Can linear programs solve NPhard problems?


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1 Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf
2 Linear programs Can linear programs solve NPhard problems? p. 2/9
3 Can linear programs solve NPhard problems? p. 2/9 Linear programs A factory produces 2 types of goods
4 Can linear programs solve NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard problems? p. 3/9
11 Can linear programs solve NPhard 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 NPhard 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 NPhard 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, worstcase exponential time
14 Can linear programs solve NPhard 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, worstcase exponential time Ellipsoid method (Khachiyan 79): takes polynomial time in the worst case, but is not practical
15 Can linear programs solve NPhard 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, worstcase 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 NPhard problems by LP? Can linear programs solve NPhard problems? p. 4/9
17 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once
18 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize LP for TSP would show P = NP
19 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize LP for TSP would show P = NP Swart claimed to have found such LPs
20 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize 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 NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize 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 NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize 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 20year open problem: what about nonsymmetric LP?
23 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize 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 20year open problem: what about nonsymmetric LP? Sometimes nonsymmetry helps a lot! (Kaibel et al 10)
24 Can linear programs solve NPhard problems? p. 4/9 Can we solve NPhard problems by LP? Famous NPhard problem: traveling salesman problem Find shortest cycle that goes through each of n cities exactly once A polynomialsize 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 20year open problem: what about nonsymmetric LP? Sometimes nonsymmetry 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 NPhard problems? p. 5/9
26 Can linear programs solve NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard problems? p. 6/9
36 Can linear programs solve NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much.
37 Can linear programs solve NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular ngon 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 NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular ngon 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 NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular ngon 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 NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular ngon 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 NPhard problems? p. 6/9 Extended formulations of polytopes Sometimes extra variables/dimensions can reduce size very much. Regular ngon 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 NPhard problems? p. 7/9
43 Can linear programs solve NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard 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 NPhard problems? p. 8/9
50 Can linear programs solve NPhard 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 NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard problems, e.g. MaxClique
52 Can linear programs solve NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 aconstraints for first 2 n rows, remaining facets for other rows S =. M ab..
55 Can linear programs solve NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 aconstraints for first 2 n rows, remaining facets for other rows xc(cor(n)) S =. M ab..
56 Can linear programs solve NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 aconstraints for first 2 n rows, remaining facets for other rows xc(cor(n)) = rank + (S) S =. M ab..
57 Can linear programs solve NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 aconstraints 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 NPhard problems? p. 8/9 Lower bound for correlation polytope Correlation polytope: COR(n) = conv{bb T b {0,1} n } Occurs naturally in NPhard 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 aconstraints 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 NPhard problems? p. 9/9
60 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope:
61 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n
62 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n Recently improved to 2 n by Rothvoss
63 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUTpolytope has extension complexity 2 n
64 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUTpolytope has extension complexity 2 n For specific graphs, the stableset polytope has extension complexity 2 n
65 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUTpolytope has extension complexity 2 n For specific graphs, the stableset polytope has extension complexity 2 n So every linear program based on extended formulations needs exponentially many constraints
66 Can linear programs solve NPhard problems? p. 9/9 Consequences for other polytopes Via reduction from the correlation polytope: TSPpolytope has extension complexity 2 n Recently improved to 2 n by Rothvoss CUTpolytope has extension complexity 2 n For specific graphs, the stableset 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 NPhard problems, and refutes all P=NP proofs à la Swart
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