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Implicit Hitting Set Problems

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Implicit Hitting Set Problems

Richard M. Karp

Harvard University August 29, 2011

- 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.

- 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.

- 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.

- 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

- 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

- 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.

- Greedy algorithm gives good approximate solutions.
- CPLEX integer programming algorithm often gives optimal solutions rapidly.

- 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.

- 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.

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

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.

- 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.

- Number of circuits generated, number of calls to solver, running time of generator.

- 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.

- 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.

- 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).

- 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.

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)

- 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.

- This is joint work with Erick Moreno Centeno