Sub-exponential Approximation Schemes: From Dense to Almost-Sparse. Dimitris Fotakis Michael Lampis Vangelis Paschos
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1 Sub-exponential Approximation Schemes: From Dense to Almost-Sparse Dimitris Fotakis Michael Lampis Vangelis Paschos NTU Athens Université Paris Dauphine Feb 18, STACS 2016
2 Motivation Sub-Exponential Approximation (50 years in a slide) Sub-Exponential Approximation Schemes 2 / 17
3 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Jack Edmonds Juris Hartmanis Richard Stearns Sub-Exponential Approximation Schemes 2 / 17
4 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Everything is NP-hard!( ) ( 70s) Stephen Cook Richard Karp Garey& Johnson Sub-Exponential Approximation Schemes 2 / 17
5 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Everything is NP-hard!( ) ( 70s) So, we should approximate ( 80s) David Johnson V. Vazirani Williamson&Shmoys Sub-Exponential Approximation Schemes 2 / 17
6 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Everything is NP-hard!( ) ( 70s) So, we should approximate ( 80s) Everything is APX-hard!( ) ( 90s) Christos Papadimitriou Sanjeev Arora Johan Håstad Sub-Exponential Approximation Schemes 2 / 17
7 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Everything is NP-hard!( ) ( 70s) So, we should approximate ( 80s) Everything is APX-hard!( ) ( 90s) More than poly-time? Everything ETH-hard ( 00s) Mike Fellows Russell Impagliazzo Fomin&Kratsch Sub-Exponential Approximation Schemes 2 / 17
8 Motivation Sub-Exponential Approximation (50 years in a slide) Efficient = Poly-time ( 60s) Everything is NP-hard!( ) ( 70s) So, we should approximate ( 80s) Everything is APX-hard!( ) ( 90s) More than poly-time? Everything ETH-hard ( 00s) Bottom line: Most problems hard to solve exactly in 2 o(n) time Most problems hard to approximate in n O(1) time Perhaps we can approximate in 2 o(n) time? Sub-Exponential Approximation Schemes 2 / 17
9 Dead on Arrival? Better than 3/2 for TSP, 4/3 for Max-3-DM, 7/8 for Max-3-SAT,..., in sub-exponential time? Sub-Exponential Approximation Schemes 3 / 17
10 Dead on Arrival? Better than 3/2 for TSP, 4/3 for Max-3-DM, 7/8 for Max-3-SAT,..., in sub-exponential time? Probably won t work (at least for Max-3-SAT) Sub-Exponential Approximation Schemes 3 / 17
11 Dead on Arrival? Better than 3/2 for TSP, 4/3 for Max-3-DM, 7/8 for Max-3-SAT,..., in sub-exponential time? Almost-linear PCPs (Moshkovitz& Raz) and P-time hardness (Håstad) give tight inapproximability for Max-3-SAT even for 2 n1 ǫ time. (Credit: Dana Moshkovitz) Sub-Exponential Approximation Schemes 3 / 17
12 Dead on Arrival? Better than 3/2 for TSP, 4/3 for Max-3-DM, 7/8 for Max-3-SAT,..., in sub-exponential time? If this is the normal behavior of APX problems, what s the point of sub-exponential approximation? Is this the normal behavior? What about problems outside APX? E.g., r-approximation in time 2 n/r for Ind. Set ([Chalermsook, Laekhanukit, Nanongkai 13]) r-approximation in time 2 n/r2 for Max Minimal VC log(n/r)-approximation in time 2 n/r for ATSP ([Bonnet,L.,Paschos (next talk!)]). What else? Sub-Exponential Approximation Schemes 3 / 17
13 Strategy We cannot get better than 7/8 for Max-3-SAT in sub-exp time (under ETH). We will therefore try to get something else: Sub-Exponential Approximation Schemes 4 / 17
14 Strategy We cannot get better than 7/8 for Max-3-SAT in sub-exp time (under ETH). We will therefore try to get something else: An island of tractability: Max-k-CSP admits a PTAS (a (1 ǫ)-approximation for all ǫ > 0) for dense instances (Arora, Karger, Karpinski 99), (de la Vega 96) Sub-Exponential Approximation Schemes 4 / 17
15 Strategy We cannot get better than 7/8 for Max-3-SAT in sub-exp time (under ETH). We will therefore try to get something else: An island of tractability: Max-k-CSP admits a PTAS (a (1 ǫ)-approximation for all ǫ > 0) for dense instances (Arora, Karger, Karpinski 99), (de la Vega 96) Extending the island: We give a version of the AKK scheme which can handle sparser instances, at the expense of needing sub-exponential time. Our scheme provides a smooth trade-off For dense instances we get a PTAS As instances gradually get more sparse, we need more time until our scheme does not work any more Sub-Exponential Approximation Schemes 4 / 17
16 Summary of results For any ǫ > 0, δ [0, 1] and fixed k 2 we have the following: Given a Max-k-CSP instance with n k 1+δ constraints We can produce a (1 ǫ)-approximate solution In time 2 O(n1 δ lnn/ǫ 3 ) Sub-Exponential Approximation Schemes 5 / 17
17 Summary of results For any ǫ > 0, δ [0, 1] and fixed k 2 we have the following: Given a Max-k-CSP instance with n k 1+δ constraints We can produce a (1 ǫ)-approximate solution In time 2 O(n1 δ lnn/ǫ 3 ) Note: This includes the AKK PTAS as a special case (δ = 1) Advantage: we provide a smooth trade-off from the easy case (dense instances) to more general cases Sub-Exponential Approximation Schemes 5 / 17
18 Summary of results For any ǫ > 0, δ [0, 1] and fixed k 2 we have the following: Given a Max-k-CSP instance with n k 1+δ constraints We can produce a (1 ǫ)-approximate solution In time 2 O(n1 δ lnn/ǫ 3 ) Note: This includes the AKK PTAS as a special case (δ = 1) Advantage: we provide a smooth trade-off from the easy case (dense instances) to more general cases We will also give some tight bounds, ruling out natural possible improvements. Sub-Exponential Approximation Schemes 5 / 17
19 Basic scheme (Max Cut) We are given a dense graph for which we want to find a large cut Sub-Exponential Approximation Schemes 6 / 17
20 Basic scheme (Max Cut) Randomly select a sample of its vertices Sub-Exponential Approximation Schemes 6 / 17
21 Basic scheme (Max Cut) Guess their correct partition Sub-Exponential Approximation Schemes 6 / 17
22 Basic scheme (Max Cut) For every vertex outside the sample, examine its neighbors in the sample Sub-Exponential Approximation Schemes 6 / 17
23 Basic scheme (Max Cut) Greedily set its value depending on this neighborhood Sub-Exponential Approximation Schemes 6 / 17
24 Basic scheme (Max Cut) The sample we select has size O(logn) (hidden constants depend on degree and ǫ) running time n O(1) (will try all partitions of sample) Sub-Exponential Approximation Schemes 6 / 17
25 Basic scheme (Max Cut) The sample we select has size O(logn) (hidden constants depend on degree and ǫ) running time n O(1) (will try all partitions of sample) Why this works (intuitively): Because graph is dense every vertex outside sample S has many neighbors in S examining N(u) S is (whp) a good representation of N(u) in the optimal solution If a vertex in V \S has >> 50% of its neighbors on one side in the optimal solution, it will (whp) have >> 50% of its neighbors on that side in S (de la Vega 96) Sub-Exponential Approximation Schemes 6 / 17
26 General scheme (Max-k-CSP) Max Cut: max (i,j) E x i (1 x j )+x j (1 x i ) Sub-Exponential Approximation Schemes 7 / 17
27 General scheme (Max-k-CSP) Max-2-SAT: max (i,j) C x i (1 x j )+x j (1 x i )+x i x j Sub-Exponential Approximation Schemes 7 / 17
28 General scheme (Max-k-CSP) Max-3-SAT: max x i (1 x j )(1 x k )+(1 x i )x j (1 x k )+...+x i x j x k (i,j,k) C Sub-Exponential Approximation Schemes 7 / 17
29 General scheme (Max-k-CSP) Max-k-CSP: where p() is a degree k polynomial. max p( x) The AKK scheme offers a PTAS that finds an assignment almost maximizing p when the polynomial has at least Ω(n k ) terms. Sub-Exponential Approximation Schemes 7 / 17
30 General scheme (Max-k-CSP) Max Cut: max (i,j) E x i (1 x j )+x j (1 x i ) Sub-Exponential Approximation Schemes 7 / 17
31 General scheme (Max-k-CSP) Max Cut: max (i,j) c ij x i x j + i c i x i +C Sub-Exponential Approximation Schemes 7 / 17
32 General scheme (Max-k-CSP) Max Cut: max i x i r i where r i ( x x i ) is the (linear) polynomial of the remaining variables I obtain if I factor out x i. Sub-Exponential Approximation Schemes 7 / 17
33 General scheme (Max-k-CSP) Max Cut: max i x i r i where r i ( x x i ) is the (linear) polynomial of the remaining variables I obtain if I factor out x i. Main idea: Estimate the values of the r i s using brute force on a small sample. Sub-Exponential Approximation Schemes 7 / 17
34 General scheme (Max-k-CSP) Max Cut: max i x i r i s.t. ˆr i ǫn j N(i) c ijx j ˆr i +ǫn where ˆr i is the estimate I have for r i. This is now a linear program. Sub-Exponential Approximation Schemes 7 / 17
35 General scheme (Max-k-CSP) Max Cut: max i x i r i Summary of algorithm: Estimate the r i values using a sample Need large enough sample to guarantee ˆr i r i This turns QIP ILP Solve fractional relaxation of ILP Round solution Sub-Exponential Approximation Schemes 7 / 17
36 Sub-exponential Extension (Max Cut) Summary of algorithm: Estimate the r i values using a sample Need large enough sample to guarantee ˆr i r i This turns QIP ILP Solve fractional relaxation of ILP Round solution Sub-Exponential Approximation Schemes 8 / 17
37 Sub-exponential Extension (Max Cut) Summary of algorithm: Estimate the r i values using a sample Need large enough sample to guarantee ˆr i r i This turns QIP ILP Solve fractional relaxation of ILP Round solution Main idea: Use larger sample Sub-Exponential Approximation Schemes 8 / 17
38 Sub-exponential Extension (Max Cut) Summary of algorithm: Estimate the r i values using a sample Need large enough sample to guarantee ˆr i r i This turns QIP ILP Solve fractional relaxation of ILP Round solution Main idea: Use larger sample Suppose graph has average degree = n δ We sample nlogn = n1 δ logn vertices whp ˆr i r i. Sub-Exponential Approximation Schemes 8 / 17
39 Sub-exponential Extension (Max Cut) Summary of algorithm: Estimate the r i values using a sample Need large enough sample to guarantee ˆr i r i This turns QIP ILP Solve fractional relaxation of ILP Round solution Main idea: Use larger sample We are almost done! Must prove sample size enough for ˆr i Pitfall: Additive error ǫn no longer negligible! Must prove rounding step still works Don t worry, it all works! Sub-Exponential Approximation Schemes 8 / 17
40 General scheme k 3 Summary so far k = 2: AKK: Average degree Ω(n), sample of O(logn) vertices Extension: Average degree n δ, sample of n 1 δ logn in time 2 n can solve Max-Cut for E n 1.5 Sub-Exponential Approximation Schemes 9 / 17
41 General scheme k 3 Summary so far k = 2: AKK: Average degree Ω(n), sample of O(logn) vertices Extension: Average degree n δ, sample of n 1 δ logn in time 2 n can solve Max-Cut for E n 1.5 How about Max-3-SAT? In poly time can solve instances with n 3 clauses In 2 n time can solve instances with... clauses? Sub-Exponential Approximation Schemes 9 / 17
42 General scheme k 3 Summary so far k = 2: AKK: Average degree Ω(n), sample of O(logn) vertices Extension: Average degree n δ, sample of n 1 δ logn in time 2 n can solve Max-Cut for E n 1.5 How about Max-3-SAT? In poly time can solve instances with n 3 clauses In 2 n time can solve instances with n 2.5 clauses Sub-Exponential Approximation Schemes 9 / 17
43 General scheme k 3 AKK scheme for k 3 Write p( x) as i x ir i Each r i has degree k 1 Write r i = j x jr ij Each r ij has degree k 2... Until we get to linear write ILP Sub-Exponential Approximation Schemes 9 / 17
44 General scheme k 3 AKK scheme for k 3 Write p( x) as i x ir i Each r i has degree k 1 Write r i = j x jr ij Each r ij has degree k 2... Until we get to linear write ILP Note: In order for this to work, all r ij... polynomials must be dense This is true if original polynomial was dense. Sub-Exponential Approximation Schemes 9 / 17
45 General scheme k 3 AKK scheme for k 3 Write p( x) as i x ir i Each r i has degree k 1 Write r i = j x jr ij Each r ij has degree k 2... Until we get to linear write ILP In our scheme, if p has n k 1+δ terms r i has n k 2+δ terms r ij has n k 3+δ terms... It seems that the right density to require is n k 1+δ? Sub-Exponential Approximation Schemes 9 / 17
46 General scheme summary Input: Max-k-CSP isntance with n k 1+δ constraints Algorithm: Sample 2 n1 δ logn/ǫ 3 variables, guess their value Write CSP as a polynomial optimization problem Estimate non-linear coefficients using sample Solve fractional LP Round solution Sub-Exponential Approximation Schemes 10 / 17
47 General scheme summary Input: Max-k-CSP isntance with n k 1+δ constraints Algorithm: Sample 2 n1 δ logn/ǫ 3 variables, guess their value Write CSP as a polynomial optimization problem Estimate non-linear coefficients using sample Solve fractional LP Round solution Works for any CSP (for fixed k) Covers all instances for k = 2 Sub-Exponential Approximation Schemes 10 / 17
48 General scheme summary Input: Max-k-CSP isntance with n k 1+δ constraints Algorithm: Sample 2 n1 δ logn/ǫ 3 variables, guess their value Write CSP as a polynomial optimization problem Estimate non-linear coefficients using sample Solve fractional LP Round solution Works for any CSP (for fixed k) Covers all instances for k = 2 Can we do better? Smaller sample/faster running time? Handle k 3 better? Sub-Exponential Approximation Schemes 10 / 17
49 Summary with a picture Complexity/Density trade-off for Max-k-CSP Sub-Exponential Approximation Schemes 11 / 17
50 Summary with a picture Possible Improvements? Faster? More general? Sub-Exponential Approximation Schemes 11 / 17
51 Density lower bound Theorem: Assuming ETH, for all k 3, r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max-k-SAT with m n k 1 in time 2 n1 ǫ Sub-Exponential Approximation Schemes 12 / 17
52 Density lower bound Theorem: Assuming ETH, for all k 3, r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max-k-SAT with m n k 1 in time 2 n1 ǫ In English: For density less than n k 1 we need exponential time to get (1 ǫ)-approximation. Sub-Exponential Approximation Schemes 12 / 17
53 Density lower bound Theorem: Assuming ETH, for all k 3, r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max-k-SAT with m n k 1 in time 2 n1 ǫ Starting Point: Max-2-SAT is APX-ETH -hard on instances with V = n and m = O( V ). Sub-Exponential Approximation Schemes 12 / 17
54 Density lower bound Theorem: Assuming ETH, for all k 3, r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max-k-SAT with m n k 1 in time 2 n1 ǫ Starting Point: Max-2-SAT is APX-ETH -hard on instances with V = n and m = O( V ). Proof: (k = 3) Add n new variables y 1,...,y n For each clause (x i x j ), for each k {1,...,n} we construct the clauses (x i x j y k ) and (x i x j y k ) Gap remains! Number of clauses n 2 Sub-Exponential Approximation Schemes 12 / 17
55 Density lower bound Theorem: Assuming ETH, for all k 3, r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max-k-SAT with m n k 1 in time 2 n1 ǫ Starting Point: Max-2-SAT is APX-ETH -hard on instances with V = n and m = O( V ). Proof: (k = 3) Add n new variables y 1,...,y n For each clause (x i x j ), for each k {1,...,n} we construct the clauses (x i x j y k ) and (x i x j y k ) Gap remains! Number of clauses n 2 Reduction similar for k > 3 Sub-Exponential Approximation Schemes 12 / 17
56 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Sub-Exponential Approximation Schemes 13 / 17
57 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ In English: Our sample size is optimal. For density n δ we need time 2 n1 δ. Sub-Exponential Approximation Schemes 13 / 17
58 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on instances with V = n and E = O( V ). Sub-Exponential Approximation Schemes 13 / 17
59 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
60 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
61 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
62 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
63 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
64 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. Sub-Exponential Approximation Schemes 13 / 17
65 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. A constant gap remains for any V = n, E = n 2, Avg. degree = If we cound do better than 2 V / then ETH Sub-Exponential Approximation Schemes 13 / 17
66 Running time lower bound Theorem: Assuming ETH, for all δ (0, 1), r < 1 s.t. ǫ > 0 no algorithm can r-approximate Max Cut with E = n 1+δ in time 2 n1 δ ǫ Starting Point: Max Cut is APX-ETH -hard on 5-regular instances with V = n. A constant gap remains for any V = n, E = n 2, Avg. degree = If we cound do better than 2 V / then ETH Bonus: The two reductions compose! Optimal running times everywhere! Sub-Exponential Approximation Schemes 13 / 17
67 Extensions: k-densest Find k vertices that induce max edges AKK scheme works for k = Θ(n) Reason: Then OPT = Θ(n 2 ) Sub-Exponential Approximation Schemes 14 / 17
68 Extensions: k-densest Find k vertices that induce max edges AKK scheme works for k = Θ(n) Reason: Then OPT = Θ(n 2 ) How to get something for any k? Simple win/win algorithm If k large, we can run our algorithm If k small, brute force ( n) k Balancing gives 2 n1 δ/3 logn/ǫ 3 time for E = n 1+δ Sub-Exponential Approximation Schemes 14 / 17
69 Extensions: k-densest Find k vertices that induce max edges AKK scheme works for k = Θ(n) Reason: Then OPT = Θ(n 2 ) How to get something for any k? Simple win/win algorithm If k large, we can run our algorithm If k small, brute force ( n) k Balancing gives 2 n1 δ/3 logn/ǫ 3 time for E = n 1+δ Can we do better? Sub-Exponential Approximation Schemes 14 / 17
70 Extensions: 3-Coloring 3-Coloring is in P for graphs with large minimum degree Dense graphs contain a dominating set of size O(logn) Guess its coloring Coloring remaining graph is 2-List Coloring (in P) Sub-Exponential Approximation Schemes 15 / 17
71 Extensions: 3-Coloring 3-Coloring is in P for graphs with large minimum degree Dense graphs contain a dominating set of size O(logn) Guess its coloring Coloring remaining graph is 2-List Coloring (in P) Can extend this to graphs with minimum degree Exists dominating set with size n logn Guess its coloring... Sub-Exponential Approximation Schemes 15 / 17
72 Conclusions Density is a crucial parameter for approximating Max-k-CSP Especially useful in sub-exponential setting Smooth trade-off between performance and generality Tight bounds Questions: Other applications? Other interesting sub-exponential approximations? Sub-Exponential Approximation Schemes 16 / 17
73 Thank you! Sub-Exponential Approximation Schemes 17 / 17
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