arXiv subjects

Yann Disser

Publications and source records attributed to Yann Disser.

At least 19 recordsLinked to original sources

arXiv2026

Online and Incremental Fractional Vertex Cover on Trees

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

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arXiv2026

Incremental Submodular Maximization: Better Than Greedy

We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

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arXiv2025

Lower bounds for ranking-based pivot rules

The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland's rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.

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arXiv2025

Incremental-Decremental Maximization

We introduce a framework for incremental-decremental maximization that captures the gradual transformation or renewal of infrastructures. In our model, an initial solution is transformed one element at a time and the utility of an intermediate solution is given by the sum of the utilities of the transformed and untransformed parts. We propose a simple randomized and a deterministic algorithm that both find an order in which to transform the elements while maintaining a large utility during all stages of transformation, relative to an optimum solution for the current stage. More specifically, our algorithms yield competitive solutions for utility functions of bounded curvature and/or generic submodularity ratio, and, in particular, for submodular functions, and gross substitute functions. Our results exhibit that incremental-decremental maximization is substantially more difficult than incremental maximization.

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arXiv2025

An unconditional lower bound for the active-set method in convex quadratic maximization

We prove that the active-set method needs an exponential number of iterations in the worst-case to maximize a convex quadratic function subject to linear constraints, regardless of the pivot rule used. This substantially improves over the best previously known lower bound [IPCO 2025], which needs objective functions of polynomial degrees $\omega(\log d)$ in dimension $d$, to a bound using a convex polynomial of degree 2. In particular, our result firmly resolves the open question [IPCO 2025] of whether a constant degree suffices, and it represents significant progress towards linear objectives, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a quadratic objective that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.

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arXiv2025

An unconditional lower bound for the active-set method on the hypercube

The existence of a polynomial-time pivot rule for the simplex method is a fundamental open question in optimization. While many super-polynomial lower bounds exist for individual or very restricted classes of pivot rules, there currently is little hope for an unconditional lower bound that addresses all pivot rules. We approach this question by considering the active-set method as a natural generalization of the simplex method to non-linear objectives. This generalization allows us to prove the first unconditional lower bound for all pivot rules. More precisely, we construct a multivariate polynomial of degree linear in the number of dimensions such that the active-set method started in the origin visits all vertices of the hypercube. We hope that our framework serves as a starting point for a new angle of approach to understanding the complexity of the simplex method.

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arXiv2024

Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial $(\alpha,\mu)$-approximation is possible, i.e., a solution that with budget $B+\alpha$ for all $B \in \mathbb{R}_{\geq 0}$ is a multiplicative $\mu$-approximation compared to the optimum solution with budget $B$. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a $(\chi,1)$-approximation, where $\chi$ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is $(\gamma,2)$-competitive where $\gamma$ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a $(\gamma,3)$-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a $(3\chi,8)$-approximation and, more generally, a $\smash{\bigl((4\ell - 1)\chi, \frac{2^{\ell + 2}}{2^{\ell}-1}\bigr)}$-approximation for every fixed $\ell \in \mathbb{N}$.

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arXiv2024

A $(5/3+ε)$-Approximation for Tricolored Non-crossing Euclidean TSP

In the Tricolored Euclidean Traveling Salesperson problem, we are given~$k=3$ sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave a PTAS based on ``patching'' for the case $k=1$ and, recently, Dross et al.~(2023) generalized this result to~$k=2$. Our contribution is a $(5/3+ε)$-approximation algorithm for~$k=3$ that further generalizes Arora's approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three tours or using an alternative approach based on a weighted solution for $k=2$.

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arXiv2023

A unified worst case for classical simplex and policy iteration pivot rules

We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule. This immediately translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig's rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.

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arXiv2023

Exploration of graphs with excluded minors

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g>0 and recovers the known tight bound for the planar case (g=0).

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arXiv2023

Fractionally Subadditive Maximization under an Incremental Knapsack Constraint

We consider the problem of maximizing a fractionally subadditive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most $\max\{3.293\sqrt{M},2M\}$, under the assumption that the values of singleton sets are in the range $[1,M]$, and we give a lower bound of $\max\{2.618,M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of $\max\{2,M\}$ and an upper bound of $2M$ for the incremental maximization of classical flows with capacities in $[1,M]$ which is tight for the unit capacity case.

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arXiv2023

Tight analysis of the lazy algorithm for open online dial-a-ride

In the open online dial-a-ride problem, a single server has to deliver transportation requests appearing over time in some metric space, subject to minimizing the completion time. We improve on the best known upper bounds on the competitive ratio on general metric spaces and on the half-line, for both the preemptive and non-preemptive version of the problem. We achieve this by revisiting the algorithm $\textsc{Lazy}$ recently suggested in [WAOA, 2022] and giving an improved and tight analysis. More precisely, we show that it has competitive ratio $2.457$ on general metric spaces and $2.366$ on the half-line. This is the first upper bound that beats known lower bounds of 2.5 for schedule-based algorithms as well as the natural $\textsc{Replan}$ algorithm.

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arXiv2023

Incremental Maximization via Continuization

We consider the problem of finding an incremental solution to a cardinality-constrained maximization problem that not only captures the solution for a fixed cardinality, but also describes how to gradually grow the solution as the cardinality bound increases. The goal is to find an incremental solution that guarantees a good competitive ratio against the optimum solution for all cardinalities simultaneously. The central challenge is to characterize maximization problems where this is possible, and to determine the best-possible competitive ratio that can be attained. A lower bound of $2.18$ and an upper bound of $φ+ 1 \approx 2.618$ are known on the competitive ratio for monotone and accountable objectives [Bernstein et al., Math. Prog., 2022], which capture a wide range of maximization problems. We introduce a continuization technique and identify an optimal incremental algorithm that provides strong evidence that $φ+ 1$ is the best-possible competitive ratio. Using this continuization, we obtain an improved lower bound of $2.246$ by studying a particular recurrence relation whose characteristic polynomial has complex roots exactly beyond the lower bound. Based on the optimal continuous algorithm combined with a scaling approach, we also provide a $1.772$-competitive randomized algorithm. We complement this by a randomized lower bound of $1.447$ via Yao's principle.

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arXiv2022

An Improved Algorithm for Open Online Dial-a-Ride

We consider the open online dial-a-ride problem, where transportation requests appear online in a metric space and need to be served by a single server. The objective is to minimize the completion time until all requests have been served. We present a new, parameterized algorithm for this problem and prove that it attains a competitive ratio of $1 + φ\approx 2.618$ for some choice of its parameter, where $φ$ is the golden ratio. This improves the best known bounds for open online dial-a-ride both for general metric spaces as well as for the real line. We also give a lower bound of~$2.457$ for the competitive ratio of our algorithm for any parameter choice.

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arXiv2022

Unified greedy approximability beyond submodular maximization

We consider classes of objective functions of cardinality constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of $γ$-$α$-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, $α$-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient - as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of $\fracαγ\cdot\frac{\mathrm{e}^α}{\mathrm{e}^α-1}$ on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for $α$-augmentable functions. In paritcular, as a by-product, we close a gap left in [Math.Prog., 2020] by obtaining a tight lower bound for $α$-augmentable functions for all $α\geq1$. For weighted rank functions of independence systems, our tight bound becomes $\fracαγ$, which recovers the known bound of $1/q$ for independence systems of rank quotient at least $q$.

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arXiv2022

On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension

We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system $(V,\mathcal{F},\mathcal{B})$ with two families $\mathcal{F},\mathcal{B}$ of subsets of $V$. The task is to find a hitting set for $\mathcal{F}$ that minimizes the maximum number of elements in any of the sets of $\mathcal{B}$. Our focus is on determining the complexity of some special cases of Sparse-HS with respect to the sparseness $k$, which is the optimum number of hitting set elements in any set of $\mathcal{B}$. For the Sparse Vertex Cover (Sparse-VC) problem, $V$ is given by the vertex set of a graph, and $\mathcal{F}$ is its edge set. We prove NP-hardness for sparseness $k\geq 2$ and polynomial time solvability for $k=1$. We also provide a polynomial-time $2$-approximation for any $k$. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family $\mathcal{B}$ is given by vertex neighbourhoods. For this problem we prove NP-hardness for constant $k$ and provide a polynomial-time $(2-\frac{1}{k})$-approximation. This is better than any approximation possible for Sparse-VC or Vertex Cover (under UGC). We then consider two problems derived from Sparse-HS related to the highway dimension, a graph parameter modelling transportation networks. Most algorithms for graphs of low highway dimension compute solutions to the $r$-Shortest Path Cover ($r$-SPC) problem, where $r>0$, $\mathcal{F}$ contains all shortest paths of length between $r$ and $2r$, and $\mathcal{B}$ contains all balls of radius $2r$. There is an XP algorithm that computes solutions to $r$-SPC of sparseness at most $h$ if the input graph has highway dimension $h$, but the existence if an FPT algorithm was open. We prove that $r$-SPC and also the related $r$-Highway Dimension ($r$-HD) problem are both W[1]-hard. Furthermore, we prove that $r$-SPC admits a polynomial-time $O(\log n)$-approximation.

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arXiv2020

An Exponential Lower Bound for Zadeh's pivot rule

The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural, deterministic pivot rules are known to yield exponential running times, the random-facet rule was shown to have a subexponential running time. For a long time, Zadeh's rule remained the most prominent candidate for the first deterministic pivot rule with subexponential running time. We present a lower bound construction that shows that Zadeh's rule is in fact exponential in the worst case. Our construction is based on a close relation to the Strategy Improvement Algorithm for Parity Games and the Policy Iteration Algorithm for Markov Decision Processes, and we also obtain exponential lower bounds for Zadeh's rule in these contexts.

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arXiv2020

Efficient fully dynamic elimination forests with applications to detecting long paths and cycles

We present a data structure that in a dynamic graph of treedepth at most $d$, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time $2^{{\cal O}(d^2)}$, which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time $f(d)$ for some non-elementary (i.e. tower-exponential) function $f$. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth $d$ from doubly-exponential in $d$ to $d^{{\cal O}(d)}$. As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter $k$ and a dynamic graph $G$, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: - Does $G$ contain a simple path on $k$ vertices? - Does $G$ contain a simple cycle on at least $k$ vertices? In the first case, the data structure achieves amortized update time $2^{{\cal O}(k^2)}$. In the second case, the amortized update time is $2^{{\cal O}(k^4)} + {\cal O}(k \log n)$. In both cases we assume access to a dictionary on the edges of $G$.

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