Can linear programs solve NP-hard problems?
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1 Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf
2 Linear programs Can linear programs solve NP-hard problems? p. 2/9
3 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods
4 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4
5 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4 It has 4 kinds of resources: 650 of R 1, 100 of R 2,...
6 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4 It has 4 kinds of resources: 650 of R 1, 100 of R 2,... - type 1 uses 34 units of R 1, type 2 uses 16 units of R 1
7 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4 It has 4 kinds of resources: 650 of R 1, 100 of R 2,... - type 1 uses 34 units of R 1, type 2 uses 16 units of R 1 Maximizing profit is linear program
8 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4 It has 4 kinds of resources: 650 of R 1, 100 of R 2,... - type 1 uses 34 units of R 1, type 2 uses 16 units of R 1 Maximizing profit is linear program: max x 1 +4x 2 s.t. 34x 1 +16x x 1,x 2 0
9 Can linear programs solve NP-hard problems? p. 2/9 Linear programs A factory produces 2 types of goods - unit of type 1 gives profite1; type 2 gives profite4 It has 4 kinds of resources: 650 of R 1, 100 of R 2,... - type 1 uses 34 units of R 1, type 2 uses 16 units of R 1 Maximizing profit is linear program: max x 1 +4x 2 Feasible region is a polytope s.t. 34x 1 +16x x 1,x 2 0
10 A bit of history of LPs Can linear programs solve NP-hard problems? p. 3/9
11 Can linear programs solve NP-hard problems? p. 3/9 A bit of history of LPs Simplex algorithm: developed by Kantorovich and Dantzig for use in WWII, published by Dantzig in 1947.
12 Can linear programs solve NP-hard problems? p. 3/9 A bit of history of LPs Simplex algorithm: developed by Kantorovich and Dantzig for use in WWII, published by Dantzig in Walks along vertices of polytope, optimizing objective
13 Can linear programs solve NP-hard problems? p. 3/9 A bit of history of LPs Simplex algorithm: developed by Kantorovich and Dantzig for use in WWII, published by Dantzig in Walks along vertices of polytope, optimizing objective Usually efficient in practice, worst-case exponential time
14 Can linear programs solve NP-hard problems? p. 3/9 A bit of history of LPs Simplex algorithm: developed by Kantorovich and Dantzig for use in WWII, published by Dantzig in Walks along vertices of polytope, optimizing objective Usually efficient in practice, worst-case exponential time Ellipsoid method (Khachiyan 79): takes polynomial time in the worst case, but is not practical
15 Can linear programs solve NP-hard problems? p. 3/9 A bit of history of LPs Simplex algorithm: developed by Kantorovich and Dantzig for use in WWII, published by Dantzig in Walks along vertices of polytope, optimizing objective Usually efficient in practice, worst-case exponential time Ellipsoid method (Khachiyan 79): takes polynomial time in the worst case, but is not practical Interior point method (Karmarkar 84): reasonably efficient in practice
16 Can we solve NP-hard problems by LP? Can linear programs solve NP-hard problems? p. 4/9
17 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once
18 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP
19 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs
20 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs Yannakakis 88: symmetric LPs for TSP are exponential
21 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs Yannakakis 88: symmetric LPs for TSP are exponential Swart s LPs were symmetric, so they couldn t work
22 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs Yannakakis 88: symmetric LPs for TSP are exponential Swart s LPs were symmetric, so they couldn t work 20-year open problem: what about non-symmetric LP?
23 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs Yannakakis 88: symmetric LPs for TSP are exponential Swart s LPs were symmetric, so they couldn t work 20-year open problem: what about non-symmetric LP? Sometimes non-symmetry helps a lot! (Kaibel et al 10)
24 Can linear programs solve NP-hard problems? p. 4/9 Can we solve NP-hard problems by LP? Famous NP-hard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomial-size LP for TSP would show P = NP Swart claimed to have found such LPs Yannakakis 88: symmetric LPs for TSP are exponential Swart s LPs were symmetric, so they couldn t work 20-year open problem: what about non-symmetric LP? Sometimes non-symmetry helps a lot! (Kaibel et al 10) Fiorini, Massar, Pokutta, Tiwary, dw (STOC 12): any LP for TSP needs exponential size
25 Polytopes and optimization problems Can linear programs solve NP-hard problems? p. 5/9
26 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d
27 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces
28 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b}
29 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P
30 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P The size of P is the minimal number of inequalities
31 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P The size of P is the minimal number of inequalities TSP polytope: each cycle in K n is vector {0,1} (n 2)
32 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P The size of P is the minimal number of inequalities TSP polytope: each cycle in K n is vector {0,1} (n 2). TSP(n) is the convex hull of all those vectors
33 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P The size of P is the minimal number of inequalities TSP polytope: each cycle in K n is vector {0,1} (n 2). TSP(n) is the convex hull of all those vectors Solving TSP w.r.t. weight function w ij : minimize the linear function i,j w ijx ij over x TSP(n)
34 Can linear programs solve NP-hard problems? p. 5/9 Polytopes and optimization problems Polytope P: convex hull of finite set of points in R d bounded intersection of finitely many halfspaces Can be written as system of linear inequalities: P = {x R d Ax b} Different systems Ax b can define the same P The size of P is the minimal number of inequalities TSP polytope: each cycle in K n is vector {0,1} (n 2). TSP(n) is the convex hull of all those vectors Solving TSP w.r.t. weight function w ij : minimize the linear function i,j w ijx ij over x TSP(n) TSP(n) has exponential size
35 Extended formulations of polytopes Can linear programs solve NP-hard problems? p. 6/9
36 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much.
37 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R 2 has size n, but is the projection of polytope Q in higher dimension, of size O(logn)
38 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R 2 has size n, but is the projection of polytope Q in higher dimension, of size O(logn) Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently!
39 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R 2 has size n, but is the projection of polytope Q in higher dimension, of size O(logn) Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently! How small can size(q) be?
40 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R 2 has size n, but is the projection of polytope Q in higher dimension, of size O(logn) Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently! How small can size(q) be? Extension complexity: xc(p) = min{size(q) Q projects to P}
41 Can linear programs solve NP-hard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R 2 has size n, but is the projection of polytope Q in higher dimension, of size O(logn) Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently! How small can size(q) be? Extension complexity: xc(p) = min{size(q) Q projects to P} Our goal: strong lower bounds on xc(p) for interesting P
42 How to bound extension complexity? Can linear programs solve NP-hard problems? p. 7/9
43 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j
44 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j NB: every entry is nonnegative; S is not unique
45 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j NB: every entry is nonnegative; S is not unique Yannakakis 88: xc(p) = positive rank of S
46 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j NB: every entry is nonnegative; S is not unique Yannakakis 88: xc(p) = positive rank of S Instead of considering all Q, can focus on slack matrix!
47 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j NB: every entry is nonnegative; S is not unique Yannakakis 88: xc(p) = positive rank of S Instead of considering all Q, can focus on slack matrix! We can use nondeterministic communication complexity to lower bound rank + (S)
48 Can linear programs solve NP-hard problems? p. 7/9 How to bound extension complexity? Slack matrix S of a polytope P = conv(v) with inequalities {A i x b i } and points V = {v j }: S ij = b i A i v j NB: every entry is nonnegative; S is not unique Yannakakis 88: xc(p) = positive rank of S Instead of considering all Q, can focus on slack matrix! We can use nondeterministic communication complexity to lower bound rank + (S) Big problem until now: which polytope to analyze, and how to analyze its slack matrix?
49 Lower bound for correlation polytope Can linear programs solve NP-hard problems? p. 8/9
50 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n }
51 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique
52 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2
53 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00)
54 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00) Take slack matrix S for COR(n), with 2 n vertices bb T for columns, 2 n a-constraints for first 2 n rows, remaining facets for other rows S =. M ab..
55 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00) Take slack matrix S for COR(n), with 2 n vertices bb T for columns, 2 n a-constraints for first 2 n rows, remaining facets for other rows xc(cor(n)) S =. M ab..
56 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00) Take slack matrix S for COR(n), with 2 n vertices bb T for columns, 2 n a-constraints for first 2 n rows, remaining facets for other rows xc(cor(n)) = rank + (S) S =. M ab..
57 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00) Take slack matrix S for COR(n), with 2 n vertices bb T for columns, 2 n a-constraints for first 2 n rows, remaining facets for other rows xc(cor(n)) = rank + (S) rank + (M) S =. M ab..
58 Can linear programs solve NP-hard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NP-hard problems, e.g. MaxClique We can find 2 n valid constraints (indexed by a {0,1} n ) whose slacks w.r.t. vertex bb T are M ab = (1 a T b) 2 Nondeterministic communication complexity of M was already analyzed in the quantum computing! (dw 00) Take slack matrix S for COR(n), with 2 n vertices bb T for columns, 2 n a-constraints for first 2 n rows, remaining facets for other rows S = xc(cor(n)) = rank + (S) rank + (M) 2 Ω(n). M ab..
59 Consequences for other polytopes Can linear programs solve NP-hard problems? p. 9/9
60 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope:
61 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n
62 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n Recently improved to 2 n by Rothvoss
63 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUT-polytope has extension complexity 2 n
64 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUT-polytope has extension complexity 2 n For specific graphs, the stable-set polytope has extension complexity 2 n
65 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUT-polytope has extension complexity 2 n For specific graphs, the stable-set polytope has extension complexity 2 n So every linear program based on extended formulations needs exponentially many constraints
66 Can linear programs solve NP-hard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSP-polytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUT-polytope has extension complexity 2 n For specific graphs, the stable-set polytope has extension complexity 2 n So every linear program based on extended formulations needs exponentially many constraints This rules out many efficient algorithms for NP-hard problems, and refutes all P=NP proofs à la Swart
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