Knapsack problems in hyperbolic groups Andrey Nikolaev (Stevens - PowerPoint PPT Presentation

Knapsack problems in hyperbolic groups Andrey Nikolaev (Stevens Institute) GAGTA, May 2013 Based on joint work with A.Miasnikov and A.Ushakov Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization Basic idea: Take a classical algorithmic problem from computer science (traveling salesman, Post correspondence, knapsack, . . . ) and translate it into group-theoretic setting. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization The classical subset sum problem ( SSP ): Given a 1 , . . . , a k , a ∈ Z decide if ε 1 a 1 + . . . + ε k a k = a for some ε 1 , . . . , ε k ∈ { 0 , 1 } . SSP for a group G : Given g 1 , . . . , g k , g ∈ G decide if g ε 1 1 . . . g ε k k = g for some ε 1 , . . . , ε k ∈ { 0 , 1 } . Elements in G are given as words in a fixed set of generators of G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization The classical subset sum problem ( SSP ): Given a 1 , . . . , a k , a ∈ Z decide if ε 1 a 1 + . . . + ε k a k = a for some ε 1 , . . . , ε k ∈ { 0 , 1 } . SSP for a group G : Given g 1 , . . . , g k , g ∈ G decide if g ε 1 1 . . . g ε k k = g for some ε 1 , . . . , ε k ∈ { 0 , 1 } . Elements in G are given as words in a fixed set of generators of G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization The classical subset sum problem ( SSP ): Given a 1 , . . . , a k , a ∈ Z decide if ε 1 a 1 + . . . + ε k a k = a for some ε 1 , . . . , ε k ∈ { 0 , 1 } . SSP for a group G : Given g 1 , . . . , g k , g ∈ G decide if g ε 1 1 . . . g ε k k = g for some ε 1 , . . . , ε k ∈ { 0 , 1 } . Elements in G are given as words in a fixed set of generators of G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization In the classical (commutative) case, SSP is pseudo-polynomial. Classical SSP If input is given in unary, SSP is in P , if input is given in binary, SSP is NP -complete. The situation is quite more involved in non-commutative case. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization In the classical (commutative) case, SSP is pseudo-polynomial. Classical SSP If input is given in unary, SSP is in P , if input is given in binary, SSP is NP -complete. The situation is quite more involved in non-commutative case. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization Group Complexity Why Nilpotent P Poly growth Z ≀ Z NP -complete ZOE Free metabelian NP -complete Z ≀ Z Thompson’s F NP -complete Z ≀ Z BS (1 , p ) NP -complete Binary SSP ( Z ) Hyperbolic P Later in the talk Note that the NP -completeness is despite unary input. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Non-commutative discrete optimization Group Complexity Why Nilpotent P Poly growth Z ≀ Z NP -complete ZOE Free metabelian NP -complete Z ≀ Z Thompson’s F NP -complete Z ≀ Z BS (1 , p ) NP -complete Binary SSP ( Z ) Hyperbolic P Later in the talk Note that the NP -completeness is despite unary input. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Knapsack problems in groups Three principle Knapsack type (decision) problems in groups: SSP subset sum, KP knapsack, SMP submonoid membership. Variations of SSP , KP , SMP : search, optimization, bounded. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Knapsack problems in groups Three principle Knapsack type (decision) problems in groups: SSP subset sum, KP knapsack, SMP submonoid membership. Variations of SSP , KP , SMP : search, optimization, bounded. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Knapsack problems in groups Three principle Knapsack type (decision) problems in groups: SSP subset sum, KP knapsack, SMP submonoid membership. Variations of SSP , KP , SMP : search, optimization, bounded. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Knapsack problems in groups Three principle Knapsack type (decision) problems in groups: SSP subset sum, KP knapsack, SMP submonoid membership. Variations of SSP , KP , SMP : search, optimization, bounded. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

The knapsack problem in groups The knapsack problem ( KP ) for G : Given g 1 , . . . , g k , g ∈ G decide if 1 . . . g ε k g ε 1 k = g for some non-negative integers ε 1 , . . . , ε k . There are minor variations of this problem, for instance, integer KP , when ε i are arbitrary integers. They are all similar, we omit them here. The subset sum problem sometimes is called 0 − 1 knapsack. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

The knapsack problem in groups The knapsack problem ( KP ) for G : Given g 1 , . . . , g k , g ∈ G decide if 1 . . . g ε k g ε 1 k = g for some non-negative integers ε 1 , . . . , ε k . There are minor variations of this problem, for instance, integer KP , when ε i are arbitrary integers. They are all similar, we omit them here. The subset sum problem sometimes is called 0 − 1 knapsack. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

The knapsack problem in groups The knapsack problems in groups is closely related to the big powers method, which appeared long before any complexity considerations (Baumslag, 1962). Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

The submonoid membership problem in groups Submonoid membership problem ( SMP ): Given a finite set A = { g 1 , . . . , g k , g } of elements of G decide if g belongs to the submonoid generated by A , i.e., if g = g i 1 , . . . , g i s for some g i j ∈ A . If the set A is closed under inversion then we have the subgroup membership problem in G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Bounded variations It makes sense to consider the bounded versions of KP and SMP , they are always decidable in groups with decidable word problem. The bounded knapsack problem ( BKP ) for G : decide, when given g 1 , . . . , g k , g ∈ G and 1 m ∈ N , if g = G g ε 1 1 . . . g ε k for some ε i ∈ { 0 , 1 , . . . , m } . k BKP is P -time equivalent to SSP in G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Bounded variations It makes sense to consider the bounded versions of KP and SMP , they are always decidable in groups with decidable word problem. The bounded knapsack problem ( BKP ) for G : decide, when given g 1 , . . . , g k , g ∈ G and 1 m ∈ N , if g = G g ε 1 1 . . . g ε k for some ε i ∈ { 0 , 1 , . . . , m } . k BKP is P -time equivalent to SSP in G . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Bounded variations Bounded submonoid membership problem ( BSMP ) for G : Given g 1 , . . . g k , g ∈ G and 1 m ∈ N (in unary) decide if g is equal in G to a product of the form g = g i 1 · · · g i s , where g i 1 , . . . , g i s ∈ { g 1 , . . . , g k } and s ≤ m . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

Hyperbolic groups Theorem Let G be a hyperbolic group then all the problems SSP ( G ) , KP ( G ) , BSMP ( G ), as well as their search and optimization versions are in P . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

KP ( G ) ∈ P , sketch of proof Draw equality g ε 1 1 . . . g ε k k = g in the Cayley graph. If one of ε i ’s is large, we can cut some powers out. Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

KP ( G ) ∈ P , sketch of proof g i g i g i d d g j g j g j Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

KP ( G ) ∈ P , sketch of proof d ε j g g ε i j i d d Now we only need to solve SSP ( G ). Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

KP ( G ) ∈ P , sketch of proof d ε j g g ε i j i d d Now we only need to solve SSP ( G ). Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

� � � � � � � SSP ( G ) ∈ P , sketch of proof w 1 , w 2 , . . . , w k , w is a positive instance of SSP iff a word equal to 1 in G is readable in the following graph: w 1 w 2 w k . . . • • • • • • α ω ε ε ε w − 1 To recognize whether a word equal to 1 in G is readable, we perform two operations, so called R-completion and folding . Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups

SSP ( G ) ∈ P , sketch of proof For a symmetrized presentation � X | R � and a graph Γ labeled by X , at each vertex of Γ we add a loop labeled by r , for each r ∈ R : r 2 r 1 R -completion C Andrey Nikolaev (Stevens Institute) Knapsack problems in hyperbolic groups