arXiv subjects

Sander Borst

Publications and source records attributed to Sander Borst.

13 recordsLinked to original sources

arXiv2026

Fair Division Meets Scheduling: Approximately Envy-Free Interval Scheduling

We study interval scheduling from the perspective of fair allocation. There are $m$ identical machines and a set of intervals, each specified by a start time, an end time, and a nonnegative weight. A schedule assigns a subset of the intervals to the machines so that no two intervals on the same machine overlap, and the goal is to maximize the total weight of scheduled intervals. Viewing machines as agents and intervals as goods, we require the schedule to be envy-free up to one item (EF1), and we measure efficiency against the offline optimum without fairness. In the offline setting, we give an algorithm that computes an EF1 schedule whose loss is at most a factor of $3/2$ in the unweighted regime, and we prove lower bounds of $\frac{3m-2}{2m-1}$, approaching $3/2$, in both the unweighted and the unit-length weighted regimes, so the price of fairness is $3/2$ in the limit. In the online setting, intervals arrive in nondecreasing order of start times; an arriving interval must be accepted or rejected, rejections are irrevocable, and an accepted interval may be revoked, and lost, at any time before it ends. For the unweighted regime we present Greedy-Balanced, a simple algorithm that maintains EF1 at every point in time and is $(2-\tfrac{1}{m})$-competitive against the offline optimum without fairness, and we prove a matching lower bound for every deterministic algorithm; the optimal deterministic fair competitive ratio is thus exactly $2-\tfrac{1}{m}$. Experiments on real-world benchmark instances show that Greedy-Balanced performs well beyond its worst-case guarantee, with an observed ratio never exceeding $1.306$.

cs.DS↗

arXiv2026

Beyond the Half-Approximation: Fair and Efficient Online Class Matching

Online bipartite matching, where agents are known in advance but items arrive sequentially and must be irrevocably assigned, is fundamental to problems ranging from ride-sharing to online advertising. When agents belong to classes such as demographic groups or geographic regions, fairness demands equitable treatment across these groups. Recent work introduced class envy-freeness (CEF), a natural extension of the classical fair division notion: an algorithm is $\alpha$-CEF if each class receives value at least an $\alpha$ fraction of what it could extract from any other class's bundle. However, all known algorithms achieving constant-factor CEF guarantees attain utilitarian social welfare (total matching value) of at most $\frac{1}{2}$ times the optimum, far below the $1-\frac{1}{e} \approx 0.632$ achievable without fairness constraints. We resolve the open question of whether fairness necessitates this efficiency loss, by introducing threshold-based algorithms parameterized by $\gamma \in [0,1]$ that equalize allocations across classes until threshold $\gamma$, then maximize efficiency. For divisible matching, this yields simultaneous $(1-e^{-\gamma})$-CEF and $(1 - \frac{e^{\gamma-1}}{\gamma+1})$-USW guarantees; for indivisible matching, $\frac{\gamma}{2}$-CEF with the same USW. Setting $\gamma > 0$ produces the first algorithms beating $\frac{1}{2}$-USW while maintaining constant CEF. We complement this with a novel upper bound construction, proving no non-wasteful $\alpha$-CEF algorithm can exceed $\frac{1 +\alpha - e^{\alpha-1}}{1+\alpha}$-USW and correcting prior bounds that were vacuous for $\alpha < 0.58$. Our upper bound nearly matches our algorithms' performance, giving the first substantive characterization of the price of fairness in online class matching.

cs.GT↗

arXiv2025

The SCIP Optimization Suite 10.0

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in SCIP Optimization Suite 10.0. The updates in SCIP 10.0 include a new solving mode for exactly solving rational mixed-integer linear programs, a new presolver for detecting implied integral variables, a novel cut-based conflict analysis and separator for flower inequalities, two new heuristics, a novel tool for explaining infeasibility, a new interface for nonlinear solvers as well as improvements in symmetry handling, branching strategies, and SCIP's Benders' decomposition framework. SCIP Optimization Suite 10.0 also includes new and improved features in the the presolving library PaPILO, the parallel framework UG, and the decomposition framework GCG. Moreover, the SCIP Optimization Suite 10.0 contains MIP-DD, the first open-source delta debugger for mixed-integer programming solvers. These additions and enhancements have resulted in an overall performance improvement of SCIP in terms of solving time, number of nodes in the branch-and-bound tree, as well as the reliability of the solver.

math.OC↗

arXiv2025

Improved Online Load Balancing in the Two-Norm

We study the online load balancing problem on unrelated machines, with the objective of minimizing the square of the $\ell_2$ norm of the loads on the machines. The greedy algorithm of Awerbuch et al. (STOC'95) is optimal for deterministic algorithms and achieves a competitive ratio of $3 + 2 \sqrt{2} \approx 5.828$, and an improved $5$-competitive randomized algorithm based on independent rounding has been shown by Caragiannis (SODA'08). In this work, we present the first algorithm breaking the barrier of $5$ on the competitive ratio, achieving a bound of $4.9843$. To obtain this result, we use a new primal-dual framework to analyze this problem based on a natural semidefinite programming relaxation, together with an online implementation of a correlated randomized rounding procedure of Im and Shadloo (SODA'20). This novel primal-dual framework also yields new, simple and unified proofs of the competitive ratio of the $(3 + 2 \sqrt{2})$-competitive greedy algorithm, the $5$-competitive randomized independent rounding algorithm, and that of a new $4$-competitive optimal fractional algorithm. We also provide lower bounds showing that the previous best randomized algorithm is optimal among independent rounding algorithms, that our new fractional algorithm is optimal, and that a simple greedy algorithm is optimal for the closely related online scheduling problem $R || \sum w_j C_j$.

cs.DS↗

arXiv2025

To buy or not to buy: deterministic rent-or-buy problems on node-weighted graphs

We study the rent-or-buy variant of the online Steiner forest problem on node- and edge-weighted graphs. For $n$-node graphs with at most $\bar{n}$ non-zero node-weights, and at most $\tilde{k}$ different arriving terminal pairs, we obtain a deterministic, $O(\log n \log \bar{n})$-competitive algorithm. This improves on the previous best, $O(\log^4 n)$-competitive algorithm obtained by the black-box reduction from (Bartal et al. 2021) combined with the previously best deterministic algorithms for the simpler 'buy-only' setting. We also obtain a deterministic, $O(\bar{n}\log \tilde{k})$-competitive algorithm. This generalizes the $O(\log \tilde{k})$-competitive algorithm for the purely edge-weighted setting from (Umboh 2015). We also obtain a randomized, $O(\log \tilde{k} \log \bar{n})$-competitive algorithm. All previous approaches were based on the randomized, black-box reduction from~\cite{AwerbuchAzarBartal96} that achieves a $O(\log \tilde{k} \log n)$-competitive ratio when combined with an algorithm for the 'buy-only' setting. Our key technical ingredient is a novel charging scheme to an instance of \emph{online prize-collecting set cover}. This allows us to extend the witness-technique of (Umboh 2015) to the node-weighted setting and obtain refined guarantees with respect to $\bar{n}$, already in the much simpler 'buy-only' setting.

cs.DS↗

arXiv2024

Stronger adversaries grow cheaper forests: online node-weighted Steiner problems

We propose a $O(\log k \log n)$-competitive randomized algorithm for online node-weighted Steiner forest. This is essentially optimal and significantly improves over the previous bound of $O(\log^2 k \log n)$ by Hajiaghayi et al. [2017]. In fact, our result extends to the more general prize-collecting setting, improving over previous works by a poly-logarithmic factor. Our key technical contribution is a randomized online algorithm for set cover and non-metric facility location in a new adversarial model which we call semi-adaptive adversaries. As a by-product of our techniques, we obtain the first deterministic $O(\log |C| \log |F|)$-competitive algorithm for non-metric facility location.

cs.DS↗

arXiv2024

Certified Constraint Propagation and Dual Proof Analysis in a Numerically Exact MIP Solver

This paper presents the integration of constraint propagation and dual proof analysis in an exact, roundoff-error-free MIP solver. The authors employ safe rounding methods to ensure that all results remain provably correct, while sacrificing as little computational performance as possible in comparison to a pure floating-point implementation. The study also addresses the adaptation of certification techniques for correctness verification. Computational studies demonstrate the effectiveness of these techniques, showcasing a 23% performance improvement on the MIPLIB 2017 benchmark test set.

math.OC↗

arXiv2024

Online Matching on $3$-Uniform Hypergraphs

The online matching problem was introduced by Karp, Vazirani and Vazirani (STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the optimal competitive ratio is $1-1/e$ for both integral and fractional versions of the problem. Since then, there has been considerable effort to find optimal competitive ratios for other related settings. In this work, we go beyond the graph case and study the online matching problem on $k$-uniform hypergraphs. For $k=3$, we provide an optimal primal-dual fractional algorithm, which achieves a competitive ratio of $(e-1)/(e+1)\approx 0.4621$. As our main technical contribution, we present a carefully constructed adversarial instance, which shows that this ratio is in fact optimal. It combines ideas from known hard instances for bipartite graphs under the edge-arrival and vertex-arrival models. For $k\geq 3$, we give a simple integral algorithm which performs better than greedy when the online nodes have bounded degree. As a corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform hypergraphs when every online node has degree at most 2. This is because the special case where every online node has degree 1 is equivalent to the edge-arrival model on graphs, for which an upper bound of 1/2 is known.

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arXiv2022

A nearly optimal randomized algorithm for explorable heap selection

Explorable heap selection is the problem of selecting the $n$th smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized $n\cdot \exp(O(\sqrt{\log{n}}))$ time algorithms using $O(\log(n)^{2.5})$ and $O(\sqrt{\log n})$ space respectively. We present a new randomized algorithm with running time $O(n\log(n)^3)$ using $O(\log n)$ space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an $\Omega(\log(n)n/\log(\log(n)))$ for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal.

cs.DS↗

arXiv2022

Integrality Gaps for Random Integer Programs via Discrepancy

We prove new bounds on the additive gap between the value of a random integer program $\max c^Tx,\ Ax\leq b,\ x\in\{0,1\}^n$ with $m$ constraints and that of its linear programming relaxation for a wide range of distributions on $(A,b,c)$ . We are motivated by the work of Dey, Dubey, and Molinaro (SODA '21), who gave a framework for relating the size of Branch-and-Bound (B&B) trees to additive integrality gaps. Dyer and Frieze (MOR '89) and Borst et al. (Mathematical Programming '22), respectively, showed that for certain random packing and Gaussian IPs, where the entries of $A,c$ are independently distributed according to either the uniform distribution on $[0,1]$ or the Gaussian distribution $\mathcal{N}(0,1)$, the integrality gap is bounded by $O_m(\log^2 n / n)$ with probability at least $1-1/n-e^{-\Omega_m(1)}$. In this paper, we generalize these results to the case where the entries of $A$ are uniformly distributed on an integer interval (e.g., entries in $\{-1,0,1\}$), and where the columns of $A$ are distributed according to an isotropic logconcave distribution. Second, we substantially improve the success probability to $1-1/poly(n)$, compared to constant probability in prior works (depending on $m$). Leveraging the connection to Branch-and-Bound, our gap results imply that for these IPs B&B trees have size $n^{poly(m)}$ with high probability (i.e., polynomial for fixed $m$), which significantly extends the class of IPs for which B&B is known to be polynomial. Our main technical contribution is a new linear discrepancy theorem for random matrices. Our theorem gives general conditions under which a target vector is equal to or very close to a $\{0,1\}$ combination of the columns of a random matrix $A$ . The proof uses a Fourier analytic approach, building on work of Hoberg and Rothvoss (SODA '19) and Franks and Saks (RSA '20).

math.OC↗

arXiv2020

Majorizing Measures for the Optimizer

The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes. One of the crowning achievements of the theory is Talagrand's tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum. In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof of the majorizing measures theorem based on two parts: (1) We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program. This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization. (2) We obtain tree-based upper and lower bound certificates by rounding, in a series of steps, the primal and dual solutions to this convex program. [...]

math.PR↗

arXiv2020

On the Integrality Gap of Binary Integer Programs with Gaussian Data

For a binary integer program (IP) ${\rm max} ~ c^\mathsf{T} x, Ax \leq b, x \in \{0,1\}^n$, where $A \in \mathbb{R}^{m \times n}$ and $c \in \mathbb{R}^n$ have independent Gaussian entries and the right-hand side $b \in \mathbb{R}^m$ satisfies that its negative coordinates have $\ell_2$ norm at most $n/10$, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by $\operatorname{poly}(m)(\log n)^2 / n$ with probability at least $1-2/n^7-2^{-\operatorname{poly}(m)}$. Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on $m$ instead of exponentially. Building upon recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), we show that the integrality gap implies that branch-and-bound requires $n^{\operatorname{poly}(m)}$ time on random Gaussian IPs with good probability, which is polynomial when the number of constraints $m$ is fixed. We derive this result via a novel meta-theorem, which relates the size of branch-and-bound trees and the integrality gap for random logconcave IPs.

math.OC↗

arXiv2020

New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees

We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of $m$ binary trees with $n$ leaves each with a running time of $O(5^k\cdot n\cdot m)$, where $k$ is the minimum temporal hybridization number. We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most $d$ and at most $k$ reticulations in $O((8k)^d5^ k\cdot n\cdot m)$ time. Lastly, we introduce a $O(6^kk!\cdot k\cdot n^2)$ time algorithm for computing a minimum temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

cs.DS↗