SAT 2020 Outline 1 Matrix Multiplication 2 Cohn-Umans Framework 3 - - PowerPoint PPT Presentation

Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles

Matthew Anderson Zongliang Ji Anthony Yang Xu

SAT 2020

Outline

1 Matrix Multiplication 2 Cohn-Umans Framework 3 Checking Strong Uniquely Solvable Puzzles 4 Searching for Strong Uniquely Puzzles 5 Conclusions & Future Work

Matrix Multiplication

Problem Input: A ∈ Fn×n, B ∈ Fn×n Output: C = A × B ∈ Fn×n. For example:

• 1

2 2

• ×
• 1

3 1 1

• =
• 1

5

• 2

6

• How many operations does it take to multiply two n-by-n matrices?
• O(n3) by naively computing n2 dot products of rows of A and

columns of B.

• Ω(n2) because there are at n2 cells to output.

Question What is the smallest ω ≤ 3 such that n-by-n matrix multiplication can be done in time O(nω)?

Progress on ω 3 Na¨ ıve 2.808 Strassen 1969 2.796 Pan 1978 2.78 Bini et al 1979 2.522 Sch¨

• nhage 1981

2.496 Coppersmith & Winograd 1982 2.479 Strassen 1986 2.375477 Coppersmith & Winograd 1987 2.374 Stothers 2010 2.3728642 Williams 2011 2.3728639 Le Gall 2014

Outline

1 Matrix Multiplication 2 Cohn-Umans Framework 3 Checking Strong Uniquely Solvable Puzzles 4 Searching for Strong Uniquely Puzzles 5 Conclusions & Future Work

Cohn-Umans Framework

In 2003, Cohn and Umans proposed an approach for improving the upper bound on ω.

• Inspired by the Θ(n log n) FFT-based algorithm for multiplying two

degree n univariate polynomial, c.f., e.g., [CLRS 2009, Chap 30]. A × B = C becomes FFT−1(FFT(A) ∗ FFT(B)) = C

Cohn-Umans Framework

In 2003, Cohn and Umans proposed an approach for improving the upper bound on ω.

• Inspired by the Θ(n log n) FFT-based algorithm for multiplying two

degree n univariate polynomial, c.f., e.g., [CLRS 2009, Chap 30]. A × B = C becomes FFT−1(FFT(A) ∗ FFT(B)) = C Idea determine a suitable group G to embed multiplication into the group algebra C[G] using sets X , Y , Z ⊆ G, with |X | = |Y | = |Z| = n. A =

• i,j∈[n]

(x −1

i

yj)Ai,j, B =

• j,k∈[n]

(y−1

j

zk)Bj,k, C =

• i,k∈[n]

(x −1

i

zk)Ci,k where triple product property holds: ∀x, x ′ ∈ X , ∀y, y′ ∈ Y , ∀z, z ′ ∈ Z, x −1yy′−1z = x ′−1z ′ iff x = x ′, y = y′, z = z ′. ω implied by G depends on |G| and aspects of its representation.

Puzzles

Definition (Puzzle) An (s, k)-puzzle is a subset P ⊆ Uk = {1, 2, 3}k with |P| = s. P =

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• P is a (5,4)-puzzle.

Puzzles

Definition (Puzzle) An (s, k)-puzzle is a subset P ⊆ Uk = {1, 2, 3}k with |P| = s. P =

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• P is a (5,4)-puzzle.
• P has five rows.
• P has four columns.

Puzzles

Definition (Puzzle) An (s, k)-puzzle is a subset P ⊆ Uk = {1, 2, 3}k with |P| = s. P =

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• P is a (5,4)-puzzle.
• P has five rows.
• P has four columns.
• Since P is a set, the rows are thought to

be unordered.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• This puzzle is not uniquely solvable.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• This puzzle is not uniquely solvable.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• This puzzle is not uniquely solvable.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• This puzzle is not uniquely solvable.
• Can be witnessed by two permutations: π2, π3.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3

2 2 3 3 3 2

• 2

3 3 3 2

• 3
• This puzzle is not uniquely solvable.
• Can be witnessed by two permutations: π2, π3.

Uniquely Solvable Puzzles – Intuition

We’re interested in puzzles that are uniquely solvable: A puzzle P is uniquely solvable if there is no way to reorganize the 1-, 2-, 3-pieces of P without overlapping into a puzzle different from P.

2 2 3 3 3 2

• 2

3 3 3 2

• 3

2 2 3 3 3 2

• 2

3 3 3 2

• 3

3 3 3 3 3 3 3 2 2 2 2 2

• This puzzle is not uniquely solvable.
• Can be witnessed by two permutations: π2, π3.
• Since the resulting puzzles is not the same as the original puzzle

(even reordering rows), the puzzle is not uniquely solvable.

Uniquely Solvable Puzzles – Formal

Definition ( Uniquely Solvable Puzzle) A puzzle P is called a uniquely solvable puzzle ( USP) if for all permutations π2, π3 of the rows of P:

1 either the permutations are identical, π2 = π3 = id, or 2 there is a row r ∈ P and column i such that at least two of the

following hold:

1 (r)i = 1, 2 (π2(r))i = 2, 3 (π3(r))i = 3.

Uniquely Solvable Puzzles – Formal

Definition ( Uniquely Solvable Puzzle) A puzzle P is called a uniquely solvable puzzle ( USP) if for all permutations π2, π3 of the rows of P:

1 either the permutations are identical, π2 = π3 = id, or 2 there is a row r ∈ P and column i such that at least two of the

following hold:

1 (r)i = 1, 2 (π2(r))i = 2, 3 (π3(r))i = 3.

The follow puzzle is uniquely solvable: 1 3 2 1

Strong Uniquely Solvable Puzzles – Formal

Definition (Strong Uniquely Solvable Puzzle) A puzzle P is called a strong uniquely solvable puzzle (SUSP) if for all permutations π2, π3 of the rows of P:

1 either the permutations are identical, π2 = π3 = id, or 2 there is a row r ∈ P and column i such that exactly two of the

following hold:

1 (r)i = 1, 2 (π2(r))i = 2, 3 (π3(r))i = 3.

Strong Uniquely Solvable Puzzles – Formal

Definition (Strong Uniquely Solvable Puzzle) A puzzle P is called a strong uniquely solvable puzzle (SUSP) if for all permutations π2, π3 of the rows of P:

1 either the permutations are identical, π2 = π3 = id, or 2 there is a row r ∈ P and column i such that exactly two of the

following hold:

1 (r)i = 1, 2 (π2(r))i = 2, 3 (π3(r))i = 3.

No good intuition for the“exactly two”part, but a useful implication.

Strong Uniquely Solvable Puzzles – Formal

Definition (Strong Uniquely Solvable Puzzle) A puzzle P is called a strong uniquely solvable puzzle (SUSP) if for all permutations π2, π3 of the rows of P:

1 either the permutations are identical, π2 = π3 = id, or 2 there is a row r ∈ P and column i such that exactly two of the

following hold:

1 (r)i = 1, 2 (π2(r))i = 2, 3 (π3(r))i = 3.

No good intuition for the“exactly two”part, but a useful implication. Lemma ([CKSU 05, Corollary 3.6]) If there is a strong uniquely solvable (s, k)-puzzle, ω ≤ min

m∈{3,4,5,...}

3 log m log(m − 1) − 3 log s! sk log(m − 1).

Useful Strong Uniquely Solvable Puzzles

Lemma ([CKSU 05, Proposition 3.8]) There is an infinite family of SUSP that achieve ω < 2.48. There are group-theoretic constructions derived from [Strassen 86] and [Coppersmith-Winograd 87] that achieve the ω’s of those works.

Useful Strong Uniquely Solvable Puzzles

Lemma ([CKSU 05, Proposition 3.8]) There is an infinite family of SUSP that achieve ω < 2.48. There are group-theoretic constructions derived from [Strassen 86] and [Coppersmith-Winograd 87] that achieve the ω’s of those works. Lemma ([BCCGU 16]) SUSP cannot show ω < 2 + ǫ, for some ǫ > 0.

• This was conditionally true if the Erd¨
• -Szemeredi sunflower

conjecture held [Alon-Shpilka-Umans 2013]. Progress on cap sets and arithmetic progressions made this unconditional [Ellenberg 2016, Croot-Lev-Pach 2016].

• More recent work has shown this type of barrier for larger classes of

approaches [Alman-Williams 2018].

Goal & Approach

Goal Find strong uniquely solvable puzzles that imply smaller ω.

Goal & Approach

Goal Find strong uniquely solvable puzzles that imply smaller ω. Approach

• For small values of width k, determine the maximum s such that

there exists an (s, k)-SUSP.

Goal & Approach

Goal Find strong uniquely solvable puzzles that imply smaller ω. Approach

• For small values of width k, determine the maximum s such that

there exists an (s, k)-SUSP.

• Algorithm Design
• Checking that a puzzle is a SUSP.
• Searching for large SUSP.

Goal & Approach

Goal Find strong uniquely solvable puzzles that imply smaller ω. Approach

• For small values of width k, determine the maximum s such that

there exists an (s, k)-SUSP.

• Algorithm Design
• Checking that a puzzle is a SUSP.
• Searching for large SUSP.
• Implementation
• Software platform to explore and experiment with SUSP.
• Focused on desktop, but have HPC implementation.

Goal & Approach

Goal Find strong uniquely solvable puzzles that imply smaller ω. Approach

• For small values of width k, determine the maximum s such that

there exists an (s, k)-SUSP.

• Algorithm Design
• Checking that a puzzle is a SUSP.
• Searching for large SUSP.
• Implementation
• Software platform to explore and experiment with SUSP.
• Focused on desktop, but have HPC implementation.
• Only need to one sufficiently large puzzle to achieve new algorithm.

Outline

1 Matrix Multiplication 2 Cohn-Umans Framework 3 Checking Strong Uniquely Solvable Puzzles 4 Searching for Strong Uniquely Puzzles 5 Conclusions & Future Work

Checking Puzzles

Problem (SUSP-Check) Input: A (s, k)-puzzle P. Output: YES if P is a SUSP, and NO otherwise. It suffices to evaluate the following formula for a puzzle P: ∀π2, π3 ∈ SymP. π2 = π3 = id ∨ ∃r ∈ P.∃i ∈ [k].(ri = 1) + ((π2(r))i = 2) + ((π3(r))i = 3) = 2

• That a P is not a SUSP is be witnessed by permutations π1, π2, π3.
• SUSP-Check is in coNP.
• When we only want to verify uniquely solvability it is reducible to

graph automorphism.

• It is not clear whether SUSP-Check is coNP-hard.

Brute Force

∀π2, π3 ∈ SymP. π2 = π3 = id ∨ ∃r ∈ P.∃i ∈ [k].(ri = 1) + ((π2(r))i = 2) + ((π3(r))i = 3) = 2

• Brute force model checking takes O((s!)2 · poly(s, k)) time.
• Impractical for puzzles with size s ≥ 6 (so k ≥ 5).
• Easy to implement reference for debugging.
• It will be more convenient to think about checking the complement

formula. ∃π2, π3 ∈ SymP. (π2 = id ∨ π3 = id) ∧ ∀r ∈ P.∀i ∈ [k].((r)i = 1) + ((π2(r))i = 2) + ((π3(r))i = 3) = 2

Preprocessing

∃π2,π3 ∈ SymP. (π2 = id ∨ π3 = id) ∧ ∀r ∈ P.∀i ∈ [k].(ri = 1) + ((π2(r))i = 2) + ((π3(r))i = 3) = 2

• The innermost ∀ can be precomputed in O(s3k) time by creating a

Boolean relation TP ∈ P × P × P, where (p, q, r) ∈ TP ⇔ ∀i ∈ [k].(pi = 1) + (qi = 2) + (ri = 3) = 2.

• This simplifies the formula we are checking to:

∃π2,π3 ∈ SymP. (π2 = id ∨ π3 = id) ∧ ∀r ∈ P.(r, π2(r), π3(r)) ∈ TP

• This makes the dominant term of the running time independent of k

and is useful for the next step.

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF

Reduction to 3D Matching

This results in the formula below which is true iff P is not a SUSP. ∃π2, π3 ∈ SymP.(π2 = id ∨ π3 = id) ∧ ∀r ∈ P.(r, π2(r), π3(r)) ∈ TP. This is an instance of a natural NP problem. Problem (3D Matching) Input: A 3-hypergraph G = V , E ⊆ V × V × V . Output: True iff ∃M ⊆ E with |M | = |V | and for all distinct e1, e2 ∈ M each coordinate of e1 and e2 are vertex disjoint. We can reduce verifying P is not a SUSP to 3D matching.

• Consider GP = P, TP.
• P is a not a SUSP iff GP has a 3D matching that isn’t the identity

matching, i.e., M = {(r1, r1, r1), . . . , (rs, rs, rs)}.

Dynamic Programming

Perfect3DM(V , E):

1: function Help3DM(i, X , Y ): 2:

if i = |V | then return true.

3:

result = false.

4:

for x ∈ X :

5:

for y ∈ Y :

6:

if (vi, x, y) ∈ E then

7:

result = result or Help3DM(i + 1, X − {x}, Y − {y}).

8:

return result.

9: return Help3DM(0, V , V ).

A recursive algorithm for determining perfect 3D matchings.

• Looks at next unmatched vertex v in first coord.
• Try matching with all x, y on 2nd & 3rd coords with edge in graph.
• Subproblems depend on # edges in matching, vertices used so far.

Dynamic Programming – Analysis

Perfect3DM(V , E):

1: function Help3DM(i, X , Y ): 2:

if i = |V | then return true.

3:

result = false.

4:

for x ∈ X :

5:

for y ∈ Y :

6:

if (vi, x, y) ∈ E then

7:

result = result or Help3DM(i + 1, X − {x}, Y − {y}).

8:

return result.

9: return Help3DM(0, V , V ).

Dynamic Programming – Analysis

Perfect3DM(V , E):

1: function Help3DM(i, X , Y ): 2:

if i = |V | then return true.

3:

result = false.

4:

for x ∈ X :

5:

for y ∈ Y :

6:

if (vi, x, y) ∈ E then

7:

result = result or Help3DM(i + 1, X − {x}, Y − {y}).

8:

return result.

9: return Help3DM(0, V , V ).

• Na¨

ıvely runs in time O((s2)s) = O(22s log s).

Dynamic Programming – Analysis

Perfect3DM(V , E):

1: function Help3DM(i, X , Y ): 2:

if i = |V | then return true.

3:

result = false.

4:

for x ∈ X :

5:

for y ∈ Y :

6:

if (vi, x, y) ∈ E then

7:

result = result or Help3DM(i + 1, X − {x}, Y − {y}).

8:

return result.

9: return Help3DM(0, V , V ).

• Na¨

ıvely runs in time O((s2)s) = O(22s log s).

• Improves to O(2s · 2s · s · s2) = O(22ss3) by memoizing.
• s = |V | options for i, and
• 2s = 2|V | options for X and Y .

Dynamic Programming – Analysis

Perfect3DM(V , E):

1: function Help3DM(i, X , Y ): 2:

if i = |V | then return true.

3:

result = false.

4:

for x ∈ X :

5:

for y ∈ Y :

6:

if (vi, x, y) ∈ E then

7:

result = result or Help3DM(i + 1, X − {x}, Y − {y}).

8:

return result.

9: return Help3DM(0, V , V ).

• Na¨

ıvely runs in time O((s2)s) = O(22s log s).

• Improves to O(2s · 2s · s · s2) = O(22ss3) by memoizing.
• s = |V | options for i, and
• 2s = 2|V | options for X and Y .
• Note that more advanced techniques can achieve O(2spoly(s))

[Bj¨

• rklund et al., 2010].

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP

Other Reductions

Reduced 3D matching to CNF satisfiability.

• Reduced satisfiability instance had 2s2 variables and

Θ(s3) constraint size.

• Used open-source clause-learning SAT solver

MapleCOMSPS that won general category of 2016 SAT Competition. Solver written by J.H. Liang, V. Ganesh, and C. Oh. http://www.satcompetition.org Reduced 3D matching to 0-1 integer programming.

• Reduced IP instance had s3 variables and Θ(s3)

formula size.

• Used a close-source optimization library Gurobi.

http://www.gurobi.com Jerry Ji Anthony Xu

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP SAT

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP SAT IP

Implementation

Hybrid Algorithm

• s ≤ 2: Brute force
• 2 < s ≤ 10: Dynamic programming
• 10 < s: SAT and IP in parallel

Implementation

Hybrid Algorithm

• s ≤ 2: Brute force
• 2 < s ≤ 10: Dynamic programming
• 10 < s: SAT and IP in parallel

Uses a number of fast, but incomplete, heuristics to terminate early.

• Answers with YES, NO, or MAYBE.

Implementation

Hybrid Algorithm

• s ≤ 2: Brute force
• 2 < s ≤ 10: Dynamic programming
• 10 < s: SAT and IP in parallel

Uses a number of fast, but incomplete, heuristics to terminate early.

• Answers with YES, NO, or MAYBE.
• Trying to randomly or greedily generate 3D matchings.
• Verifying that all pairs (or triples) of rows form a SUSP.
• Testing whether the puzzle is uniquely solvable using the graph

isomorphism library Nauty: http://users.cecs.anu.edu.au/~bdm/nauty/

• Simplifying the 3D matching instance using properties of the puzzle,

e.g., using that a column only contains two of {1, 2, 3}.

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP SAT IP

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP SAT IP Hybrid

Experimental Running Time

5 10 15 20 25 30 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Puzzle Size Average Time (sec) Average checking time versus puzzle height for 10000 random width 8 puzzles BF DP SAT IP Hybrid 20 40 60 80 100 Percent SUSP

Experimental Running Time

Width 5

1 10−1 10−2 10−3 10−4 10−5 10−6 10−7 Average Time (sec)

BF DP SAT IP HYB SUSP %

Width 6 Width 7

20 40 60 80 100 Percent SUSP

Width 8

10 20 30 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 Puzzle Size Average Time (sec)

Width 9

10 20 30 Puzzle Size

Width 10

10 20 30 Puzzle Size 20 40 60 80 100 Percent SUSP

Outline

1 Matrix Multiplication 2 Cohn-Umans Framework 3 Checking Strong Uniquely Solvable Puzzles 4 Searching for Strong Uniquely Puzzles 5 Conclusions & Future Work

Search

Problem (SUSP-Search) Input: k ∈ N Output: The maximum s ∈ N such that there exists a (s, k)-puzzle that is a strong uniquely solvable puzzle.

Search

Problem (SUSP-Search) Input: k ∈ N Output: The maximum s ∈ N such that there exists a (s, k)-puzzle that is a strong uniquely solvable puzzle.

• Considered constructive approaches to solving this problem that use

SUSP-Check as a subroutine.

• (s, k)-puzzles have sk entries and there are 3sk such puzzles.
• Searching the full space for k > 4 is infeasible.
• Density of SUSPs quickly approaches 0.
• Our approaches are ad hoc and use domain knowledge for heuristics.
• SUSP do not form a matroid, augmentation property fails.

Tree Search

Lemma If P is a SUSP and P′ ⊆ P, then P′ is a SUSP.

Tree Search

Lemma If P is a SUSP and P′ ⊆ P, then P′ is a SUSP.

• This lemma allows us to construct SUSP from the bottom up.

Tree Search

Lemma If P is a SUSP and P′ ⊆ P, then P′ is a SUSP.

• This lemma allows us to construct SUSP from the bottom up.
• BFS allowed us to explore the set of all SUSP for k ≤ 5.
• Initial implementation was parallel and used MPI and Map-Reduce to

maintain the search frontier on Union’s 900-node HPC cluster.

• SAT / IP improvements lead it to be feasible on a single desktop.

Tree Search

Lemma If P is a SUSP and P′ ⊆ P, then P′ is a SUSP.

• This lemma allows us to construct SUSP from the bottom up.
• BFS allowed us to explore the set of all SUSP for k ≤ 5.
• Initial implementation was parallel and used MPI and Map-Reduce to

maintain the search frontier on Union’s 900-node HPC cluster.

• SAT / IP improvements lead it to be feasible on a single desktop.
• For k > 6 we used DFS and A∗ search with a variety of heuristics.
• Unable to explore full search space.
• Accurate heuristics are themselves computationally intensive.
• Ongoing work.

Strong USP Found – Examples

(1,1): 1 (2,2): 1 3 2 1 (3,3): 1 1 1 3 2 1 3 3 2 (5,4): 3 1 3 2 1 2 3 2 1 1 1 3 3 2 1 3 3 3 2 3 (8,5): 3 3 3 1 1 1 1 2 2 1 2 1 3 3 2 3 2 2 2 3 2 1 2 1 3 2 2 3 1 2 3 2 3 2 1 3 1 2 1 1 (14,6): 2 3 3 1 1 1 2 1 1 2 1 1 3 3 1 2 1 1 3 2 2 2 1 1 2 3 1 1 2 1 2 2 3 1 2 1 3 3 1 3 2 1 3 2 3 3 2 1 2 1 1 3 1 2 2 3 1 3 2 2 3 1 1 1 1 3 3 3 2 3 1 3 3 3 2 1 2 3 2 2 3 2 2 3

Strong USP Found – Trends and Comparison

[Cohen et al 2005] This work k Maximum s ω Maximum s ω 1 1 = s 3.000 1 = s 3.000 2 2 ≤ s ≤ 3 2.875 2 = s 2.875 3 3 ≤ s ≤ 6 2.849 3 = s 2.849 4 4 ≤ s ≤ 12 2.850 5 = s 2.806 5 4 ≤ s ≤ 24 8 = s 2.777 6 10 ≤ s ≤ 45 2.792 14 ≤ s 2.733 7 10 ≤ s ≤ 86 21 ≤ s 2.722 8 16 ≤ s ≤ 162 30 ≤ s 2.719 9 36 ≤ s ≤ 307 2.739 42 ≤ s 2.718 10 36 ≤ s ≤ 581 64 ≤ s 2.706 11 36 ≤ s ≤ 1098 112 ≤ s 2.678 12 136 ≤ s ≤ 2075 2.696 196 ≤ s 2.653

• [CKSU05]’s construction asymptotically implies ω < 2.48.
• Suggests that there is substantial room for improvement.

Outline

1 Matrix Multiplication 2 Cohn-Umans Framework 3 Checking Strong Uniquely Solvable Puzzles 4 Searching for Strong Uniquely Puzzles 5 Conclusions & Future Work

Future Work

Check

• Look for reductions with o(s3) size – no more 3D matching.
• Verify P is a SUSP by trying to multiply random matrices using P.
• Apply machine learning.

Future Work

Check

• Look for reductions with o(s3) size – no more 3D matching.
• Verify P is a SUSP by trying to multiply random matrices using P.
• Apply machine learning.

Search

• Symmetry reduction.
• Iterated local search.
• Repair from concatenated puzzles.
• Derive better upper bounds.
• Develop theory of dimensionality for puzzles.

Future Work

Check

• Look for reductions with o(s3) size – no more 3D matching.
• Verify P is a SUSP by trying to multiply random matrices using P.
• Apply machine learning.

Search

• Symmetry reduction.
• Iterated local search.
• Repair from concatenated puzzles.
• Derive better upper bounds.
• Develop theory of dimensionality for puzzles.

Conjecture There is a construction that takes SUSPs of size (s1, k1) and (s2, k2) and produces a (s1 + s2, max(k1, k2) + 1)-puzzle that is a SUSP.

Future Work

Check

• Look for reductions with o(s3) size – no more 3D matching.
• Verify P is a SUSP by trying to multiply random matrices using P.
• Apply machine learning.

Search

• Symmetry reduction.
• Iterated local search.
• Repair from concatenated puzzles.
• Derive better upper bounds.
• Develop theory of dimensionality for puzzles.

Conjecture There is a construction that takes SUSPs of size (s1, k1) and (s2, k2) and produces a (s1 + s2, max(k1, k2) + 1)-puzzle that is a SUSP.

Thanks for listening!