Implicit Hitting Set Problems

1. Implicit Hitting Set Problems Richard M. Karp Harvard University August 29, 2011

2. Worst-case Analysis of NP-Hard Problems • Exact solution methods: exponential running time in worst case. • Polynomial-time approximation algorithms for optimization problems. Approximation ratios are usually unrealistically high. • Parametrized complexity: polynomial-time complexity for instances with fixed parameter, but dependence on parameter is usually adverse.

3. Probabilistic Analysis and Heuristics • In probabilistic analysis problem instances are drawn from simple probability distributions. Often one can prove excellent performance on the average. However, the probability distributions may not correspond to real-life instances. • Heuristics are often “unreasonably effective,” for reasons not well understood. • We seek systematic methods for tuning heuristics and validating them by empirical testing on training sets of representative instances.

4. Unreasonably Effective Heuristics • Large traveling-salesman problems can be solved by quick tour construction methods, local improvement methods or cutting plane methods. • Local improvement methods find near-optimal solutions to graph bisection problems. • Huge satisfiability problems are routinely solved rapidly by branch-and-bound methods. • The greedy set cover algorithm typically gives solutions within a few percent of optimal.

5. Implicit Optimization Problems • Set of constraints defined implicitly by a generation algorithm rather than by an explicit list. -- Linear and convex programming: equivalence of separation and optimization -- Integer programming: cutting-plane methods -- Linear programming: column generation

6. Hitting Set Problem • Ground set V • For every v in V, a positive weight c(v). • C*: collection of subsets of V (circuits) • Goal: Find a set of minimum weight that hits every set in C* • Equivalent to set cover problem

7. Complexity of the Hitting Set Problem • NP-hard and hard to approximate within ratio o(log | C*|). • Greedy algorithm achieves approximation ratio O(log | C*|): Repeat: Choose element v in V that minimizes ratio of c(v) to number of sets hit; Delete sets hit by v.

8. Hitting Set Problem in Practice • Greedy algorithm gives good approximate solutions. • CPLEX integer programming algorithm often gives optimal solutions rapidly.

9. Implicit Hitting Set Problem • The collection of circuits C* has a compact implicit description. • There is a polynomial-time separation oracle which, given a subset H of the ground set, either determines that H is a hitting set or produces a circuit that H does not hit. Example: in the feedback vertex set problem, the separation oracle produces vertex set of a shortest cycle in the subgraph induced by V\H.

10. Examples • Feedback vertex set in a graph or digraph: vertex sets of cycles • Feedback edge set in a digraph: edge sets of cycles • Max cut: edge sets of odd cycles • Steiner tree: edge sets of cycles that partition the required vertices • Maximum 2-sat: minimal contradictory sets of 2-element clauses • Intersection of k matroids: circuits of each matroid • Maximal feasible subset of set of linear inequalities; minimal infeasible subsets.

11. Naïve Algorithm for Solving Implicit Hitting Set Problem 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 Return C

12. Circuit-Finding Subroutine 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.

13. Refined Algorithm • Input: set of circuits C and hitting set H for C (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) Return H.

14. Metrics • Number of circuits generated, number of calls to solver, running time of generator.

15. Application: Multi-Genome Alignment • Highly similar sequences in two genomes constitute an anchor pair. The individual sequences are called anchors. • A genome is a linearly ordered sequence of anchors. • An alignment is a matrix with a row for each genome, and an assignment of each anchor to a column, respecting the linear orders. • An anchor pair is synchronized if its two anchors lie in the same column. • Goal: maximize the sum of the weights of the synchronized anchor pairs.

17. Complexity Bounds • The 2-genome problem is equivalent to the maximum-weight increasing subsequence problem and is solvable in time O(n log n), where n is the cardinality of the ground set. The k-genome problem can be solved in time O(nk) by dynamic programming.

18. Alignment as a Hitting Set Problem • Ground set: anchor pairs • Goal: delete a minimum-weight set of anchor pairs such that the remaining anchor pairs can be simultaneously synchronized. • Directed edge (u,v): u precedes v . • undirected edge (u,v) : u and v are an anchor pair • Mixed cycle: contains directed and undirected edges, but at least one directed edge. • An edge must be deleted from the set of undirected edges of each mixed cycle (Kececioglu).

20. Solving the Alignment Problem • Run the generic implicit hitting set algorithm, with the elements as anchors and the undirected edge sets of mixed cycles as circuits. • Separation oracles: given a putative hitting set H, search for a mixed cycle in the graph induced by the edges not in H. Two methods: (1) a variant of depth-first search; (2) attempt to align the remaining edges until blocked by the occurrence of a mixed cycle.

21. Performance on 4085 Problems of Aligning Five Worm Genome 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)

22. Tuning the Algorithm • Within the general algorithmic strategy there are many possible choices of the separation oracle, greedy algorithm, versions of CPLEX, parameter choices etc. By tuning these choices on a training set of real-world examples we improved the performance by a factor of several hundred.

23. Acknowledgment • This is joint work with Erick Moreno Centeno