Implicit Hitting Set Problems. Richard M. Karp Harvard University August 29, 2011. Worst-case Analysis of NP-Hard Problems. Exact solution methods: exponential running time in worst case.
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Implicit Hitting Set Problems
Richard M. Karp
Harvard University August 29, 2011
validating them by empirical testing on training sets
of representative instances.
-- Linear and convex programming: equivalence of separation and optimization
-- Integer programming: cutting-plane methods
-- Linear programming: column generation
Repeat: Choose element v in V that minimizes ratio of c(v) to number of sets hit; Delete sets hit by v.
Example: in the feedback vertex set problem, the separation oracle produces vertex set of a shortest cycle in the subgraph induced by V\H.
Repeat until a feasible hitting set His found:
(1) Given C, a subset of C*, find a minimum-weight hitting set Hfor C.
(2) Using the separation oracle, find a minimum-cardinality circuit c not hit by H.
(3) Add c to C
Input: C, a set of circuits and H, a hitting set for C
Repeat until H hits every circuit in C*
find a circuit c not hit by H and choose an element x in c; add c to C and add x to H.
(1)Execute the circuit-finding subroutine
(2) Repeat until k iterations yield no circuits: construct a greedy hitting set H for C and execute the circuit-finding subroutine.
(3) Using CPLEX, construct an optimal hitting set H for C.
If H is infeasible, go to (1)
cycles as circuits.
(1) a variant of depth-first search;
(2) attempt to align the remaining edges until blocked by the occurrence of a mixed cycle.
Time (sec.) # solved # edges
0 to 0.01 1311 (1; 52; 399)
0.01 to 0.1 764 (20; 203; 549)
0.1 to 1 1086 (26; 450; 1837)
1 to 10 632 (44; 1104; 4645)
10 to 60 151 (65; 1351; 12313)
60 to 600 75 (103; 1136; 14690)
600 to 3600 36 (166; 1236; 13916)