Abstract
We prove that integer programming with three alternating quantifiers is NP-complete, even for a fixed number of variables. This complements earlier results by Lenstra [ 16 ] [Lenstra H ( 1983 ) Integer programming with a fixed number of variables. Math. Oper. Res. 8(4):538–548.] and Kannan [ 13, 14 ] [Kannan R ( 1990 ) Test sets for integer programs, ∀ ∃ sentences. Polyhedral Combinatorics (American Mathematical Society, Providence, RI), 39–47. Kannan R ( 1992 ) Lattice translates of a polytope and the Frobenius problem. Combinatorica 12(2):161–177.], which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes [Formula: see text], counting the projections of integer points in Q\P is #P-complete. This contrasts the 2003 result by Barvinok and Woods [ 5 ] [Barvinok A, Woods K ( 2003 ) Short rational generating functions for lattice point problems. J. Amer. Math. Soc. 16(4):957–979.], which allows counting in polynomial time the projections of integer points in P and Q separately.
Highlights
In a pioneer paper [19], Lenstra showed that Integer Programming in a bounded dimension can be solved in polynomial time
Theorem 2 refutes the possibility of any reduction from (1.2) to an easier form with m, n and q bounded for which decision could be in polynomial time, unless P = NP
Given a polytope P ⊂ Rd, the number of integer points in P ∩ Zd can be computed in polynomial time
Summary
In a pioneer paper [19], Lenstra showed that Integer Programming in a bounded dimension can be solved in polynomial time. Given a polyhedron P ⊆ Rd1 , a matrix A ∈ Zm×(d1+d2) and a vector b ∈ Zm, the following sentence can be decided in polynomial time:. By an easy application of the Doignon–Bell–Scarf theorem, (1.1) is polynomial time reducible to the case with m and n fixed. This simple reduction breaks down when there are more than two quantifiers (see Section 7.1) as in (1.2). Theorem 2 refutes the possibility of any reduction from (1.2) to an easier form with m, n and q bounded for which decision could be in polynomial time, unless P = NP.
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