LKPBound: p-value Through a Continuous Relaxation
In elec.strat: Functions for election audits using stratified random samples

Description Usage Arguments Author(s) References See Also Examples

Finds a p-value through the LKP Bound: a continuous relaxation bound of the original 0-1 knapsack problem. Offers an option to include a lower-bound in output, thus computing an upper and lower bound on the exact p-value. See Higgins, Rivest, Stark for more details.

	LKPBound(Z, t = NULL, asTaint = FALSE, asNumber = FALSE, 
		M = NULL, takeOutZeroMMB = TRUE, LKP.lower.bound = FALSE, 
		bound.col = "e.max",  
		calc.e_p=calc.pairwise.e_p, w_p = weight.function("no.weight"))

`Z`	A `strat.elec.data` object.
`t`	Value of the observed maximum, either as the MRO, as taint, or as the overstatement of the margin in votes.
`asTaint`	Set `asTaint = TRUE` if `t` is the maximum observed taint.
`asNumber`	Set `asNumber` if `t` is the maximum observed overstatement of the margin in votes.
`M`	A priori margin. If NULL, `M` defaults to 1.
`takeOutZeroMMB`	Setting `takeOutZeroMMB = TRUE` will consider batches with a `maximumMarginBound` of zero as having no chance of being sampled.
`LKP.lower.bound`	Set `LKP.lower.bound = TRUE` to compute a lower-bound of the exact p-value in addition to the upper-bound computed from the continuous relaxation. Lower-bound computed according to Higgins, Rivest, Stark.
`bound.col, calc.e_p, w_p`	Arguments used to compute `t` from audit data, instead of passing `t` directly. These arguments are ignored if `t` is not NULL. See `compute.stark.t` for details.

Mike Higgins, Hua Yang

M. Higgins, R. L. Rivest, P. B. Stark. Sharper p-Values for Stratified Election Audits

See eqValBound and withReplaceBound for finding a p-value through other relaxations. See BaB for finding an exact p-value through solving a 0-1 knapsack problem. See compute.stark.t for computing t through audit data.

	data(MN_Senate_2006)
	LKPBound(MN_Senate_2006.strat, takeOutZeroMMB = FALSE)
	LKPBound(MN_Senate_2006.strat, t = 2, asNumber = TRUE, 
		takeOutZeroMMB = FALSE, LKP.lower.bound = TRUE)
	
	data(CA_House_2008)	
	CA_House_2008.strat[[1]]$strat$audit <- 1
	LKPBound(CA_House_2008.strat[[1]], t = 0, LKP.lower.bound = TRUE)