pushkar tripathi georgia institute of technology approximability of combinatorial optimization...
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- Slide 1
- Pushkar Tripathi Georgia Institute of Technology Approximability of Combinatorial Optimization Problems with Submodular Cost Functions Based on joint work with Gagan Goel, Chinmay Karande, and Wang Lei
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- Motivation Network Design Problem Objective: Find minimum spanning tree that can be built collaboratively by these agents f h g
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- Additive Cost Function Functions which capture economies of scale cost(a) = 1 cost(b) = 1 cost(a,b) = 2 cost(a) = 1 cost(b) = 1 cost(a,b) = 1.5 How to mathematically model these functions? - We use Submodular Functions as a starting point. Can one design efficient approximation algorithms under Submodular Cost Functions? How to mathematically model these functions? - We use Submodular Functions as a starting point. Can one design efficient approximation algorithms under Submodular Cost Functions?
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- Assumptions over cost functions Normalized: Monotone: Decreasing Marginal: Submodularity + + Submodular Functions
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- General Framework Ground set X and collection C 2 X C: set of all tours, set of all spanning trees k agents, each specifies f i : 2 X R + f i is submodular and monotone Find S 1, , S k such that: [ S i 2 C i f i ( S i ) is minimized ORACLE S f(S)
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- Our Results Lower Bounds : Information theoretic Upper Bounds : Rounding of configurational LPs, Approximating sumdodular functions and Greedy Problem Upper BoundLower BoundUpper BoundLower Bound Vertex Cover 2 2 - 2. log n (log n) Shortest Path O (n 2/3 )(n 2/3 ) O (n 2/3 )(n 2/3 ) Spanning Treen (n) n Perfect Matchingn (n) n Multiple AgentsSingle Agent
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- Selected Related Work [Grtschel, Lovsz, Schrijver 81] Minimizing non-monotone submodular function is poly-time [Feige, Mirrokni, Vondrak 07] Maximizing non-monotone function is hard. 2/5-Approximation Algorithm. [Calinescu, Chekuri, Pal, Vondrak 08] Maximizing monotone function subject to Matroid constraint: 1-1/e Approximation. [Svitkina, Fleischer 09] Upper and lower bounds for Submodular load balancing, Sparsest Cut, Balanced Cut [Iwata, Nagano 09] Bounds for Submodular Vertex Cover, Set Cover [Chekuri, Ene 10] Bounds for Submodular Multiway Partition
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- In this talk Submodular Shortest Path with single agent O(n 2/3 ) approximation algorithm Matching hardness of approximation
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- In this talk Submodular Shortest Path with single agent O(n 2/3 ) approximation algorithm Matching hardness of approximation
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- Submodular Shortest Path s t Given: Graph G, Two nodes s and t f : 2 E R + Submodular, Monotone Goal: Find path P s.t. f(P) is minimized G=(V,E) |V| =n, |E| =m
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- t Attempt 1: Approximate by Additive function Let w e = f({e}) Idea : w e OPT w e s 2. Pruning: Remove edges costlier than e* 1. Guess e* = argmax{ w e | e 2 OPT } 3. Search: Find the shortest length s-t path in the residual graph ALG diameter(G). w e* diameter(G).OPT e 2 OPT
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- Attempt 2: Ellipsoid Approximation Johns theorem : For every polytope P, there exists an ellipsoid contained in it that can be scaled by a factor of O( n) to contain P [GHIM 09]: If the convex body is a polymatroid, then there is a poly-time algorithm to compute the ellipse. P
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- Attempt 2: Ellipsoid Approximation P [GHIM 09]: If the convex body is a polymatroid, then there is a poly-time algorithm to compute the ellipse. S: e 2 S x(e) f(S) e: x(e) 0 f: Submodular, monotone Polymatroid
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- Approximating Submodular Functions X f : Monotone submodular function g(S) = d e e 2 S g(S) f(S) n g(S) d1d1 d2d2 d6d6 d4d4 d5d5 d3d3 Polynomial time |X| = n
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- Attempt 2: Ellipsoid Approximation f: 2 E R + Submodular, Monotone STEP 1: STEP 2: Min g(S) s.t. S 2 PATH(s,t) * Minimizing over g(S) is equivalent to minimizing just the additive part [GHIM 09] Analysis : f(P) g(P) g(O) f(O) g(S): = d e P: Optimum path under g O: Optimum path under f {de}{de} EE EE EE
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- Recap. Approximating by linear functions : Works for graphs with small diameter Approximating by ellipsoid functions : Works for sparse graphs n/2 Dense Graph with large diameter
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- Algorithm for Shortest Path STEP 1: Pruning - Guess edge e* = argmax {w e | e OPT path} - Remove edges costlier than w e*
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- Algorithm for Shortest Path STEP 1: Pruning - Guess edge e* = argmax {w e | e OPT path} - Remove edges costlier than w e* STEP 2 : Contraction - if v, s.t. degree(v) > n 1/3, contract neighborhood of v - repeat
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- s t s t Dense connected component
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- Algorithm for Shortest Path STEP 1: Pruning - Let w e = f({e}) - Guess edge e* = argmax {w e | e OPT path} - Remove edges costlier than w e* STEP 2 : Contraction - if v, s.t. degree(v) < n 1/3, contract neighborhood of v - repeat STEP 3 : Ellipsoid Approximation - Calculate ellipsoidal approximation (d,g) for the residual graph
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- Algorithm for Shortest Path STEP 1: Pruning - Let w e = f({e}) - Guess edge e* = argmax {w e | e OPT path} - Remove edges costlier than w e* STEP 2 : Contraction - if v, s.t. degree(v) < n 1/3, contract neighborhood of v - repeat STEP 3 : Ellipsoid Approximation - Calculate ellipsoidal approximation (d,g) for the residual graph STEP 4 : Search - Find shortest s-t path according to g.
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- s t
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- Algorithm for Shortest Path STEP 1: Pruning - Let w e = f({e}) - Guess edge e* = argmax {w e | e OPT path} - Remove edges costlier than w e* STEP 2 : Contraction - if v, s.t. degree(v) < n 1/3, contract neighborhood of v - repeat STEP 3 : Ellipsoid Approximation - Calculate ellipsoidal approximation (d,g) for the residual graph STEP 4 : Search - Find shortest s-t path according to g. STEP 5 : Reconstruction - Replace the path through each contracted vertex with one having the fewest edges.
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- s t Path having fewest edges
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- Analysis s t P1 P2 R
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- Bounding the cost of P1 s t P1 P2 Has at most n 4/3 edges R f(P 1 ) E(R).g(P 1 ) E(R).g(OPT) E(R).f(OPT) n 2/3 f(OPT)
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- Bounding the cost of P2 s t Diam(G i ) | G i |/ n 1/3 f(P 2 ) (dia(G 1 ) +.. +dia(G k ) ) w e* (|G 1 | / n 1/3 + . ) w e* (n / n 1/3 ) w e* n 2/3 f(OPT) G1G1 G2G2 G3G3
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- In this talk Submodular Shortest Path with single agent O(n 2/3 ) approximation algorithm Matching hardness of approximation
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- Information Theoretic Lower Bound Polynomial number of queries to the oracle Algorithm is allowed unbounded amount of time to process the results of the queries Not contingent on P vs NP f S1 f(S1) S2 f(S2) S3 f(S2)
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- General Technique Cost functions f, g satisfying OPT( f ) >> OPT( g ) f (S) = g(S) for most sets S A any randomized algorithm f(Q ) = g( Q ) with high probability for every query Q made by A. Probability over random bits in A.
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- Yaos Lemma f(Q) = g(Q) with high probability for every query Q made by randomized algorithm A. f and a distribution D from which we choose g, such that for an arbitrary query Q, f(Q) = g(Q) with high probability
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- Non-combinatorial Setting X : Ground set f(S) = min{ |S|, } D : R X, |R| = g R (S) = min{| S R c | + min( S R, ) } D : R X, |R| = g R (S) = min{| S R c | + min( S R, ) }
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- Optimal Query Claim : Optimal query has size Case 1 : |Q| < Probability can only increase if we increase |Q|
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- Case 2 : |Q| > Probability can only increase if we decrease |Q| Optimal query size to distinguish f and g R is
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- Distinguishing f and g R = (1+ ) E[|Q R|] f and g are hard to distinguish Chernoff Bounds
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- Hardness of learning submodular functions Set = n 1/2 log n Optimal query size = = n 1/2 log n |R| = = n 1/2 log n E[ Q R] = log 2 n = (1+ ) E[ Q R] = (1+ ) log 2 n Super logarithmic Corollary : Hard to learn a submodular function to a factor better than n 1/2 /log n in polynomial value queries. f and g are indistinguishable f(R) = min{ |R|, } = |R| = = n 1/2 log n g R (R ) = min{| R R c | + min( R R, ) } = = log 2 n
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- Randomly chosen set may not be a feasible solution in the combinatorial setting. Eg. Randomly chosen set of edges rarely yield a s-t path. Difficulty in Combinatorial Setting Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution. Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution.
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- Base Graph G ...st n 2/3 levels n 1/3 vertices
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- Functions f and g . st Y B f(S) = f( S B ) & g(S) = g( S B )
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- Functions f and g . st Y B f(S) = min( |S B|, )
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- Functions f and g Y B .. st .. g R (S) = min{| S R B| + min( S R B, )} Uniform random subset of B of size Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution. Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution.
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- Functions f and g . st Y B g R (S) = min{| S R B| + min( S R B, ) Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution. Solution : 1.Do not choose R randomly from the entire domain X. 2.Use a subset of R as a proxy for the solution. R = n 2/3 log 2 n
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- Setting the constants Set = n 2/3 log 2 n Optimal Query size = = n 2/3 log 2 n = log 2 n f and g are indistinguishable f(OPT) = min{ |R|, } = |R| = = O( n 2/3 log 2 n) g R (OPT ) = min{| R R c | + min( R R, ) } = = log 2 n Theorem : Submodular Shortest Path problem is hard to approximate to a factor better than O(n 2/3 )
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- Problem Upper BoundLower BoundUpper BoundLower Bound Vertex Cover 2 2 - 2. log n (log n) Shortest Path O (n 2/3 )(n 2/3 ) O (n 2/3 )(n 2/3 ) Spanning Treen (n) n Perfect Matchingn (n) n Multiple AgentsSingle Agent n: # of vertices in graph G Whats the right model to study economies of scale? Summary
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- Newer Models Discount Models f h g E R
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- Task: Minimize sum of payments Cost Payment f(a) + f(b) + f(c) . Sub modular functions
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- Approximability under Discounted Costs[GTW 09] ProblemLower BoundUpper bound Edge CoverO(log n) Spanning TreeO(log n) Shortest Pathn Minimum Perfect Matching O(poly log n)n O(log n) O(poly log n) O(log n)
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- Shortest Path : O(log c n) hardness Set Cover Instance U S s t Agents - Cost of every edge is 1 1 1 Claim : Set cover of size |S| Shortest path of length |S|
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- Hardness Gap Amplification s t s t Original Instance Harder Instance Replace each edge by a copy of the original graph. Edges of the same color get the same copy. Edges of different colors gets copies with new colors(agents)
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- Claim : The new instance has a solution of cost 2 iff the original instance has a solution of cost . For any fixed constant c iterate this construction c times to further amplify the lower bound to O(log c n).
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- Q.E.F
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- Why is it so hard to distinguish f and g ? Observation: f R (S ) is at most g(S ) for any set S. Case 1: Small size queries - |Q | n This probability can only increase if we increase |Q |
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- Case 2: Large size queries - |Q | n This probability can only increase if we decrease |Q |
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- Combinatorial Optimization C - Ground set f - Valuation function over subsets of C X - Collection of some subsets C having a special property Task - Find the set in X that has minimum cost under a given valuation function.
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- General Technique cont. f S2 S3 S1f(S1) = g(S1) f(S2) = g(S2) f(S3) = g(S3) A cannot distinguish between f and g Output is at least OPT( g ) OPT( g ) OPT( f )
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- Plan Fix a cost function f Fix a distribution D of functions such that for every g in D OPT(f ) >> OPT (g) For an arbitrary query Q, f(Q) = g(Q) with high probability
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- Optimal size queries Queries of size n- |Q | = n