arXiv subjects

Jiří Fink

Publications and source records attributed to Jiří Fink.

12 recordsLinked to original sources

arXiv2026

Faster and simpler traversal of 0/1-polytopes

Recently, Merino and Mütze (FOCS'23+SICOMP'24) presented an algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope ${\rm conv}(X)$, where $X\subseteq\{0,1\}^n$. The algorithm uses as a black box an algorithm for solving the classical linear optimization problem $\min\{w\cdot x\mid x\in X\}$ for some weight vector $w\in\mathbb{R}^n$. The resulting delay per visited vertex on the Hamilton path is only by a $\log n$ factor larger than the time to solve one instance of the optimization algorithm. In this paper, we make the Hamilton path algorithm simpler and faster. Namely, we obtain an amortized delay that is only by a constant factor larger than the running time of the optimization algorithm, thus removing the $\log n$ factor. As concrete results, this yields improved algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings and maximum matchings in a graph, vertex covers, minimum vertex covers, independent sets and maximum independent sets in a bipartite graph, and antichains, maximum antichains and ideals in a poset. All of these listings correspond to Hamilton paths on the corresponding polytopes. Furthermore, we obtain an $\mathcal{O}(t_{\rm LP})$ amortized delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and $t_{\rm LP}$ is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the $\mathcal{O}(t_{\rm LP} \log n)$ delay algorithm of Merino and Mütze, and the previous $\mathcal{O}(t_{\rm LP}\,n)$ delay algorithm of Bussieck and Lübbecke from 1998.

cs.DS↗

arXiv2026

The realization graph of every degree sequence has a Hamilton path

For a degree sequence $d$, the realization graph $\mathcal{G_F}(d)$ is the graph whose vertices are the labeled realizations of $d$, two of which are adjacent if they differ by a single $2$-switch. We prove that for every degree sequence $d$ and every realization $S$ of $d$, the graph $\mathcal{G_F}(d)$ contains a Hamilton path starting at $S$. This answers a question of Barrus (2016), which was also raised independently by Mütze (2023) in his survey of combinatorial Gray codes. As a consequence, an embedding observation of Arikati and Peled (1999) implies that for any vectors $R$ and $C$ of non-negative integers, the interchange graph $\mathcal{A}_{\mathcal F}(R,C)$ of $(0,1)$-matrices with row sums $R$ and column sums $C$ contains a Hamilton path starting at any prescribed matrix, thereby resolving a question of Brualdi (1980).

math.CO↗

arXiv2026

Constant delay Gray code enumeration of ideals and antichains in posets

We present an algorithm that enumerates all ideals of an input poset with constant delay in Gray code order, i.e., such that consecutively visited ideals differ in at most three elements. This answers a long-standing open problem posed by Pruesse and Ruskey, and improves upon previous algorithms by Pruesse and Ruskey, Squire, Habib, Medina, Nourine and Steiner, as well as Abdo. Using the same techniques, we also obtain an algorithm that enumerates all antichains of an input poset with constant delay such that successively visited antichains differ in at most three elements. As a key technical ingredient, we introduce a new potential-based analysis framework for recursive algorithms, which we call the Pyramid method. We show that this method subsumes the Push-out method of Uno. Beyond the present application, the Pyramid method is a general framework to analyze recursive algorithms and may thus be of independent interest.

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arXiv2026

Intermediate Results on the Complexity of STRIPS$_{1}^{1}$

This paper is based on Bylander's results on the computational complexity of propositional STRIPS planning. He showed that when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. While NP-hardness is settled, it is unknown whether propositional STRIPS with operators that only have one precondition and one effect is NP-complete. We shed light on the question whether this small solution hypothesis for STRIPS$^1_1$ is true, calling a SAT solver for small instances, introducing the literal graph, and mapping it to Petri nets.

cs.AI↗

arXiv2025

Constant time enumeration of perfect bipartite matchings

We present an algorithm that enumerates all the perfect matchings in a given bipartite graph G = (V,E). Our algorithm requires a constant amortized time to visit one perfect matching of G, in contrast to the current fastest algorithm, published 25 years ago by Uno, which requires O(log |V|) time. To facilitate the listing of all edges in a visited perfect matching, we develop a variant of arithmetic circuits, which may have broader applications in future enumeration algorithms. Consequently, a visited perfect matching is represented within a binary tree. Although it is more common to provide visited objects in an array, we present a class of graphs for which achieving constant amortized time is not feasible in this case.

cs.DS↗

arXiv2025

Matchings in hypercubes extend to long cycles

The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of $Q_d$, $d\ge 2$, can be extended to a cycle that visits at least a $2/3$-fraction of all vertices.

math.CO↗

arXiv2025

Minimum maximal matchings in permutahedra

We prove that the minimal size $M(π_n)$ of a maximal matching in the permutahedron $π_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(π_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(π_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(π_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.

math.CO↗

arXiv2025

Matchings in Hypercubes Extend to Long Cycles

The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional hypercube $Q_n$ can be extended into a Hamilton cycle. We prove that matchings of $Q_n$ containing edges spanning at most $d = 5$ directions can be extended into a Hamilton cycle. We also characterize when these matchings of most $d = 5$ directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary $d$ and $n$ where $d \le n$ assuming some extension properties hold in $Q_d$ which we verified by a computer for $d=5$.

math.CO↗

arXiv2020

Arc-routing for winter road maintenance

The arc-routing problems are known to be notoriously hard. We study here a natural arc-routing problem on trees and more generally on bounded tree-width graphs and surprisingly show that it can be solved in a polynomial time. This implies a sub-exponential algorithm for the planar graphs and small number of maintaining cars, which is of practical relevance.

cs.DS↗

arXiv2020

A Greedy algorithm for local heating

This paper studies a planning problem for supplying hot water in domestic environment. Hereby, boilers (e.g. gas or electric boilers, heat pumps or microCHPs) are used to heat water and store it for domestic demands. We consider a simple boiler which is either turned on or turned off and is connected to a buffer of limited capacity. The energy needed to run the boiler has to be bought e.g. on a day-ahead market, so we are interested in a planning which minimizes the cost to supply the boiler with energy in order to fulfill the given heat demand. We present a greedy algorithm for this heating problem whose time complexity is O(T α(T )) where T is the number of time intervals and α is the inverse of Ackermann function.

cs.DS↗

arXiv2020

Approximation algorithms for scheduling a group of heat pumps

This paper studies planning problems for a group of heating systems which supply the hot water demand for domestic use in houses. These systems (e.g. gas or electric boilers, heat pumps or microCHPs) use an external energy source to heat up water and store this hot water for supplying the domestic demands. The latter allows to some extent a decoupling of the heat production from the heat demand. We focus on the situation where each heating system has its own demand and buffer and the supply of the heating systems is coming from a common source. In practice, the common source may lead to a coupling of the planning for the group of heating systems. The bottleneck to supply the energy may be the capacity of the distribution system (e.g. the electricity networks or the gas network). As this has to be dimensioned for the maximal consumption, it is important to minimize the maximal peak. This planning problem is known to be \NP-hard. We present polynomial-time approximation algorithms for four variants of peak minimization problems, and we determine the worst-case approximation error.

cs.DS↗

arXiv2020

A New Arc-Routing Algorithm Applied to Winter Road Maintenance

This paper studies large scale instances of a fairly general arc-routing problem as well as incorporate practical constraints in particular coming from the scheduling problem of the winter road maintenance (e.g. different priorities for and methods of road maintenance). We develop a new algorithm based on a bin-packing heuristic which is well-scalable and able to solve road networks on thousands of crossroads and road segments in few minutes. Since it is impossible to find an optimal solution for such a large instances to compare it with a result of our algorithm, we also develop techniques to compute lower bounds which are based on Integer Linear Programming and Lazy Constraints.

cs.AI↗