Covering Graphs using Trees and Stars p.1
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1 Covering Graphs using Trees and Stars Amitabh Sinha GSIA, Carnegie Mellon University (Work done while visiting MPII Saarbrucken, Germany) Covering Graphs using Trees and Stars p.1
2 Covering Graphs using Trees and Stars Joint work with G. Even Amitabh Sinha GSIA, Carnegie Mellon University (Work done while visiting MPII Saarbrucken, Germany) Covering Graphs using Trees and Stars p.1
3 Covering Graphs using Trees and Stars Amitabh Sinha Joint work with G. Even N. Garg GSIA, Carnegie Mellon University (Work done while visiting MPII Saarbrucken, Germany) Covering Graphs using Trees and Stars p.1
4 Covering Graphs using Trees and Stars Amitabh Sinha Joint work with G. Even N. Garg J. Könemann GSIA, Carnegie Mellon University (Work done while visiting MPII Saarbrucken, Germany) Covering Graphs using Trees and Stars p.1
5 Covering Graphs using Trees and Stars Amitabh Sinha Joint work with G. Even N. Garg J. Könemann R. Ravi GSIA, Carnegie Mellon University (Work done while visiting MPII Saarbrucken, Germany) Covering Graphs using Trees and Stars p.1
6 Motivation: Nurse station location Hospital; k nurses (each with her own station); n patients in various beds. Covering Graphs using Trees and Stars p.2
7 Motivation: Nurse station location Hospital; k nurses (each with her own station); n patients in various beds. At 8 am, each nurse begins her morning round of patients under her care. Covering Graphs using Trees and Stars p.2
8 Motivation: Nurse station location Hospital; k nurses (each with her own station); n patients in various beds. At 8 am, each nurse begins her morning round of patients under her care. Morning round ends when all nurses have returned to their bases. Covering Graphs using Trees and Stars p.2
9 Motivation: Nurse station location Hospital; k nurses (each with her own station); n patients in various beds. At 8 am, each nurse begins her morning round of patients under her care. Morning round ends when all nurses have returned to their bases. Objective: Assign patients to nurses so that morning rounds end ASAP. Covering Graphs using Trees and Stars p.2
10 Problem definition Input: Graph G = (V, E), edge weights w, integer k. INPUT Covering Graphs using Trees and Stars p.3
11 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. INPUT Covering Graphs using Trees and Stars p.3
12 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. FEASIBLE SOLUTION Covering Graphs using Trees and Stars p.3
13 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. Objective: Minimize max i w(t i ). FEASIBLE SOLUTION Covering Graphs using Trees and Stars p.3
14 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. Objective: Minimize max i w(t i ). Rooted version: Given roots R V, find a k-tree cover with each tree using a distinct root in R. Roots INPUT Covering Graphs using Trees and Stars p.3
15 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. Objective: Minimize max i w(t i ). Rooted version: Given roots R V, find a k-tree cover with each tree using a distinct root in R. Roots INPUT Covering Graphs using Trees and Stars p.3
16 Problem definition Input: Graph G = (V, E), edge weights w, integer k. k-tree cover: Set of trees {T 1, T 2,..., T k } such that k i=1 V (T i) = V. Objective: Minimize max i w(t i ). Rooted version: Given roots R V, find a k-tree cover with each tree using a distinct root in R. Star cover: Cover with stars, same objective; may be rooted or unrooted. Roots INPUT Covering Graphs using Trees and Stars p.3
17 Hardness (of rooted k-star cover) Reduction from BIN-PACK: Given elements U with sizes s u, k bins of size B. Can we pack elements in k bins? BINS OBJECTS SIZES BIN PACK 8 Covering Graphs using Trees and Stars p.4
18 Hardness (of rooted k-star cover) Reduction from BIN-PACK: Given elements U with sizes s u, k bins of size B. Can we pack elements in k bins? Convert to Rooted k-star cover: Complete bipartite graph between elements and bins, edge weights = element sizes, bins = roots ROOTS VERTICES (Complete bipartite) Rooted k Star cover Covering Graphs using Trees and Stars p.4
19 Hardness (of rooted k-star cover) Reduction from BIN-PACK: Given elements U with sizes s u, k bins of size B. Can we pack elements in k bins? Convert to Rooted k-star cover: Complete bipartite graph between elements and bins, edge weights = element sizes, bins = roots. Claim: BIN-PACK is identical to this special case of Rooted k-star cover. ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.4
20 Hardness of others Also by reduction from BIN-PACK Key: Poly time algorithm to convert any solution to Rooted k- star cover solution without increasing cost. (In BIN-PACK graph.) ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.5
21 Hardness of others Also by reduction from BIN-PACK Key: Poly time algorithm to convert any solution to Rooted k- star cover solution without increasing cost. (In BIN-PACK graph.) ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.5
22 Hardness of others Also by reduction from BIN-PACK Key: Poly time algorithm to convert any solution to Rooted k- star cover solution without increasing cost. (In BIN-PACK graph.) ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.5
23 Hardness of others Also by reduction from BIN-PACK Key: Poly time algorithm to convert any solution to Rooted k- star cover solution without increasing cost. (In BIN-PACK graph.) ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.5
24 Hardness of others Also by reduction from BIN-PACK Key: Poly time algorithm to convert any solution to Rooted k- star cover solution without increasing cost. (In BIN-PACK graph.) ROOTS Solution VERTICES Covering Graphs using Trees and Stars p.5
25 Algorithm for Rooted k-tree cover Guess-and-check type algorithm. Covering Graphs using Trees and Stars p.6
26 Algorithm for Rooted k-tree cover Guess-and-check type algorithm. Guess optimal solution cost B. Let true optimum be B. If fail, then proof that B < B. If success, then find solution of cost no more than 4B. Covering Graphs using Trees and Stars p.6
27 Algorithm for Rooted k-tree cover Guess-and-check type algorithm. Guess optimal solution cost B. Let true optimum be B. If fail, then proof that B < B. If success, then find solution of cost no more than 4B. Binary search yields (weakly) polynomial time 4-approximation algorithm. Covering Graphs using Trees and Stars p.6
28 Algorithm for Rooted k-tree cover Guess-and-check type algorithm. Guess optimal solution cost B. Let true optimum be B. If fail, then proof that B < B. If success, then find solution of cost no more than 4B. Binary search yields (weakly) polynomial time 4-approximation algorithm. Can be made strongly polynomial; approximation ratio worsens to 4 + ɛ. Covering Graphs using Trees and Stars p.6
29 Algorithm: Overview Given B, set of roots R, and G. 1. Remove all edges with w e > B. Covering Graphs using Trees and Stars p.7
30 Algorithm: Overview Given B, set of roots R, and G. 1. Remove all edges with w e > B. 2. Contract R; compute MST M. {T i } i := forest obtained by expanding R. Covering Graphs using Trees and Stars p.7
31 Algorithm: Overview Given B, set of roots R, and G. 1. Remove all edges with w e > B. 2. Contract R; compute MST M. {T i } i := forest obtained by expanding R. 3. Decompose each T i into trees {S j i }j + L i s.t. w(s j i ) [B, 2B) and w(l i) < B. Covering Graphs using Trees and Stars p.7
32 Algorithm: Overview Given B, set of roots R, and G. 1. Remove all edges with w e > B. 2. Contract R; compute MST M. {T i } i := forest obtained by expanding R. 3. Decompose each T i into trees {S j i }j + L i s.t. w(s j i ) [B, 2B) and w(l i) < B. 4. Match trees {S j i }j i to roots in R within distance B from it. If possible, return success. If impossible, return fail. Covering Graphs using Trees and Stars p.7
33 Algorithm: Demonstration 1. Prune. Roots INPUT Covering Graphs using Trees and Stars p.8
34 Algorithm: Demonstration 1. Prune. 2. Contract R, compute MST. CONTRACTED ROOTS Roots Covering Graphs using Trees and Stars p.8
35 Algorithm: Demonstration 1. Prune. 2. Contract R, compute MST. Roots MST Covering Graphs using Trees and Stars p.8
36 Algorithm: Demonstration 1. Prune. 2. Contract R, compute MST. Expanded MST Roots Covering Graphs using Trees and Stars p.8
37 Algorithm: Demonstration 1. Prune. 2. Contract R, compute MST. 3. Decompose. Decompose into light trees Roots Covering Graphs using Trees and Stars p.8
38 Algorithm: Demonstration 1. Prune. 2. Contract R, compute MST. 3. Decompose. 4. Match. Match success? Roots Covering Graphs using Trees and Stars p.8
39 Algorithm: Success Claim: On success, each tree has cost no more than 4B. Covering Graphs using Trees and Stars p.9
40 Algorithm: Success Claim: On success, each tree has cost no more than 4B. Proof: Each tree in our solution has 3 components: Decomposed tree S j i, cost 2B. Covering Graphs using Trees and Stars p.9
41 Algorithm: Success Claim: On success, each tree has cost no more than 4B. Proof: Each tree in our solution has 3 components: Decomposed tree S j i, cost 2B. Edge to root, cost B. Covering Graphs using Trees and Stars p.9
42 Algorithm: Success Claim: On success, each tree has cost no more than 4B. Proof: Each tree in our solution has 3 components: Decomposed tree S j i, cost 2B. Edge to root, cost B. Leftover tree L i, cost B. Covering Graphs using Trees and Stars p.9
43 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Covering Graphs using Trees and Stars p.10
44 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Alternatively: If B B, matching exists. Covering Graphs using Trees and Stars p.10
45 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Alternatively: If B B, matching exists. Proof: Hall s Theorem: We show N(S) S for all S {S j i }j i. Covering Graphs using Trees and Stars p.10
46 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Alternatively: If B B, matching exists. Proof: Hall s Theorem: We show N(S) S for all S {S j i }j i. Consider optimal solution T = {T1,..., T k }. Let T (S) = T S. Hence N(S) T (S). Covering Graphs using Trees and Stars p.10
47 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Alternatively: If B B, matching exists. Proof: Hall s Theorem: We show N(S) S for all S {S j i }j i. Consider optimal solution T = {T1,..., T k }. Let T (S) = T S. Hence N(S) T (S). Deleting all edges in S and adding all edges in T (S) also yields a spanning tree of G, and since our tree was MST, w(t (S)) w(s). Covering Graphs using Trees and Stars p.10
48 Algorithm: Failure Lemma: On failure (matching does not exist), B < B. Alternatively: If B B, matching exists. Proof: Hall s Theorem: We show N(S) S for all S {S j i }j i. Consider optimal solution T = {T1,..., T k }. Let T (S) = T S. Hence N(S) T (S). Deleting all edges in S and adding all edges in T (S) also yields a spanning tree of G, and since our tree was MST, w(t (S)) w(s). B N(S) B T (S) w(t (S)) w(s) B S. Covering Graphs using Trees and Stars p.10
49 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. Covering Graphs using Trees and Stars p.11
50 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. If algorithm says w m = B < B, then contract all edges of weight at most ɛw m n 2. Now binary search in range, nw m ], which is polynomial. [ ɛw m n 2 Covering Graphs using Trees and Stars p.11
51 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. If algorithm says w m = B < B, then contract all edges of weight at most ɛw m n 2. Now binary search in range, nw m ], which is polynomial. [ ɛw m n 2 Otherwise, find i such that B (w i, 4w i+1 ]. Covering Graphs using Trees and Stars p.11
52 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. If algorithm says w m = B < B, then contract all edges of weight at most ɛw m n 2. Now binary search in range, nw m ], which is polynomial. [ ɛw m n 2 Otherwise, find i such that B (w i, 4w i+1 ]. If w i+1 w i n2 ɛ, binary search in above range is polynomial. Covering Graphs using Trees and Stars p.11
53 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. If algorithm says w m = B < B, then contract all edges of weight at most ɛw m n 2. Now binary search in range, nw m ], which is polynomial. [ ɛw m n 2 Otherwise, find i such that B (w i, 4w i+1 ]. If w i+1 w i n2 ɛ, binary search in above range is polynomial. If not, set w = n 2 w i /ɛ. If B [w i, w ], then polynomial. Covering Graphs using Trees and Stars p.11
54 Strongly polynomial algorithm Fix ɛ > 0. Sort edges w 1 w 2... w m. If algorithm says w m = B < B, then contract all edges of weight at most ɛw m n 2. Now binary search in range, nw m ], which is polynomial. [ ɛw m n 2 Otherwise, find i such that B (w i, 4w i+1 ]. If w i+1 w i n2 ɛ, binary search in above range is polynomial. If not, set w = n 2 w i /ɛ. If B [w i, w ], then polynomial. If not, then contract all edges of weight at most w i. Now binary search in [w i+1, 4w i+1 ] is polynomial. Covering Graphs using Trees and Stars p.11
55 Algorithm for Unrooted k-tree cover 1. Prune edges w e > B. Let {G i } i be components. Long edges pruned Covering Graphs using Trees and Stars p.12
56 Algorithm for Unrooted k-tree cover 1. Prune edges w e > B. Let {G i } i be components. 2. MST i = MST of G i. k i = w(mst i) 2B. MST Covering Graphs using Trees and Stars p.12
57 Algorithm for Unrooted k-tree cover 1. Prune edges w e > B. Let {G i } i be components. 2. MST i = MST of G i. k i = w(mst i) 2B. 3. If i (k i + 1) > k, return fail. MST Covering Graphs using Trees and Stars p.12
58 Algorithm for Unrooted k-tree cover 1. Prune edges w e > B. Let {G i } i be components. 2. MST i = MST of G i. k i = w(mst i) 2B. 3. If i (k i + 1) > k, return fail. 4. Decompose each MST i into at most k i +1 trees Si Sk i i +L i such that w(s j i ) [2B, 4B) and w(l i ) < 2B. Return success. Final solution Covering Graphs using Trees and Stars p.12
59 Analysis Claim: On success, each tree has weight no more than 4B. Covering Graphs using Trees and Stars p.13
60 Analysis Claim: On success, each tree has weight no more than 4B. Claim: On failure, B < B. Covering Graphs using Trees and Stars p.13
61 Analysis Claim: On success, each tree has weight no more than 4B. Claim: On failure, B < B. Alternatively, if B B, then k i + 1 k i for all i. Covering Graphs using Trees and Stars p.13
62 Analysis Claim: On success, each tree has weight no more than 4B. Claim: On failure, B < B. Alternatively, if B B, then k i + 1 k i for all i. Proof: Let optimal solution cover G i with {T1,..., T k }. We i can make it span G i by adding at most ki + 1 edges, so:. k i j=1 w(t i ) + (k i 1)B w(mst i) Covering Graphs using Trees and Stars p.13
63 Analysis Claim: On success, each tree has weight no more than 4B. Claim: On failure, B < B. Alternatively, if B B, then k i + 1 k i for all i. Proof: Let optimal solution cover G i with {T1,..., T k }. We i can make it span G i by adding at most ki + 1 edges, so:. k i j=1 w(t i ) + (k i 1)B w(mst i) Therefore k i w(mst i) 2B > k i. Covering Graphs using Trees and Stars p.13
64 Extensions and conclusion Rooted k-star cover: Reduces to Generalized Assignment problem, yields a 2-approximation. Covering Graphs using Trees and Stars p.14
65 Extensions and conclusion Rooted k-star cover: Reduces to Generalized Assignment problem, yields a 2-approximation. Unrooted k-star cover: LP rounding gives bicriteria approximation: Covers with 2k stars, each costing no more than twice the optimum. Covering Graphs using Trees and Stars p.14
66 Extensions and conclusion Rooted k-star cover: Reduces to Generalized Assignment problem, yields a 2-approximation. Unrooted k-star cover: LP rounding gives bicriteria approximation: Covers with 2k stars, each costing no more than twice the optimum. Tree cover algorithms also yield constant factor approximations for tour cover, the original nursing station location problem. Covering Graphs using Trees and Stars p.14
67 Extensions and conclusion Rooted k-star cover: Reduces to Generalized Assignment problem, yields a 2-approximation. Unrooted k-star cover: LP rounding gives bicriteria approximation: Covers with 2k stars, each costing no more than twice the optimum. Tree cover algorithms also yield constant factor approximations for tour cover, the original nursing station location problem. Questions? Covering Graphs using Trees and Stars p.14
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