On the complexity of quasiconvex integer minimization problem

Abstract

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered, assuming that the optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on log R to optimize quasiconvex functions in the ball of radius R using only the comparison oracle. On the other hand, if the optimized function is conic, then we show that there is a polynomial on log R algorithm (the dimension is fixed). We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

“…Since K is convex, we have 1 A ∞ • B 1 ⊆ K, and, consequently, r ≥ 1 A ∞ . Finally, let us prove the inequality (7). Again, we need to show that r ≥ 1 A 1 with respect to the l ∞ -norm.…”

“…The state of the art algorithm, due to [9,11] (see also [7,18,34] for more general formulations), for the Feasibility and Optimization Problems 1, 3 (optimization without counting) has the complexity bound O(n) n •poly(φ), where φ = size(A, b) is the bit-encoding length of the system Ax ≤ b. It is a long-standing open problem to break the O(n) n dimension-dependence in the complexity of ILP algorithms.…”

“…In the Integer Feasibility Problem, we need to decide whether P ∩ Z n = ∅ or to find some x ∈ P ∩ Z n in the opposite case. Currently, its state of the art algorithm, due to [9,11] (see also [7,18,34] for more general formulations), has the complexity bound O(n) n • poly(φ), where φ = size(A, b). It is a long-standing open problem to break the O(n) n dimension-dependence in the complexity of ILP algorithms.…”

A faster algorithm for counting the integer points number in $Δ$-modular polyhedra

“…The second and third bounds can be used to obtain a faster algorithm for the ILP feasibility problem, when the parameters m and ∆ are relatively small. For example, taking m = O(d) and ∆ = 2 O(d) in the second bound, it becomes 2 O(d) , which is faster, than the state of the art algorithm, due to [10,11] (see also [7,17,37] Using the Hadamard's inequality, we can write trivial complexity estimates in terms of ∆ 1 := ∆ 1 (A) = A max for Problem 1 in the standard form.…”

Characterizations of mixed binary convex quadratic representable sets

“…(i) they require the construction of an extension f ′ of f to R p or to a ball in R p , such that f ′ is convex (theorem 1 in [23]), conic (theorem 14 in [4]) or discrete convic (theorem 1 in [26]); (ii) they show that the running time is FPT only in expectation, as opposed to worst-case (theorem 7.5.1 in [6], theorem 10 in [12]), or polynomial time in fixed dimension, which is weaker than FPT (theorem 1 in [23]). Furthermore, most of these results require a subgradient oracle for f (theorem 7.5.1 in [6], theorem 10 in [12]) or for an extension of f (theorem 1 in [23]).…”