The Clever Shopper Problem
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(2) Introduction Clever Shopper.
(3) Introduction Clever Shopper b1. b2. b3. b4. Input: a set of books B,.
(4) Introduction Clever Shopper b1. s1. b2. s2. b3. s3. b4. s4. s5. Input: a set of books B, a set of shops S,.
(5) Introduction Clever Shopper b1. s1. b2. s2. b3. s3. b4. s4. s5. Input: a set of books B, a set of shops S, edges E ⊆ B × S,.
(6) Introduction Clever Shopper b1. s1. b2. s2. b3. s3. b4. s4. s5. Input: a set of books B, a set of shops S, edges E ⊆ B × S,.
(7) Introduction Clever Shopper b1. 5 6. kRUB. s1. kRUB. b2. s2. b3. s3. b4. s4. s5. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ ,.
(8) Introduction Clever Shopper b1. b2. 5 6 7 8 8 6. s2. s3. b3. b4. s1. 8 7 2 3. s4. s5. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ ,.
(9) Introduction Clever Shopper b1. b2. 5 6 7 8 8 6. s2. s3. b3. b4. s1. 8 7 2 3. s4. s5. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ ,. Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • minimum total cost.
(10) Introduction Total: 20 b1. b2. 5 6 7 8 8 6. s1 12 s2. b4. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ ,. 6 s3. b3 8 7 2 3. Clever Shopper. 0 s4 2 s5 0. Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • minimum total cost.
(11) Introduction Total: 20 b1. b2. 5 6 7 8 8 6. s1 12 s2 6 s3. b3. b4. 8 7 2 3. Clever Shopper. 0 s4 2 s5 0. unt! Disco 20kRUB For ≥ UB! kR Get -3. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ , a discount function D, Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • minimum total cost.
(12) Introduction Total: 19 b1. b2. 5 6 7 8 8 6. s1 0 s2 17 s3. b3. b4. 8 7 2 3. Clever Shopper. 0 s4 2 s5 0. unt! Disco 20kRUB For ≥ UB! kR Get -3. Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ , a discount function D, Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • minimum total cost.
(13) Introduction. b1. b2. 5 6 7 8 8 6. s2. s3. b3. b4. s1. 8 7 2 3. unt! Disco 12kRUB For ≥ UB! kR Get -2 unt! Disco 20kRUB For ≥ UB! kR Get -3 unt! Disco 9kRUB For ≥ UB! kR Get -1 ! count. s4. s5. Dis 10kRUB For ≥ UB! kR Get -4 unt! Disco 8kRUB For ≥ UB! kR Get -1. Clever Shopper Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ , a discount function D, Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • minimum total cost.
(14) Introduction. b1. b2. 5 6 7 8 8 6. s2. s3. b3. b4. s1. 8 7 2 3. unt! Disco 12kRUB For ≥ UB! kR Get -2 unt! Disco 20kRUB For ≥ UB! kR Get -3 unt! Disco 9kRUB For ≥ UB! kR Get -1 ! count. s4. s5. Dis 10kRUB For ≥ UB! kR Get -4 unt! Disco 8kRUB For ≥ UB! kR Get -1. Clever Shopper Input: a set of books B, a set of shops S, edges E ⊆ B × S, with weights w : E → N+ , a discount function D, a budget K ∈ N+ Output: E 0 ⊆ E such that: • each b ∈ B has 1 incident edge • total price ≤ K.
(15) Introduction I. Variant of Internet Shopping problem [Blazewicz et al., 2010] I. I. Discounts ↔ free shipping depending on specific sets of purchased items Strongly NP-hard, even with free items and unit shipping costs.
(16) Introduction I. Variant of Internet Shopping problem [Blazewicz et al., 2010] I. I. I. Discounts ↔ free shipping depending on specific sets of purchased items Strongly NP-hard, even with free items and unit shipping costs. We seek a complete picture of the tractability of Clever Shopper, with respect to: I I I. Number of books (n) Number of shops (m) Price range Constant? Polynomially bounded? Unconstrained?. I. Degree Few books per shops ? Few shops selling each book?.
(17) Introduction I. Variant of Internet Shopping problem [Blazewicz et al., 2010] I. I. I. Discounts ↔ free shipping depending on specific sets of purchased items Strongly NP-hard, even with free items and unit shipping costs. We seek a complete picture of the tractability of Clever Shopper, with respect to: I I I. Number of books (n) Number of shops (m) Price range Constant? Polynomially bounded? Unconstrained?. I. Degree Few books per shops ? Few shops selling each book?. I. Any approximation algorithm?.
(18) Results Sparse instances:.
(19) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. 3-Sat.
(20) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2. 3-Sat Matching.
(21) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching.
(22) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m): NP-hard with 2 shops and unbounded prices. 3-Sat Matching. Partition.
(23) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming.
(24) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing.
(25) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs.
(26) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code.
(27) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation:.
(28) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0.
(29) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat.
(30) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy.
(31) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n):.
(32) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT. dynamic programming.
(33) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT No polynomial kernel. dynamic programming or-composition of x3c.
(34) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT No polynomial kernel. dynamic programming or-composition of x3c.
(35) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal (x1 ∨ x2 ∨ . . .). x1. (x2 ∨ . . .). x2. (x2 ∨ x3 ∨ . . .) (x2 ∨ x3 ∨ x4 ). x2. m clauses, n variables. x3. x4.
(36) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. C2. x1. x2. C3. x2. m clauses, n variables. C4. x3. x4.
(37) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. 3 Buy ≥ 2 get. x1. C1. C2. x2. C3. x2. m clauses, n variables. C4. x3. x4.
(38) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. All prices = 1. m clauses, n variables. C4. x3. x4.
(39) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. set x2 = true. All prices = 1. m clauses, n variables. C4. x3. x4.
(40) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. set x2 = true. All prices = 1. m clauses, n variables. C4. x3. x4.
(41) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. set x2 = true. All prices = 1. m clauses, n variables. C4. x3. x4.
(42) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. set x2 = true. All prices = 1. m clauses, n variables. C4. x3. x4.
(43) NP-hard with book-degree 2, shop-degree 3, unit prices Reduction from Max 3-SAT, with 2 occurrences per literal 1 Buy ≥ 1 t e g. C1. 3 Buy ≥ 2 get. x1. C2. x2. C3. x2. C4. x3. set x2 = true. All prices = 1. Total discount: 2n + (1 per satisfied clause). m clauses, n variables. x4.
(44) Polynomial for shop-degree ≤ 2 Algorithm I. Subtract minimum price for each book from its incident edges. 12. 10 9 11 7. 4. 5 8. 8. 7. 10 Buy ≥ 3 t e g. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37.
(45) Polynomial for shop-degree ≤ 2 Algorithm I I. Subtract minimum price for each book from its incident edges Connect books sold by same shop with remaining cost. 12. 0. 9. 1. 0. 4. 2 3. 0. 8. 1 4. 0. 7. 0. 10 Buy ≥ 3 t e g. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37.
(46) Polynomial for shop-degree ≤ 2 Algorithm I. Subtract minimum price for each book from its incident edges Connect books sold by same shop with remaining cost. I. Subtract discount wherever threshold is reached. I. 1 12. 0. 9. 1. 0. 4. 2 3 2 3. 4. 0. 1 1 4. 8. 0. 7. 0. 10 Buy ≥ 3 t e g. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37.
(47) Polynomial for shop-degree ≤ 2 Algorithm I. Subtract minimum price for each book from its incident edges Connect books sold by same shop with remaining cost. I. Subtract discount wherever threshold is reached. I. Find max weight matching (on graph with opposite weights). I. -1 9 0 -2 0 -1 3. 1. -2 12. -3. 4. 0. -2 1 4. 8. 0. 7. 0. 10 Buy ≥ 3 t e g. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37.
(48) Polynomial for shop-degree ≤ 2 Algorithm I. Subtract minimum price for each book from its incident edges Connect books sold by same shop with remaining cost. I. Subtract discount wherever threshold is reached. I. Find max weight matching (on graph with opposite weights). I. Matched edges yield a solution. I. -2 10 Buy ≥ 3 t e g. -3. -1. 0. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37.
(49) Polynomial for shop-degree ≤ 2 Algorithm I. Subtract minimum price for each book from its incident edges Connect books sold by same shop with remaining cost. I. Subtract discount wherever threshold is reached. I. Find max weight matching (on graph with opposite weights). I. Matched edges yield a solution. I. 12. 10 9 11 7. 4. 5 8. 8. 7. 10 Buy ≥ 3 t e g. greedy solution = (12 − 3) + 9 + 4 + 8 + 7 = 37 optimal solution = (12 − 3) + (11 − 3) + (5 + 8 − 3) + 7 = 34.
(50) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT No polynomial kernel. dynamic programming or-composition of x3c.
(51) NP-hard with 2 shops, unbounded prices -hard weakly NP. Reduction from Partition P Input: X = {x1 , . . . , xn } with xi ∈X xi = 2A. P Question: ∃?X 0 ⊆ X such that xi ∈X 0 xi = A I. 2 shops, n books. I. book i has cost xi (in both shops). I. discounts: buy A get −1. I. budget: 2A − 2 b1. b2. b3. s1. b4. b5. b6. s2. b7.
(52) NP-hard with 2 shops, unbounded prices -hard weakly NP. Reduction from Partition P Input: X = {x1 , . . . , xn } with xi ∈X xi = 2A. P Question: ∃?X 0 ⊆ X such that xi ∈X 0 xi = A I. 2 shops, n books. I. book i has cost xi (in both shops). I. discounts: buy A get −1. I. budget: 2A − 2 b1. 20 For ≥ t e G -1. b2. b3. 6 4 2 s1. b4. 8. b5. 3. b6. 9 8 s2. b7.
(53) NP-hard with 2 shops, unbounded prices -hard weakly NP. Reduction from Partition P Input: X = {x1 , . . . , xn } with xi ∈X xi = 2A. P Question: ∃?X 0 ⊆ X such that xi ∈X 0 xi = A I. 2 shops, n books. I. book i has cost xi (in both shops). I. discounts: buy A get −1. I. budget: 2A − 2 b1. 20 For ≥ t e G -1. b2. b3. 6 4 2 s1. b4. 8 19. b5. 3. b6. 9 8 s2. 19. b7.
(54) W[1]-hard for m shops, polynomial prices [1]-hard Reduction from Bin-Packing P strongly W Input: X = {x1 , . . . , xn } with xi ∈X xi = mB. P Question: ∃?(X1 , . . . , Xm ) partition of X with xi ∈Xj xi = B. I. m shops, n books. I. book i has cost xi (in all shops). I. discounts: buy B get −1. I. budget: m(B − 1) b1. b2. s1. b3. b4. s2. b5. b6. s3. b7.
(55) W[1]-hard for m shops, polynomial prices [1]-hard Reduction from Bin-Packing P strongly W Input: X = {x1 , . . . , xn } with xi ∈X xi = mB. P Question: ∃?(X1 , . . . , Xm ) partition of X with xi ∈Xj xi = B. I. m shops, n books. I. book i has cost xi (in all shops). I. discounts: buy B get −1. I. budget: m(B − 1) b1. 10 For ≥ t e G -1. b2. 6. b3. 4 s1. b4. b5. 5 1. 4 s2. b6. b7. 7. 3 s3.
(56) W[1]-hard for m shops, polynomial prices [1]-hard Reduction from Bin-Packing P strongly W Input: X = {x1 , . . . , xn } with xi ∈X xi = mB. P Question: ∃?(X1 , . . . , Xm ) partition of X with xi ∈Xj xi = B. I. m shops, n books. I. book i has cost xi (in all shops). I. discounts: buy B get −1. I. budget: m(B − 1) b1. 10 For ≥ t e G -1. b2. 6. b3. 4 s1. 9. b4. b5. 5 1. 4 s2. 9. b6. b7. 7. 3 s3. 9.
(57) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT No polynomial kernel. dynamic programming or-composition of x3c.
(58) The problem with approximations When minimising the total cost: no approximation is possible..
(59) The problem with approximations When minimising the total cost: no approximation is possible. I. Take the (commercially questionable) discount function: Buy ≥ A Get −A. I. Partition, Bin-Packing or Perfect Code reductions yield: Clever Shopper is NP-hard, even with K = 0.
(60) The problem with approximations When minimising the total cost: no approximation is possible. I. Take the (commercially questionable) discount function: Buy ≥ A Get −A. I. Partition, Bin-Packing or Perfect Code reductions yield: Clever Shopper is NP-hard, even with K = 0. Other optimisation strategy: maximise the total discount I. Meaningful only if each book has a uniform price. I. Max 3-SAT reduction yields APX-hardness..
(61) Results Sparse instances: NP-hard with book-degree 2, shop-degree 3 and unit prices. Polynomial if shop degree ≤ 2 Few shops (parameter m):. 3-Sat Matching. NP-hard with 2 shops and unbounded prices Partition XP for m with polynomial prices dynamic programming W[1]-hard for m with polynomial prices Bin-Packing FPT for m with unit prices f -star subgraphs W[1]-hard for “selected shops” with unit prices Perfect Code Approximation: No approximation is possible NP-hard with K = 0 APX-hard to maximise the total discount... Max 3-Sat ... but k-approximable for shop-degree k greedy Few books (parameter n): FPT No polynomial kernel. dynamic programming or-composition of x3c.
(62) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5.
(63) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5.
(64) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5.
(65) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5.
(66) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5.
(67) FPT for number of books n Dynamic Programming Table: ∀B 0 ⊆ B, j ≤ m, p≤j (B 0 ) := Lowest possible price when buying books of B 0 from shops {s1 , . . . , sj }. Recurrence: p≤j (B 0 ) := min p≤j−1 (B 0 \ B 00 ) + cost for books B 00 in sj 00 0 B ⊆B. b1. s1. b2. s2. b3. s3. b4. s4 s5. For each j: enumerate every B 00 ⊆ B 0 ⊆ B −→ O(m3n ).
(68) Open questions. I. Constant-factor approximation (maximising total discount)?. I. Kernel for parameter m with unit prices?. I. FPT for number of shops + max. price?. I. What if all books are available everywhere at constant price?. Thank you!.
(69)
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