Mgr. Michal Opler, Ph.D.

Stať ve sborníku

Katedra teoretické informatiky

The Mutliagent Path Finding (MAPF) problem consists of identifying the trajectories that a set of agents should follow inside a given network in order to reach their desired destinations as soon as possible, but without colliding with each other. We aim to minimize the maximum time any agent takes to reach their goal, ensuring optimal path length. In this work, we complement a recent thread of results that aim to systematically study the algorithmic behavior of this problem, through the parameterized complexity point of view. First, we show that MAPF is NP-hard when the given network has a star-like topology (bounded vertex cover number) or is a tree with 11 leaves. Both of these results fill important gaps in our understanding of the tractability of this problem that were left untreated in the recent work of [Fioravantes et al. Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology. AAAI'24]. Nevertheless, our main contribution is an exact algorithm that scales well as the input grows (FPT) when the topology of the given network is highly centralized (bounded distance to clique). This parameter is significant as it mirrors real-world networks. In such environments, a bunch of central hubs (e.g., processing areas) are connected to only few peripheral nodes.

Stať ve sborníku

10.1109/FOCS61266.2024.00049

Katedra teoretické informatiky

We present a deterministic comparison-based algorithm that sorts sequences avoiding a fixed permutation $\pi$ in linear time, even if $\pi$ is a priori unkown. Moreover, the dependence of the multiplicative constant on the pattern $\pi$ matches the information-theoretic lower bound. A crucial ingredient is an algorithm for performing efficient multi-way merge based on the Marcus-Tardos theorem. As a direct corollary, we obtain a linear-time algorithm for sorting permutations of bounded twin-width.

Stať ve sborníku

10.1609/aaai.v38i16.29686

Katedra teoretické informatiky

In the Multiagent Path Finding (MAPF for short) problem, we focus on efficiently finding non-colliding paths for a set of k agents on a given graph G, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule l, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our Fixed-Parameter Tractability (FPT) results. We show that MAPF is W[1]-hard with respect to k (even if k is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan l are fixed constants. On the positive side, we show an FPT algorithm for k+l. As we continue, the structure of G comes into play. We give an FPT algorithm for parameter k plus the diameter of the graph G. The MAPF problem is W[1]-hard for cliquewidth of G plus l while it is FPT for treewidth of G plus l.

Článek

10.1007/s00453-024-01220-9

Katedra teoretické informatiky

A permutation π is a merge of a permutation σ and a permutation τ, if we can color the elements of π red and blue so that the red elements have the same relative order as σ and the blue ones as τ. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation π is a merge of an element of C with an element of D. We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of streaming recognizability of permutations via polynomially constructible nondeterministic automata, as well as a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

Stať ve sborníku

10.1145/3618260.3649631

Katedra teoretické informatiky

Permutation pattern-avoidance is a central concept of both enumerative and extremal combinatorics. In this paper we study the effect of permutation pattern-avoidance on the complexity of optimization problems. In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC 1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015) conjectured that the amortized search cost of an optimal binary search tree (BST) is constant whenever the search sequence is pattern-avoiding. The best known bound to date is 2α(n)(1+o(1)) recently obtained by Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); here n is the BST size and α(·) the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight (1) bound. This indicates a barrier to dynamic optimality: any candidate online BST (e.g., splay trees or greedy trees) must match this optimum, but current analysis techniques only give superconstant bounds. More broadly, we argue that the easiness of pattern-avoiding input is a general phenomenon, not limited to BSTs or even to data structures. To illustrate this, we show that when the input avoids an arbitrary, fixed, a priori unknown pattern, one can efficiently compute: (1) a k-server solution of n requests from a unit interval, with total cost n(1/logk), in contrast to the worst-case Θ(n/k) bound, and (2) a traveling salesman tour of n points from a unit box, of length (logn), in contrast to the worst-case Θ(√n) bound; similar results hold for the euclidean minimum spanning tree, Steiner tree, and nearest-neighbor graphs. We show both results to be tight. Our techniques build on the Marcus-Tardos proof of the Stanley-Wilf conjecture, and on the recently emerging concept of twin-width.

Stať ve sborníku

10.1137/1.9781611977912.59

Katedra teoretické informatiky

We consider the following general model of a sorting procedure: we fix a hereditary permutation class C, which corresponds to the operations that the procedure is allowed to perform in a single step. The input of sorting is a permutation π of the set [n] = {1, 2,…,n}, i.e., a sequence where each element of [n] appears once. In every step, the sorting procedure picks a permutation σ of length n from C, and rearranges the current permutation of numbers by composing it with σ. The goal is to transform the input π into the sorted sequence 1, 2,…,n in as few steps as possible. Formally, for a hereditary permutation class C and a permutation π of [n], we say that C can sort π in k steps, if the inverse of π can be obtained by composing k (not necessarily distinct) permutations from C. The C-sorting time of π, denoted st(C; π), is the smallest k such that C can sort π in k steps; if no such k exists, we put st(C; π) = +∞. For an integer n, the worst-case C-sorting time, denoted wst(C; n), is the maximum of st(C; π) over all permutations π of [n]. This model of sorting captures not only classical sorting algorithms, like insertion sort or bubble sort, but also sorting by series of devices, like stacks or parallel queues, as well as sorting by block operations commonly considered, e.g., in the context of genome rearrangement. Our goal is to describe the possible asymptotic behavior of the function wst(C; n), and relate it to structural properties of C. As the main result, we show that any hereditary permutation class C falls into one of the following five categories: • wst(C; n) = +∞ for n large enough, • wst(C; n) = Θ(n2), • Ω(√n) ≤ wst(C; n) ≤ O(n), • Ω(log n) ≤ wst(C; n) ≤ O(log2 n), or • wst(C; n) = 1 for all n ≥ 2. In addition, we characterize the permutation classes in each of the five categories.

Článek

10.46298/DMTCS.10802

Katedra teoretické informatiky

Consider the following hat guessing game. A bear sits on each vertex of a graph G, and a demon puts on each bear a hat colored by one of h colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess g colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number μ^, arising from the hat guessing game. The parameter μ^ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of G, and to compute the exact value of μ^ of cliques, paths, and cycles.

Stať ve sborníku

10.1007/978-3-031-43380-1_22

Katedra teoretické informatiky

[Alecu et al.: Graph functionality, JCTB2021] define functionality, a graph parameter that generalizes graph degeneracy. They research the relation of functionality to many other graph parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we prove logarithmic lower bound for functionality of the random graph~$G(n,1/2)$. Previously known graphs have functionality logarithmic in number of vertices. We show that for every graph~$G$ on~$n$ vertices we have $\mathrm{fun}(G) \le O(\sqrt{n \log n})$ and we give a nearly matching $\Omega(\sqrt{n})$-lower bound provided by projective planes. Further, we study a related graph parameter \emph{symmetric difference}, the minimum of $|N(u) \Delta N(v)|$ over all pairs of vertices of the ``worst possible'' induced subgraph. It was observed by Alecu et al. that $\mathrm{fun}(G) \le \mathrm{sd}(G)+1$ for every graph~$G$. We compare $\mathrm{fun}$ and $\mathrm{sd}$ for the class~$\mathrm{INT}$ of interval graphs and $\mathrm{CA}$ of circular-arc graphs. We let $\mathrm{INT}_n$ denote the $n$-vertex interval graphs, similarly for $\mathrm{CA}_n$. Alecu et al. ask, whether $\mathrm{fun}(\mathrm{INT})$ is bounded. Dallard et al. answer this positively in a recent preprint. On the other hand, we show that $\Omega(\sqrt[4]{n}) \leq \sd(\mathrm{INT}_n) \leq O(\sqrt[3]{n})$. For the related class $\mathrm{CA}$ we show that $\mathrm{sd}(\mathrm{CA}_n) = \Theta(\sqrt{n})$. We propose a follow-up question: is $\mathrm{fun}(\mathrm{CA})$ bounded?

Stať ve sborníku

10.1007/978-3-031-49193-1_16

Katedra teoretické informatiky

We improve bounds for the binary paint shop problem posed by Meunier and Neveu [Computing solutions of the paintshop-necklace problem. Comput. Oper. Res. 39, 11 (2012), 2666-2678]. In particular, we disprove their conjectured upper bound for the number of color changes by giving a linear lower bound. We show that the recursive greedy heuristics is not optimal by providing a tiny improvement. We also introduce a new heuristics, recursive star greedy, that a preliminary analysis shows to be 10% better.