomial-Time Problems

Complexity Theory of Polynomial-Time Problems Lecture 5: Subcubic Equivalences Karl Bringmann Reminder: Relations = Reductions transfer hardness of one problem to another one by reductions problem P instance