arXiv subjects

Kurt Mehlhorn

Publications and source records attributed to Kurt Mehlhorn.

At least 19 recordsLinked to original sources

arXiv2026

Complete EFX Allocations Exist for Four Additive Agents and Up to Nine Goods

We prove that every fair-division instance with four agents, additive valuations over the non-negative reals, and at most nine indivisible goods admits a \emph{complete} allocation that is envy-free up to any good in the strong, zero-tolerant sense ($\EFXo$). The case $m=9=n+5$ lies beyond the previously known frontier for complete EFX with four agents ($m\le n+3$). The proof combines a small set of hand-proven reduction lemmas with a machine-verified certificate corpus. The valuation polytope is covered by a collection of smaller polytopes. For each smaller polytope $P$, a family $F$ of allocations is found that contains an $\EFXo$ allocation for every valuation in $P$. The check that $F$ suffices for $P$ is a quantifier-free linear-arithmetic unsatisfiability verdict, re-derived and solved from scratch by an independent certifier, corroborated per clause, and re-verifiable by a independent small third implementation. The $m=8$ case is established twice: by an earlier independent project at that size and as a one-paragraph padding corollary of the $m=9$ theorem. We additionally give a possible explanation why the problem is hard: difficulty concentrates on near-identical valuations, where only ${\approx}0.14\%$ of all $4^9$ allocations are $\EFXo$, and explicit valuation pairs inside a single region force opposite mandatory allocation structure, evidence relevant to the general conjecture independently of any solver stack.

cs.GT↗

arXiv2026

A Simple Polynomial-Time EFX Repair for Cancelable Valuations

The leximin++ proof of Plaut and Roughgarden for agents with identical monotone valuations gives a natural EFX-repair procedure: starting from an arbitrary partition, repeatedly transfer an eligible item to a minimum-valued bundle. The procedure terminates, but the standard argument gives no polynomial bound on the number of transfers, even for additive valuations. We show that a single deterministic tie-breaking rule makes this repair procedure polynomial for the broader class of cancelable valuations. Fix an ordering of the items consistent with their singleton values and always transfer the highest-ranked eligible item. Consecutive transferred items strictly decrease in this ordering, and hence the algorithm performs at most $m$ transfers, where $m$ is the number of items. Moreover, the repair procedure does not decrease the minimum bundle value or increase the maximum bundle value. As an application, for every fixed $\varepsilon>0$, we compute in polynomial time an allocation of restricted additive chores that is simultaneously EFX, $(1+\varepsilon)$-MMS, and a $2$-approximation to the optimal social cost. This improves upon the previous polynomial-time $4/3$-MMS guarantee. Finally, we exhibit a monotone cancelable ordering on five items with no additive representation, showing that the extension beyond additivity is genuine.

cs.GT↗

arXiv2026

The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees

We study the problem of fairly allocating indivisible goods among agents with monotone valuations. We introduce a new share-based fairness notion, the residual maximin share (RMMS), and show that it provides a unified framework for several existing lone-divider style techniques in fair division. RMMS satisfies two key properties: feasibility and self-maximization. Using RMMS, we give simple proofs of the existence of partial allocations that are both RMMS and envy-free up to any good (EFX), and complete allocations that are both RMMS and envy-free up to one good (EF1), in fact satisfying the stronger notion of EFL. This unifies and strengthens several previously known results. We further demonstrate the power of the share-based approach by studying the compatibility of fairness notions related to the long-standing EFX problem. While allocations satisfying either epistemic EFX (EEFX) or EF1 are known to exist for general monotone valuations, whether they can always be achieved simultaneously has remained open in every setting where EFX existence itself is unresolved. For additive valuations, we resolve this question affirmatively by proving the existence of allocations that satisfy both EEFX and EFL. Our proof introduces the strong EEFX share, a new share notion implying EEFX feasibility of bundles. We show that the strong EEFX share is upper bounded by RMMS, enabling us to derive EEFX+EFL allocations via the RMMS framework. This answers the main open question of Akrami and Rathi (2025). Finally, although our algorithm for computing EEFX and EF1 allocations may take exponential time in general, we develop a polynomial-time algorithm for restricted additive valuations. Unlike the lone-divider approach, our algorithm exploits the structural properties of restricted additive valuations to compute allocations satisfying both EEFX and EF1.

cs.GT↗

arXiv2026

EFX Allocations Exist on Multi-Graphs

We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central fairness notion is envy-freeness up to any good (EFX), which requires that any envy toward another agent vanishes after the removal of any single good from the latter's bundle. The existence of EFX allocations is considered a major open problem in fair division. So far, it has only been established in limited settings. Christodoulou et al. [2023] proved the existence of EFX allocations for graphical valuations. In this setting, the agents correspond to the nodes of an underlying graph, and the goods correspond to the edges, and any good has positive value only for the endpoints of the corresponding edge. Their proof crucially relies on the restriction that the graph is simple, meaning that for any pair of agents, there is at most one good that has value to both. For multigraph valuations, where multiple goods may be valued by the same pair of agents, only partial results are known. Amanatidis et al. [2024] and Kaviani et al. [2025] obtained 2/3 and sqrt(2)/2 approximations of EFX, respectively; Kaviani et al. [2024] established existence under restricted additive valuations; and Afshinmehr et al. [2025a] proved existence under the assumption that the shortest cycle containing non-parallel edges has length at least 4. In this paper, we resolve this open problem by proving the existence of EFX allocations for multigraph instances under cancelable valuations, a strict superclass of additive valuation functions. Our proof is algorithmic and computes such allocations in polynomial time when the valuation functions are cancelable. This work contributes to the small number of EFX existence results that apply to an arbitrary number of agents.

cs.GT↗

arXiv2026

A Counterexample to EFX $n \ge 3$ Agents, $m \ge n + 5$ Items, Submodular Valuations via SAT-Solving

The existence of EFX allocations is a central open problem in discrete fair division. An allocation is EFX (envy-free up to any good) if no agent envies another agent after the removal of any single good from the other agent's bundle. We resolve this longstanding question by providing the \textbf{first-ever counterexample} to the existence of EFX allocations for agents with monotone valuations, which in turn immediately implies a counterexample for submodular valuations. Specifically, we show that EFX allocations need not exist for instances with $n \ge 3$ agents and $m \ge n+5$ goods. In contrast, we prove that every instance with three agents and seven goods admits an EFX allocation. Both results are obtained via SAT solving. We encode the negation of EFX existence as a SAT instance: satisfiability yields a counterexample, while unsatisfiability establishes universal existence. The correctness of the encoding is formally verified in Lean. Finally, we establish positive guarantees for fair allocations with three agents and an arbitrary number of goods. Although EFX allocations may fail to exist, we prove that every instance with three agents and monotone valuations admits at least one of two natural relaxations of EFX: tEFX, or EF1 and EEFX.

cs.GT↗

arXiv2026

Gabow's $O(\sqrt{n}m)$ Maximum Cardinality Matching Algorithm, Revisited

We revisit Gabow's $O(\sqrt{n} m)$ maximum cardinality matching algorithm (The Weighted Matching Approach to Maximum Cardinality Matching, Fundamenta Informaticae, 2017). It adapts the weighted matching algorithm of Gabow and Tarjan~\cite{GT91} to maximum cardinality matching. Gabow's algorithm works iteratively. In each iteration, it constructs a maximal number of edge-disjoint shortest augmenting paths with respect to the current matching and augments them. It is well-known that $O(\sqrt{n})$ iterations suffice. Each iteration consists of three parts. In the first part, the length of the shortest augmenting path is computed. In the second part, an auxiliary graph $H$ is constructed with the property that shortest augmenting paths in $G$ correspond to augmenting paths in $H$. In the third part, a maximal set of edge-disjoint augmenting paths in $H$ is determined, and the paths are lifted to and augmented to $G$. We give a new algorithm for the first part. Gabow's algorithm for the first part is derived from Edmonds' primal-dual algorithm for weighted matching. We believe that our approach is more direct and will be easier to teach. We have implemented the algorithm; the implementation is available at the companion webpage (https://people.mpi-inf.mpg.de/~mehlhorn/CompanionPageGenMatchingImplementation.html).

cs.DS↗

arXiv2026

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.

cs.GT↗

arXiv2025

EFX Allocations Exist on Triangle-Free Multi-Graphs

We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central open question in this area is the existence of EFX allocations-allocations in which any envy of any agent i towards any agent j vanishes upon the removal of any single good from j's bundle. Establishing the existence of such allocations has proven notoriously difficult in general, but progress has been made for restricted valuation classes. Christodoulou et al. [2023] proved existence for graphical valuations, where goods correspond to edges in a graph, agents to nodes, and each agent values only incident edges. The graph was required to be simple, i.e., for any pair of agents, there could be at most one good that both agents value. The problem remained open, however, for multi-graph valuations, where for a pair of agents several goods may have value to both. In this setting, Sgouritsa and Sotiriou [2025] established existence whenever the shortest cycle with non-parallel edges has length at least six, while Afshinmehr et al. [2025] proved existence when the graph contains no odd cycles. In this paper, we strengthen these results by proving that EFX allocations always exist in multi-graphs that contain no cycle of length three. Assuming monotone valuations, we further provide a pseudo-polynomial time algorithm for computing such an allocation, which runs in polynomial time when agents have cancelable valuations, a strict superclass of additive valuation functions. Accordingly, our results stand as one of the only cases where EFX allocations exist for an arbitrary number of agents.

cs.GT↗

arXiv2025

A Formal Correctness Proof of Edmonds' Blossom Shrinking Algorithm

We present the first formal correctness proof of Edmonds' blossom shrinking algorithm for maximum cardinality matching in general graphs. We focus on formalising the mathematical structures and properties that allow the algorithm to run in worst-case polynomial running time. We formalise Berge's lemma, blossoms and their properties, and a mathematical model of the algorithm, showing that it is totally correct. We provide the first detailed proofs of many of the facts underlying the algorithm's correctness.

cs.LO↗

arXiv2025

EFX Allocations and Orientations on Bipartite Multi-graphs: A Complete Picture

We consider the fundamental problem of fairly allocating a set of indivisible items among agents having valuations that are represented by a multi-graph -- here, agents appear as vertices and items as edges between them and each vertex (agent) only values the set of its incident edges (items). The goal is to find a fair, i.e., envy-free up to any item (EFX) allocation. This model has recently been introduced by Christodoulou et al. (EC-23) where they show that EFX allocations always exist on simple graphs for monotone valuations, i.e., where any two agents can share at most one edge (item). A natural question arises as to what happens when we go beyond simple graphs and study various classes of multi-graphs? We answer the above question affirmatively for the valuation class of bipartite multi-graphs and multi-cycles. The main contribution of this work is to establish the existence of EFX allocations on bipartite multi-graphs for monotone valuations and on multi-cycles for MMS-feasible valuations. We also present pseudo-polynomial time algorithms to compute EFX allocations for the above settings. Furthermore, we show that for bipartite multi-graphs with cancelable valuations, EFX allocations can be computed in polynomial time. We thus widen the spectrum where EFX allocations are guaranteed to exist. Next, we study EFX orientations (allocations where every item is assigned to one of its two endpoint agents) and provide a complete characterization of their existence on bipartite multi-graphs in terms of two key parameters: (i) the number of edges shared between any two agents and (ii) the diameter of the graph. Finally, we prove that it is NP-complete to determine whether a given fair division instance on a bipartite multi-graph admits an EFX orientation, even with a constant number of agents.

cs.GT↗

arXiv2025

Improving Order with Queues

Given a sequence of $n$ numbers and $k$ parallel First-in-First-Out (FIFO) queues, how close can one bring the sequence to sorted order? It is known that $k$ queues suffice to sort the sequence if the Longest Decreasing Subsequence (LDS) of the input sequence is at most $k$. But, what if the number of queues is too small for sorting completely? - We give a simple algorithm, based on Patience Sort, that reduces the LDS by $k - 1$. We also show, that the algorithm is optimal, i.e., for any $L > 0$ there exists a sequence of LDS $L$ such that the LDS cannot be reduced below $L - k + 1$ with $k$ queues. - Merging two sorted queues is at the core of Merge Sort. In contrast, two sequences of LDS two cannot always be merged into a sequence of LDS two. We characterize when it is possible and give an algorithm to decide whether it is possible. Merging into a sequence of LDS three is always possible. - A down-step in a sequence is an item immediately followed by a smaller item. We give an optimal algorithm for reducing the number of down-steps. The algorithm is online. Our research was inspired by an application in car manufacturing.

cs.DS↗

arXiv2025

Optimizing Car Resequencing on Mixed-Model Assembly Lines: Algorithm Development and Deployment

The mixed-model assembly line (MMAL) is a production system used in the automobile industry to manufacture different car models on the same conveyor, offering a high degree of product customization and flexibility. However, the MMAL also poses challenges, such as finding optimal sequences of models satisfying multiple constraints and objectives related to production performance, quality, and delivery -- including minimizing the number of color changeovers in the Paint Shop, balancing the workload and setup times on the assembly line, and meeting customer demand and delivery deadlines. We propose a multi-objective algorithm to solve the MMAL resequencing problem under consideration of all these aspects simultaneously. We also present empirical results obtained from recorded event data of the production process over $4$ weeks following the deployment of our algorithm in the Saarlouis plant of Ford-Werke GmbH. We achieved an improvement of the average batch size of about $30\%$ over the old control software translating to a $23\%$ reduction of color changeovers. Moreover, we reduced the spread of cars planned for a specific date by $10\%$, reducing the risk of delays in delivery. We discuss effectiveness and robustness of our algorithm in improving production performance and quality as well as trade-offs and limitations.

eess.SY↗

arXiv2024

Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case

A set of $m$ indivisible goods is to be allocated to a set of $n$ agents. Each agent $i$ has an additive valuation function $v_i$ over goods. The value of a good $g$ for agent $i$ is either $1$ or $s$, where $s$ is a fixed rational number greater than one, and the value of a bundle of goods is the sum of the values of the goods in the bundle. An \emph{allocation} $X$ is a partition of the goods into bundles $X_1$, \ldots, $X_n$, one for each agent. The \emph{Nash Social Welfare} ($\NSW$) of an allocation $X$ is defined as \[ \NSW(X) = \left( \prod_i v_i(X_i) \right)^{\sfrac{1}{n}}.\] The \emph{$\NSW$-allocation} maximizes the Nash Social Welfare. In~\cite{NSW-twovalues-halfinteger} it was shown that the $\NSW$-allocation can be computed in polynomial time, if $s$ is an integer or a half-integer, and that the problem is NP-complete otherwise. The proof for the half-integer case is quite involved. In this note we give a simpler and shorter proof

cs.GT↗

arXiv2024

Approximate EFX and Exact tEFX Allocations for Indivisible Chores: Improved Algorithms

We explore the fair distribution of a set of $m$ indivisible chores among $n$ agents, where each agent's costs are evaluated using a monotone cost function. Our focus lies on two fairness criteria: envy-freeness up to any item (EFX) and a relaxed notion, namely envy-freeness up to the transfer of any item (tEFX). We demonstrate that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous $(2 + \sqrt{6})$ approximation for additive cost functions. This result requires extensive case analysis. Christoforidis et al. (IJCAI'24) independently claim the same approximation for additive cost functions; however, we provide a counter-example to their algorithm. We expand the number of agents to any number to get the same approximation guarantee with the assumption of partially identical ordering (IDO) for the cost functions. Additionally, we establish that a tEFX allocation is achievable for three agents if one has an additive 2-ratio bounded cost function, while the others may have general monotone cost functions. This is an improvement from the prior requirement of two agents with additive 2-ratio bounded cost functions. This allocation can also be extended to agent groups with identical valuations. Further, we show various analyses of EFX allocations for chores, such as the relaxations for additive $α$-ratio-bounded cost functions.

cs.GT↗

arXiv2024

Self-organized transport in noisy dynamic networks

We present a numerical study of multi-commodity transport in a noisy, nonlinear network. The nonlinearity determines the dynamics of the edge capacities, which can be amplified or suppressed depending on the local current flowing across an edge. We consider network self-organization for three different nonlinear functions: For all three we identify parameter regimes where noise leads to self-organization into more robust topologies, that are not found by the sole noiseless dynamics. Moreover, the interplay between noise and specific functional behavior of the nonlinearity gives rise to different features, such as (i) continuous or discontinuous responses to the demand strength and (ii) either single or multi-stable solutions. Our study shows the crucial role of the activation function on noise-assisted phenomena.

cond-mat.stat-mech↗

arXiv2024

MMS Approximations Under Additive Leveled Valuations

We study the problem of fairly allocating indivisible goods to a set of agents with additive leveled valuations. A valuation function is called leveled if and only if bundles of larger size have larger value than bundles of smaller size. The economics literature has well studied such valuations. We use the maximin-share (MMS) and EFX as standard notions of fairness. We show that an algorithm introduced by Christodoulou et al. ([11]) constructs an allocation that is EFX and $\frac{\lfloor \frac{m}{n} \rfloor}{\lfloor \frac{m}{n} \rfloor + 1}\text{-MMS}$. In the paper, it was claimed that the allocation is EFX and $\frac{2}{3}\text{-MMS}$. However, the proof of the MMS-bound is incorrect. We give a counter-example to their proof and then prove a stronger approximation of MMS.

cs.GT↗

arXiv2024

Gabow's Cardinality Matching Algorithm in General Graphs: Implementation and Experiments

It is known since 1975 (\cite{HK75}) that maximum cardinality matchings in bipartite graphs with $n$ nodes and $m$ edges can be computed in time $O(\sqrt{n} m)$. Asymptotically faster algorithms were found in the last decade and maximum cardinality bipartite matchings can now be computed in near-linear time~\cite{NearlyLinearTimeBipartiteMatching, AlmostLinearTimeMaxFlow,AlmostLinearTimeMinCostFlow}. For general graphs, the problem seems harder. Algorithms with running time $O(\sqrt{n} m)$ were given in~\cite{MV80,Vazirani94,Vazirani12,Vazirani20,Vazirani23,Goldberg-Karzanov,GT91,Gabow:GeneralMatching}. Mattingly and Ritchey~\cite{Mattingly-Ritchey} and Huang and Stein~\cite{Huang-Stein} discuss implementations of the Micali-Vazirani Algorithm. We describe an implementation of Gabow's algorithm~\cite{Gabow:GeneralMatching} in C++ based on LEDA~\cite{LEDAsystem,LEDAbook} and report on running time experiments. On worst-case graphs, the asymptotic improvement pays off dramatically. On random graphs, there is no improvement with respect to algorithms that have a worst-case running time of $O(n m)$. The performance seems to be near-linear. The implementation is available open-source.

cs.DS↗

arXiv2024

Welfare-Optimal Serial Dictatorships have Polynomial Query Complexity

Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi [10], we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (n^5) of queries. This solves the main open problem stated by Caragiannis and Rathi [CR23]. Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.

cs.GT↗