arXiv subjects

Koen Ligthart

Publications and source records attributed to Koen Ligthart.

3 recordsLinked to original sources

arXiv2026

Value Functions of Separable Convex Integer Programs are Periodically Convex

We consider the periodic behavior of the value functions $b\mapsto\min\{f(x)\ \vert\ Ax=b,\,x\in\mathbb Z_{\ge0}^n\}$ of integer programs. We show that there exists a positive integer $M$ depending only on the constraint matrix $A\in\mathbb Z^{m\times n}$ so that the value function is convex extensible on any subdomain of the form $r+M\mathbb Z^m$ for any $r\in\mathbb Z^m$ and any separable convex objective function $f$. With this, we extend the known periodic convexity of such functions for linear objective functions $f$, as established by Eisenbrand and Rothvoss (SODA 25), to the broader class of separable convex objective functions. We derive our main periodic convexity result by first showing that periodic convexity along lines is equivalent to the integer decomposition property of dilated polyhedra. Subsequently, we use Graver basis techniques to extend the $1$-dimensional periodic convexity to domains of arbitrary fixed dimension. We apply this periodic convexity to show that value function reformulations of block-structured integer programs become periodically convex, which yields fixed-parameter tractable (FPT) algorithms. More specifically, we optimize two-stage stochastic integer programs and $n$-fold integer programs in FPT time when parameterized by the block dimensions and coefficient size of the local blocks of the constraint matrix, allowing the coefficients of the global blocks to be large. In the setting of this parameterization, which was recently introduced by Cslovjecsek, Kouteck\'y, Lassota, Pilipczuk, and Polak (TheoretiCS 2025), our algorithms exponentially improve on the running times of the previous optimization algorithms and extend the class of objective functions that can be optimized from linear to separable convex.

math.OC↗

arXiv2026

Solving 4-Block Integer Linear Programs Faster Using Affine Decompositions of the Right-Hand Sides

We present a new and faster algorithm for the 4-block integer linear programming problem, overcoming the long-standing runtime barrier faced by previous algorithms that rely on Graver complexity or proximity bounds. The 4-block integer linear programming problem asks to compute $\min\{c_0^\top x_0+c_1^\top x_1+\dots+c_n^\top x_n\ \vert\ Ax_0+Bx_1+\dots+Bx_n=b_0,\ Cx_0+Dx_i=b_i\ \forall i\in[n],\ (x_0,x_1,\dots,x_n)\in\mathbb Z_{\ge0}^{(1+n)k}\}$ for some $k\times k$ matrices $A,B,C,D$ with coefficients bounded by $\overline\Delta$ in absolute value. Our algorithm runs in time $f(k,\overline\Delta)\cdot n^{k+\mathcal O(1)}$, improving upon the previous best running time of $f(k,\overline\Delta)\cdot n^{k^2+\mathcal O(1)}$ [Oertel, Paat, and Weismantel (Math. Prog. 2024), Chen, Kouteck\'y, Xu, and Shi (ESA 2020)]. Further, we give the first algorithm that can handle large coefficients in $A, B$ and $C$, that is, it has a running time that depends only polynomially on the encoding length of these coefficients. We obtain these results by extending the $n$-fold integer linear programming algorithm of Cslovjecsek, Kouteck\'y, Lassota, Pilipczuk, and Polak (SODA 2024) to incorporate additional global variables $x_0$. The central technical result is showing that the exhaustive use of the vector rearrangement lemma of Cslovjecsek, Eisenbrand, Pilipczuk, Venzin, and Weismantel (ESA 2021) can be made \emph{affine} by carefully guessing both the residue of the global variables modulo a large modulus and a face in a suitable hyperplane arrangement among a sufficiently small number of candidates. This facilitates a dynamic high-multiplicy encoding of a \emph{faithfully decomposed} $n$-fold ILP with bounded right-hand sides, which we can solve efficiently for each such guess.

cs.CC↗

arXiv2025

Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns

We study integer linear programs (ILP) of the form $\min\{c^\top x\ \vert\ Ax=b,l\le x\le u,x\in\mathbb Z^n\}$ and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the well-established approach of capturing the hardness of a problem by the distance to triviality. The generalized matching problem is an ILP where each column of the constraint matrix has a $1$-norm of at most $2$. It captures several well-known polynomial time solvable problems such as matching and flow problems. We parameterize by the size of variable and constraint backdoors, which measure the least number of columns or rows that must be deleted to obtain a generalized matching ILP. We present the following results: (i) a fixed-parameter tractable (FPT) algorithm for ILPs parameterized by the size $p$ of a minimum variable backdoor to generalized matching; (ii) a randomized slice-wise polynomial (XP) time algorithm for ILPs parameterized by the size $p+h$ of a mixed variable plus constraint backdoor to generalized matching as long as $c$ and $A$ are encoded in unary; (iii) we complement (ii) by proving that solving ILPs is W[1]-hard when parameterized by the size of a minimum constraint backdoor $h$ even when all coefficients are bounded. To obtain (i), we prove a variant of lattice-convexity of the degree sequences of weighted $b$-matchings, which we study in the light of SBO jump M-convex functions. This allows us to model the matching part as a polyhedral constraint on the integer backdoor variables. The resulting ILP is solved using an FPT integer programming algorithm. For (ii), the randomized XP time algorithm is obtained by pseudo-polynomially reducing the problem to the exact matching problem. To prevent an exponential blowup in terms of the encoding length of $b$, we bound the proximity of the ILP through a subdeterminant based circuit bound.

cs.CC↗