arXiv subjects

Cinar Ari

Publications and source records attributed to Cinar Ari.

4 recordsLinked to original sources

arXiv2026

Bounded Cubic Integer Programming in Fixed Dimension

We consider the exact minimization of a rational cubic polynomial over the integer points of a rational polytope in fixed dimension. Del Pia, Hildebrand, Weismantel, and Zemmer proved polynomial-time solvability in dimension two, while quartic polynomial minimization is already NP-hard in dimension two. We show that the bounded cubic result extends to every fixed dimension. The proof combines two exact identities for cubic polynomials with three ingredients from the recent fixed-dimensional algorithms for integer quadratic programming of Ari and Hildebrand: symmetric displacement covers, negative-displacement search for quadratic forms, and integer-query convex feasibility. At an integer query point, a negative Hessian direction yields a linear curvature cut valid for every global minimizer in the current cell, while absence of such a direction yields a linear objective cut through an exact endpoint-Hessian identity. These cuts define an integer-query separation oracle for the convex hull of the curvature-admissible sublevel points. Integer convex feasibility via ellipsoids and lattice algorithms then gives exact optimization without computing lattice centerpoints. A real-algebraic localization bound extends the result to unbounded polyhedra with a bounded real improving sublevel, including all coercive objectives. The argument also explains why degree three is a natural boundary for this approach: the symmetric second difference of a cubic is exactly its directional Hessian, which is affine in the basepoint; for quartics an additional fourth-order term appears.

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arXiv2026

Curvature batching for integer and mixed-integer quadratic programming

We give an explicit algorithm for integer quadratic programming with running time $(nL)^{O(n^2)} \mathrm{poly}(ϕ)$, where $n$ is the number of variables, $L$ bounds the absolute entries of the constraint and quadratic matrices, and $ϕ$ is the total input encoding length. The linear objective and constraint right-hand sides enter only through $ϕ$. The algorithm refines Lokshtanov's branching framework. Small integer kernel directions identify optima near constraint boundaries. When all chosen basis directions have nonnegative curvature, their gradient values can be imposed in one batch. The quadratic form then vanishes on the remaining kernel, and the residual problem is an integer linear program. Thus determinant growth after the batch does not feed into further branching. Over a polytope, the method extends to mixed-integer quadratic programming with $q$ continuous variables in time $(nL)^{O(n^2 (q+1)^2)} \mathrm{poly}(ϕ)$, without a convexity assumption on the continuous block. We also decide and certify integer and mixed-integer unboundedness in $(m+n)^{O(n)} \mathrm{poly}(ϕ)$ time for arbitrary rational polyhedra with $m$ inequalities. Noncopositivity on polyhedral cones and integer unboundedness are W[1]-hard in the dimension and admit no $(m+ϕ)^{o(n)}$ algorithm under ETH, matching the linear order of dimension in the recession-test exponent. Companion results give a right-hand-side-independent bound for concave integer minimization and explicit runtimes for applications of IQP.

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arXiv2026

Bounded Integer Quadratic Programming through Parallelepiped Covers and Discrete Convic Optimization

We give an exact algorithm for minimizing an arbitrary rational quadratic polynomial $x^T Q x + c^T x + γ$ over the integer points of a bounded rational polyhedron $\{x : Ax \le b\}$. For $n$ variables and $m$ inequalities, the running time is $$2^{O(n \log(n+1))} (m+1)^{O(n)} β^{O(n)} (1+L)^{O(1)},$$ where $β$ is one plus the maximum binary encoding length of an entry of $A$ or $Q$, and $L$ is the full input encoding length. The polynomial degree in $L$ is absolute; in particular, the encoding of $b$, $c$, and $γ$ enters only this fixed-degree factor. Our algorithm refines the integral parallelepiped cover of Goemans and Rothvoss into cells with controlled displacement sets. On each cell, either an integer displacement of negative quadratic value rules out every point as a global minimizer, or the quadratic identity supplies a separation oracle for the convex hull of each integer objective sublevel set. The integer-query feasibility theorem of Hildebrand and Goess then finds the cell minimum. A refinement through scaled lattices solves the displacement search, and a localization argument separates the encoding dependence of the constraint matrix and quadratic part from that of the remaining data. Combining the bounded algorithm with the unboundedness test of Ari and Hildebrand extends the method to arbitrary rational polyhedra.

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arXiv2026

Integer Maximization over $\ell_p$ Balls: Hardness and Exact Algorithms

We study the problem of maximizing a linear function over the integer points of an origin-centered $\ell_p$ ball, which we call \BallIPp{p}. For every fixed integer $p\ge2$, we prove that the decision problem over an $\ell_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \BallIP{} instance in dimension $d+k\le2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.

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