Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max Karl Bringmann, Marvin Künnemann, Karol Węgrzycki Max Planck Institute Saarbrücken, University of Warsaw STOC 2019
All-Pairs Shortest Path Problem APSP: Given a directed graph G with positive edge weights in { 1 , . . . , W } . Compute shortest path distance between all pairs of vertices. 1 2 3 1 1 1 3 3 3 1 � n 3 � n 3 / 2 Ω( √ log n ) [Williams’14] Algorithms: O [Floyd’62,Warshall’62] , � n 3 − δ � APSP-hypothesis: ∀ δ > 0 : APSP has no O algorithm (even for W = poly( n ) ). [Vassilevska-Williams, Williams ’10]
All-Pairs Shortest Path Problem APSP: Given a directed graph G with positive edge weights in { 1 , . . . , W } . Compute shortest path distance between all pairs of vertices. 1 1 2 2 3 1 1 1 1 3 3 3 1 1 � n 3 � n 3 / 2 Ω( √ log n ) [Williams’14] Algorithms: O [Floyd’62,Warshall’62] , � n 3 − δ � APSP-hypothesis: ∀ δ > 0 : APSP has no O algorithm (even for W = poly( n ) ). [Vassilevska-Williams, Williams ’10]
Approximate APSP (1 + ε ) approximate APSP: Compute (1 + ε ) -approximation for all pairwise distances � n ω � Algorithm: � O ε log W [Zwick’02] This yields the same upper bound for approximate graph characteristics , such as: Diameter, Radius, Median, Minimum Weight Triangles, Minimum Weight Cycle, . . . This is state-of-the-art for: ◮ directed graphs for any constant ε > 0 , ◮ undirected graphs for any ε ∈ (0 , 1) . � n 2 . 373 � O ( n ω ) ≤ O is the running time of fast matrix multiplication.
� n ω � Zwick’s � O Algorithm, part I ε log W MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . � n 3 � � O ( Wn ω ) Algorithms: O ,
� n ω � Zwick’s � O Algorithm, part I ε log W MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . � n 3 � � O ( Wn ω ) Algorithms: O , Theorem [Zwick’02] If approximate MinPlusProduct in time � O ( n c /ε ) then approximate APSP in time � O ( n c /ε ) . ( c is some constant) Sketch of the reduction: Given adjacency matrix A 1. Add self-loops with cost 0 , 2. Square ⌈ log n ⌉ times using approximate MinPlusProduct 3. D := A 2 ⌈ log n ⌉ 4. Then D i,j is a distance between i and j .
� n ω � Zwick’s � O Algorithm, part II ε log W MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . � n 3 � � O ( Wn ω ) Algorithms: O , � n ω � Theorem ( (1 + ε ) -approximate MinPlusProduct in � O ε log W ) [Zwick’02] For q := 1 , 2 , 4 , . . . , W : Search for all outputs C i,j ∈ [ q, 2 q ] ◮ Remove all entries > 2 q of A, B , (by setting to ∞ ) ◮ Round all entries in A, B to multiples of εq , 1 ◮ Scale all entries by εq , (hence, W = O (1 /ε ) ) ◮ Run exact � O ( Wn ω ) algorithm for MinPlusProd.
Approximate APSP (1 + ε ) approximate APSP: Compute (1 + ε ) -approximation for all pairwise distances � n ω � Algorithm: � O ε log W [Zwick’02] Is this optimal? ◮ Improving dependence on n to n ω − 0 . 01 would yield improved algorithm for BMM. ◮ Improving dependence on ε to 1 /ε o (1) would violate APSP-hypothesis. ◮ What about log W factor? � n ω � Can we get strongly polynomial time � O ? ε We care because for floating-point numbers log W factors can be large and because of theoretical interest in strongly polynomial algorithms.
Computational Model � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Are we cheating? Input numbers already consist of log W bits. . .
Computational Model � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Are we cheating? Input numbers already consist of log W bits. . . Model for this talk : Number of arithmetic operations on RAM machine � n ω � � n ω � [Zwick’02] uses � arithmetic operations; � O ε log W O could be possible. ε In the paper we also consider more restrictive models, i.e., bit complexity with log W � n ω � ε log 2 W bit integers ( [Zwick’02] runs in � O time), bit complexity with floating-point approximation . In principle log W improvement could be possible in all of them.
Computational Model � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Are we cheating? Input numbers already consist of log W bits. . . Model for this talk : Number of arithmetic operations on RAM machine � n ω � � n ω � [Zwick’02] uses � arithmetic operations; � O ε log W O could be possible. ε In the paper we also consider more restrictive models, i.e., bit complexity with log W � n ω � ε log 2 W bit integers ( [Zwick’02] runs in � O time), bit complexity with floating-point approximation . In principle log W improvement could be possible in all of them. Input Format : Floating-Point Representation � � We approximate each input number by floating point (1 + x ) · 2 y with O log 1 -bit ε mantissa and O (log log W ) -bit exponent. Note, that changing any input integer by a factor (1 + ε ) still yields approximation.
Main Algorithmic Results � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Theorem [Bringmann, Künnemann, W’19] ◮ Computing graph characteristics (e.g., Diameter, Radius, Median,. . . ) in � n ω � � O time. ε Note that n ω ≤ n 2 . 373 and n 3+ ω ≤ n 2 . 687 . 2
Main Algorithmic Results � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Theorem [Bringmann, Künnemann, W’19] ◮ Computing graph characteristics (e.g., Diameter, Radius, Median,. . . ) in � n ω � � O time. ε � n ω � ◮ Approximate APSP on undirected graphs in � O time. ε Note that n ω ≤ n 2 . 373 and n 3+ ω ≤ n 2 . 687 . 2
Main Algorithmic Results � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Theorem [Bringmann, Künnemann, W’19] ◮ Computing graph characteristics (e.g., Diameter, Radius, Median,. . . ) in � n ω � � O time. ε � n ω � ◮ Approximate APSP on undirected graphs in � O time. ε � � 3+ ω ◮ Approximate APSP on directed graphs in � 2 /ε O n time and is equivalent to exact MinMaxProduct. Note that n ω ≤ n 2 . 373 and n 3+ ω ≤ n 2 . 687 . 2
Focus of this talk � n ω � � n ω � Approximate APSP: � [Zwick’02] . Can we get � O ε log W O time? ε Theorem [Bringmann, Künnemann, W’19] � � 3+ ω Approximate APSP on directed graphs in � 2 /ε O n time and is equivalent to exact MinMaxProduct. Why should you care? ◮ First strongly polynomial approximation for APSP (subcubic in n ), ◮ Equivalence between approximate and exact problem, ◮ Conditional lower bound (under nonstandard conjecture).
Basic Ingredients MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . Ingredient 1 [Zwick’02] : If approximate MinPlusProduct in time � O ( n c /ε ) , then approximate APSP in time � O ( n c /ε ) (and vice versa).
Basic Ingredients MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . Ingredient 1 [Zwick’02] : If approximate MinPlusProduct in time � O ( n c /ε ) , then approximate APSP in time � O ( n c /ε ) (and vice versa). MinMaxProduct : Given A, B ∈ R n × n . Their (min , max) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] max { A i,k , B k,j }. � � 3+ ω Ingredient 2 [Duan,Pettie’09] : MinMaxProduct can be computed in � O n . 2 MinMaxProduct is equivalent to All-Pairs Bottleneck Paths (APBP) [VVWY’07] .
Basic Ingredients MinPlusProduct : Given A, B ∈ R n × n . Their (min , +) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] A i,k + B k,j . Ingredient 1 [Zwick’02] : If approximate MinPlusProduct in time � O ( n c /ε ) , then approximate APSP in time � O ( n c /ε ) (and vice versa). MinMaxProduct : Given A, B ∈ R n × n . Their (min , max) -product is matrix C ∈ R n × n with C i,j = min k ∈ [ n ] max { A i,k , B k,j }. � � 3+ ω Ingredient 2 [Duan,Pettie’09] : MinMaxProduct can be computed in � O n . 2 MinMaxProduct is equivalent to All-Pairs Bottleneck Paths (APBP) [VVWY’07] . Goal: Prove equivalence between approximate MinPlusProduct and exact MinMaxProduct
Overview of previous slide Apx APSP ≡ Apx MinPlusProd ≡ MinMaxProd ≡ APBP Next Goal: For any c ≥ 2 , the following are equivalent : � � ◮ Approximate MinPlusProd in strongly polynomial � n c O time. poly( ε ) ◮ Exact MinMaxProd in � O ( n c ) time.
Goal 1: If MinMax in time � O ( n c ) then ApxMinPlus in � O ( n c /ε ) Clearly, A i,k + B k,j ≈ max { A i,k , B k,j } up to a factor 2 . In particular, MinPlusProd ( A, B ) ≈ MinMaxProd ( A, B ) up to a factor 2 . Can we reduce the factor 2 to (1 + ε ) ?
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