Approximating Cumulative Pebbling Cost is Unique Games Hard Jeremiah - PowerPoint PPT Presentation

Approximating Cumulative Pebbling Cost is Unique Games Hard Jeremiah Blocki 1 , Seunghoon Lee 1 , Samson Zhou 2 1 Department of Computer Science, Purdue University 2 School of Computer Science, Carnegie Mellon University November 6, 2019 Jeremiah Blocki, Seunghoon Lee, Samson Zhou Approximating Cumulative Pebbling Cost is Unique Games Hard 1/40 1 = 40

Contents Summary of Our Work 2/40 Approximating Cumulative Pebbling Cost is Unique Games Hard Jeremiah Blocki, Seunghoon Lee, Samson Zhou Open Questions The Main Result and Concluding Remark Superconcentrators / Superconcentrators Overlay Svensson’s Result of Unique Games Hardness Technical Ingredients Depth Robustness of a Graph Unique Games Conjecture Preliminaries The Main Result Graph Pebbling and Cumulative Pebbling Cost Introduction Reducing the Indegree: � -Extreme Depth Robust Graphs Main Theorem: Unique Games Hardness of cc ( G ) 2 = 40

Goal. Place pebbles on all sink nodes. Pebbling Rules. (informal) Technical Ingredients. constant factor 3/40 take minimum Results. take minimum memory-hard functions (Parallel) Pebbling Example. Jeremiah Blocki, Seunghoon Lee, Samson Zhou minimum cost pebbling take minimum Problem Statement. take minimum Overview (Parallel) Graph Pebbling. We Are Here Approximating Cumulative Pebbling Cost is Unique Games Hard (Parallel) Graph Pebbling and Cumulative Pebbling Cost ( cc ( G ) ) � Initially, the graph is unpebbled and start with the root nodes. � We can add a new pebble only if its parents were all pebbled. � Pebbling example � (Parallel) We can place multiple pebbles at the same time. � Cumulative Pebbling Cost of G � We can discard pebbles at any time if not needed. � Given a DAG G fjnd the (approx.) 2 2 2 Signifjcance of cc ( G ) . � Analysis of data-independent � Amortization / Parallelism 1 1 1 4 4 4 5 5 � Unique Games Hard to 3 3 3 approximate cc ( G ) for any P 1 = f 1 g ; P 2 = f 2 ; 3 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g � Indegree reduction using � P t � P t � P t � P t � -extreme depth robust graphs ∴ ∴ ∴ ∴ cc ( G ) cc ( G ) cc ( G ) cc ( G ) i =1 j P i j = 1 + 2 + 2 + 1 = 6 : i =1 j P i j = 1 + 2 i =1 j P i j = 1 i =1 j P i j = 1 + 2 + 2 � Superconcentrator overlay | {z } | {z } | {z } | {z } 3 = 40

Goal. Place pebbles on all sink nodes. Pebbling Rules. (informal) Technical Ingredients. constant factor 3/40 take minimum Results. take minimum memory-hard functions (Parallel) Pebbling Example. Jeremiah Blocki, Seunghoon Lee, Samson Zhou minimum cost pebbling take minimum Problem Statement. take minimum Overview (Parallel) Graph Pebbling. We Are Here Approximating Cumulative Pebbling Cost is Unique Games Hard (Parallel) Graph Pebbling and Cumulative Pebbling Cost ( cc ( G ) ) � Initially, the graph is unpebbled and start with the root nodes. � We can add a new pebble only if its parents were all pebbled. � Pebbling example � (Parallel) We can place multiple pebbles at the same time. � Cumulative Pebbling Cost of G � We can discard pebbles at any time if not needed. � Given a DAG G fjnd the (approx.) 2 2 2 Signifjcance of cc ( G ) . � Analysis of data-independent � Amortization / Parallelism 1 1 1 4 4 4 5 5 � Unique Games Hard to 3 3 3 approximate cc ( G ) for any P 1 = f 1 g ; P 2 = f 2 ; 3 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g � Indegree reduction using � P t � P t � P t � P t � -extreme depth robust graphs ∴ ∴ ∴ ∴ cc ( G ) cc ( G ) cc ( G ) cc ( G ) i =1 j P i j = 1 + 2 + 2 + 1 = 6 : i =1 j P i j = 1 + 2 i =1 j P i j = 1 i =1 j P i j = 1 + 2 + 2 � Superconcentrator overlay | {z } | {z } | {z } | {z } 3 = 40

Goal. Place pebbles on all sink nodes. Pebbling Rules. (informal) Technical Ingredients. constant factor 3/40 take minimum Results. take minimum memory-hard functions (Parallel) Pebbling Example. Jeremiah Blocki, Seunghoon Lee, Samson Zhou minimum cost pebbling take minimum Problem Statement. take minimum Overview (Parallel) Graph Pebbling. We Are Here Approximating Cumulative Pebbling Cost is Unique Games Hard (Parallel) Graph Pebbling and Cumulative Pebbling Cost ( cc ( G ) ) � Initially, the graph is unpebbled and start with the root nodes. � We can add a new pebble only if its parents were all pebbled. � Pebbling example � (Parallel) We can place multiple pebbles at the same time. � Cumulative Pebbling Cost of G � We can discard pebbles at any time if not needed. � Given a DAG G fjnd the (approx.) 2 2 2 Signifjcance of cc ( G ) . � Analysis of data-independent � Amortization / Parallelism 1 1 1 4 4 4 5 5 � Unique Games Hard to 3 3 3 approximate cc ( G ) for any P 1 = f 1 g ; P 2 = f 2 ; 3 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g � Indegree reduction using � P t � P t � P t � P t � -extreme depth robust graphs ∴ ∴ ∴ ∴ cc ( G ) cc ( G ) cc ( G ) cc ( G ) i =1 j P i j = 1 + 2 + 2 + 1 = 6 : i =1 j P i j = 1 + 2 i =1 j P i j = 1 i =1 j P i j = 1 + 2 + 2 � Superconcentrator overlay | {z } | {z } | {z } | {z } 3 = 40

Goal. Place pebbles on all sink nodes. Pebbling Rules. (informal) Technical Ingredients. constant factor 3/40 take minimum Results. take minimum memory-hard functions (Parallel) Pebbling Example. Jeremiah Blocki, Seunghoon Lee, Samson Zhou minimum cost pebbling take minimum Problem Statement. take minimum Overview (Parallel) Graph Pebbling. We Are Here Approximating Cumulative Pebbling Cost is Unique Games Hard (Parallel) Graph Pebbling and Cumulative Pebbling Cost ( cc ( G ) ) � Initially, the graph is unpebbled and start with the root nodes. � We can add a new pebble only if its parents were all pebbled. � Pebbling example � (Parallel) We can place multiple pebbles at the same time. � Cumulative Pebbling Cost of G � We can discard pebbles at any time if not needed. � Given a DAG G fjnd the (approx.) 2 2 2 Signifjcance of cc ( G ) . � Analysis of data-independent � Amortization / Parallelism 1 1 1 4 4 4 5 5 � Unique Games Hard to 3 3 3 approximate cc ( G ) for any P 1 = f 1 g ; P 2 = f 2 ; 3 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g � Indegree reduction using � P t � P t � P t � P t � -extreme depth robust graphs ∴ ∴ ∴ ∴ cc ( G ) cc ( G ) cc ( G ) cc ( G ) i =1 j P i j = 1 + 2 + 2 + 1 = 6 : i =1 j P i j = 1 + 2 i =1 j P i j = 1 i =1 j P i j = 1 + 2 + 2 � Superconcentrator overlay | {z } | {z } | {z } | {z } 3 = 40

Goal. Place pebbles on all sink nodes. Pebbling Rules. (informal) Technical Ingredients. constant factor 3/40 take minimum Results. take minimum memory-hard functions (Parallel) Pebbling Example. Jeremiah Blocki, Seunghoon Lee, Samson Zhou minimum cost pebbling take minimum Problem Statement. take minimum Overview (Parallel) Graph Pebbling. We Are Here Approximating Cumulative Pebbling Cost is Unique Games Hard (Parallel) Graph Pebbling and Cumulative Pebbling Cost ( cc ( G ) ) � Initially, the graph is unpebbled and start with the root nodes. � We can add a new pebble only if its parents were all pebbled. � Pebbling example � (Parallel) We can place multiple pebbles at the same time. � Cumulative Pebbling Cost of G � We can discard pebbles at any time if not needed. � Given a DAG G fjnd the (approx.) 2 2 2 Signifjcance of cc ( G ) . � Analysis of data-independent � Amortization / Parallelism 1 1 1 4 4 4 5 5 � Unique Games Hard to 3 3 3 approximate cc ( G ) for any P 1 = f 1 g ; P 2 = f 2 ; 3 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g ; P 4 = f 5 g P 1 = f 1 g P 1 = f 1 g ; P 2 = f 2 ; 3 g ; P 3 = f 3 ; 4 g � Indegree reduction using � P t � P t � P t � P t � -extreme depth robust graphs ∴ ∴ ∴ ∴ cc ( G ) cc ( G ) cc ( G ) cc ( G ) i =1 j P i j = 1 + 2 + 2 + 1 = 6 : i =1 j P i j = 1 + 2 i =1 j P i j = 1 i =1 j P i j = 1 + 2 + 2 � Superconcentrator overlay | {z } | {z } | {z } | {z } 3 = 40