Linear Programming and Halfplane Intersection

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1 CMPS 3130/6130 Computational Geometry Spring 2015 Linear Programming and Halfplane Intersection Carola Wenk CMPS 3130/6130 Computational Geometry 1

2 Word Problem A company produces tables and chairs. The profit for a chair is $2, and for a table $4. Machine group A needs 4 hours to produce a chair, and 6 hours to produce a table. Machine group B needs 2 hours to produce a chair, and 6 hours to produce a table. Per day there are at most 120 working hours for group A and at most 72 hours for group B. How can the company maximize profit? Variables: c A = # chairs produced on machine group A c B = # chairs produced on machine group B t A = # tables produced on machine group A t B = # tables produced on machine group B Objective function (profit): Maximize 2(c A +c B )+4(t A +t B ) Constraints: 4c A +6t A 120 2c B +6t B 72 CMPS 3130/6130 Computational Geometry 2

3 Linear Programming Variables: x 1,,x d Constraints: h 1 : a 11 x 1 + +a 1d x d b 1 h 2 : a 21 x 1 + +a 2d x d b 2... h n : a n1 x 1 + +a nd x d b n Objective function: Maximize f c (x) = c 1 x 1 + +c d x d Linear program in d variables with n constraints Each constraint h i is a half-space in R d is the feasible region of the linear program Maximizing f c (x) corresponds to finding a point x that is extreme in direction c. CMPS 3130/6130 Computational Geometry 3

4 Sub-Problem: Halfspace Intersection (in R 2 : Halfplane Intersection) Given: A set H={h 1, h 2,, h n } of halfplanes h i : a i x + b i y c i with constants a i, b i, c i ; for i=1,,n. Find:, i.e., the feasible region of all points (x,y)r 2 satisfying all n constraints at the same time. This is a convex polygonal region bounded by at most n edges. intersection bounded intersection unbounded intersection empty intersection degenerated to a point CMPS 3130/6130 Computational Geometry 4

5 D&C Halfplane Intersection Algorithm Intersect_Halfplanes(H): Input: A set H of n halfplanes in R 2 Output: The convex polygonal region C= if H =1 then C = h, where H={h} else split H into two sets H 1 and H 2 of size n/2 each C 1 = Intersect_Halfplanes(H 1 ) C 2 = Intersect_Halfplanes(H 2 ) C = Intersect_Convex_Regions(C 1, C 2 ) return C Use a plane-sweep to develop an O(n)-time algorithm for Intersect_Convex_Regions T(n) = 2T(n/2)+n T(n)O(n log n)

6 Incremental Linear Programming 2D linear program (LP) Assume the LP is bounded (otherwise add constraints) Assume there is one unique solution (if any); take the lexicographically smallest solution Incremental approach: Add one halfplane after the other.,, Let v i = unique optimal vertex for feasible region C i, for 2. Then, and hence if for some then for all.

7 Incremental Linear Programming Lemma: Let 2 i n. (i) If then (ii) If then or the line bounding Handling case (ii) involves solving a 1-dimensional LP on : The feasible region is just an interval, that can be computed in linear time [rightmost left-bounded halfplane, leftmost right-bounded halfplane] We can compute a new, or decide that the LP is infeasible, in O(i) time

8 2D_Bounded_LP Algorithm 2D_Bounded_LP(H, c ): Input: A two-dimensional LP (H, c ) Output: Report if (H, c ) is infeasible. Otherwise report the lexicographically smallest point that maximizes f c. Let,, be the halfplanes of H Let be the corner of, which exists because LP is bounded for i=3 to n do if then else // Case (ii) point on that maximizes f c subject to constraints in if such a point does not exist then Report that the LP is infeasible break; return Runtime: Storage:

9 Randomized Incremental LP Depending on the insertion order of the halfplanes the runtime varies between O(n) and O(n 2 ). Randomize the input order of the halfplanes. Theorem: 2D_Randomized_Bounded_LP runs in O(n) expected time and O(n) deterministic space. Proof: Define a random variable 1, 0, The total time spent to resolve case (ii), over all,, is

10 Randomized Incremental LP We now need to bound the expected value E( and we know that. Apply backwards analysis to bound : Fix,, which determines. Analyze what happened in last step when was added. P(had to compute new optimal vertex when adding ) = P(optimal vertex changes when we remove a halfplane from ) 2 lines defining v i Total expected runtime is

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