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Cooperative mobile guards in grids
PublikacjaPraca dotyczy problemu strzeżenia dwuwymiarowych krat ortogonalnych, przy założeniu, że obszar widoczności strażnika obejmuje jedną ulicę oraz wszystkie ulice ją przecinające. Rozważano wariant straży słabo współpracujących, w którym dodatkowo każdy strażnik musi widzieć przynajmniej jednego innego strażnika. Podano dowód NP-trudności problemu optymalizacyjnego w przypadku ogólnym, algorytm dokładny o złożoności O(n log n) dla...
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Compact cyclic edge-colorings of graphs
PublikacjaArtykuł jest poświęcony modelowi zwartego cyklicznego kolorowania krawędzi grafów. Ten wariant kolorowania jest stosowany w modelowaniu uszeregowań w systemach produkcyjnych, w których proces produkcyjny ma charakter cykliczny. W pracy podano konstrukcje grafów, które nie zezwalają na istnienie pokolorowania w rozważanym modelu. Wykazano także kilka własności teoretycznych, takich jak ograniczenia górne na liczbę kolorów w optymalnym...
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Equitable coloring of graphs. Recent theoretical results and new practical algorithms
PublikacjaIn this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.
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Robust amplification of Santha-Vazirani sources with three devices
PublikacjaWe demonstrate that amplification of arbitrarily weak randomness is possible using quantum resources. We present a randomness amplification protocol that involves Bell experiments. We find a Bell inequality which can amplify arbitrarily weak randomness and give a detailed analysis of the protocol involving it. Our analysis includes finding a sufficient violation of Bell inequality as a function of the initial quality of randomness....
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Rendezvous of Distance-Aware Mobile Agents in Unknown Graphs
PublikacjaWe study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size...
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Distributed Evacuation in Graphs with Multiple Exits
PublikacjaWe consider the problem of efficient evacuation using multiple exits. We formulate this problem as a discrete problem on graphs where mobile agents located in distinct nodes of a given graph must quickly reach one of multiple possible exit nodes, while avoiding congestion and bottlenecks. Each node of the graph has the capacity of holding at most one agent at each time step. Thus, the agents must choose their movements strategy...
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Parallel processing subsystems with redundancy in a distributed environment
PublikacjaW pracy rozważano problem podziału systemu rozproszonego na spójne podsystemy złożone z przynajmniej trzech jednostek, pozwalające na detekcję i skorygowanie pojedynczych błędów. Wykazano, że problem maksymalizacji liczby takich jednostek jest NP-trudny nawet dla dwuspójnych kubicznych topologii sieci. Podano też nowe algorytmy przybliżone.
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On greedy graph coloring in the distributed model
PublikacjaArtykuł traktuje o zachłannym kolorowaniu grafów w modelu rozproszonym. Zaprezentowano nowy probabilistyczny algorytm dający w wyniku pokolorowanie LF. Udowodniono, że jakakolwiek rozproszona implementacja LF wymaga co najmniej D rund, gdzie D jest maksymalnym stopniem wierzchołka w grafie.
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Three-fast-searchable graphs
PublikacjaIn the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates...
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Fast Collaborative Graph Exploration
PublikacjaWe study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure...
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Quantum randomness protected against detection loophole attacks
PublikacjaDevice and semi-device-independent private quantum randomness generators are crucial for applications requiring private randomness. However, they are vulnerable to detection inefficiency attacks and this limits severely their usage for practical purposes. Here, we present a method for protecting semi-device-independent private quantum randomness generators in prepare-and-measure scenarios against detection inefficiency attacks....
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The complexity of the L(p,q)-labeling problem for bipartite planar graphs of small degree
PublikacjaW pracy pokazano, że problem L(p,q)-kolorowania przy użyciu ''t'' kolorów jest NP-zupełny nawet w wersji ograniczonej do grafów planarnych dwudzielnych małego stopnia, nawet dla stosunkowo niewielkich wartości ''t''. Jako wniosek z uzyskanych wyników stwierdzono, że problem L(2,1)-kolorowania grafów planarnych przy użyciu 4 kolorów jest NP-zupełny, a także że problem L(p,q)-kolorowania grafów o maksymalnym stopniu 4 jest NP-zupełny...
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A FPTAS for minimizing total completion time in a single machine time-dependent scheduling problem
PublikacjaIn this paper a single machine time-dependent scheduling problem with total completion time criterion is considered. There are given n jobs J1,…,Jn and the processing time pi of the ith job is given by pi=a+bisi, where si is the starting time of the ith job (i=1,…,n),bi is its deterioration rate and a is the common base processing time. If all jobs have deterioration rates different and not smaller than a certain constant u>0,...
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Bortezomib induces methylation changes in neuroblastoma cells that appear to play a significant role in resistance development to this compound
PublikacjaThe anticancer activity of bortezomib (BTZ) has been increasingly studied in a number of indications and promising results for the use of this treatment have been shown in neuroblastoma. As BTZ treatment is usually administered in cycles, the development of resistance and side effects in patients undergoing therapy with BTZ remains a major challenge for the clinical usage of this compound. Common resistance development also means...
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2-outer-independent domination in graphs
PublikacjaWe initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D, and the set V(G)\D is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two,...
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An efficient algorithm for finding ideal schedules
PublikacjaPodejmujemy problem szeregowania zadań jednostkowych z zadanymi czasamy przybycia i zależnościami kolejnościowymi. Uszeregowanie jest idealne jeśli jednocześnie minimalizuje maksymalny oraz średni czas zakończenia zadania. Podajemy przyklad pokazujący, że uszeregowania idealne nie istnieją dla relacji zależności zadań będącej drzewem, gdy dopuścimy możliwość wystąpienia przerwań. Z drugiej strony podajemy algorytm o złożoności...
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Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
PublikacjaThe problem of scheduling n identical jobs on 4 uniform machines with speeds s1>=s2>=s3>=s4 is considered.The aim is to find a schedule with minimum possible length. We assume that jobs are subject to mutual exclusion constraints modeled by a bipartite incompatibility graph of degree delta. We show that the general problem is NP-hard even if s1=s2=s3. If, however, delta<5 and s1>12s2 s2=s3=s4, then the problem can be solved to...
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Approximate search strategies for weighted trees
PublikacjaW pracy podajemy 3-przybliżony algorytm dla problemu spójnego przeszukiwania drzew ważonych.
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Equitable and semi-equitable coloring of cubic graphs and its application in batch scheduling
PublikacjaIn the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP- complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize...
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A note on mixed tree coloring
PublikacjaZaproponowano liniowy algorytm dla problemu kolorowania mieszanego w drzewach, uzyskując tym samym poprawę w stosunku do algorytmu o złożoności O(n^2) podanego w pracy [P. Hansen, J. Kuplinsky, D. de Werra, Mixed graph colorings, Math. Methods Oper. Res. 45 (1997) 145-160].
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Packing [1,Delta]-factors in graphs of small degree
PublikacjaRozważano problem znalezienia w grafie zadanej liczby k krawędziowo rozłącznych [1,Delta]-faktorów, gdzie Delta oznacza stopień grafu. Problem ten można rozwiązać w czasie liniowym dla k=2, jest on jednak NP-trudny dla każdego k>=3. Pokazano, że wariant minimalizacjny problemu dla k=2 jest NP-trudny dla grafów planarnych podkubicznych, jednak w ogólności istnieje algorytm (42 Delta - 30) / (35 Delta - 21) - aproksymacyjny.
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Shared multi-processor scheduling
PublikacjaWe study shared multi-processor scheduling problem where each job can be executed on its private processor and simultaneously on one of many processors shared by all jobs in order to reduce the job’s completion time due to processing time overlap. The total weighted overlap of all jobs is to be maximized. The problem models subcontracting scheduling in supply chains and divisible load scheduling in computing. We show that synchronized...
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Bounds on the cover time of parallel rotor walks
PublikacjaThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node successively propagates walkers visiting it along its outgoing arcs in round-robin fashion, according to a fixed ordering. We consider...
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Bounds on the Cover Time of Parallel Rotor Walks
PublikacjaThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node maintains a cyclic ordering of its outgoing arcs, and successively propagates walkers which visit it along its outgoing arcs in...
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Scheduling with precedence constraints: mixed graph coloring in series-parallel graphs.
PublikacjaW pracy rozważono problem kolorowania grafów mieszanych, opisujący zagadnienie szeregowania zadań, w którym zależności czasowe zadań mają charakter częściowego porządku lub wzajemnego wykluczania. Dla przypadku, w którym graf zależności jest szeregowo-równoległy, podano algorytm rozwiązujący problem optymalnie w czasie $O(n^3.376 * log n)$.
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Trade-offs in multiparty Bell-inequality violations in qubit networks
PublikacjaTwo overlapping bipartite binary input Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signalling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations...
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Robustness of the Rotor-router Mechanism
PublikacjaW pracy rozważano model eksploracji grafu nieskierowanego przez pojedynczego agenta, w którym sterowanie agentem odbywa się zgodnie z zasadą ''rotor-router'' (inaczej: ''Propp machine''). Przeanalizowano czas stabilizacji agenta do trajektorii w postaci cyklu Eulera w przypadku wystąpienia zaburzeń w grafie: usunięcie krawędzi, dodanie krawędzi, lokalna zamiana portów
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Rendezvous of heterogeneous mobile agents in edge-weighted networks
PublikacjaWe introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal...
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A lower bound on the total outer-independent domination number of a tree
PublikacjaA total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V(G)D is independent. The total outer-independent domination number of a graph G, denoted by gamma_t^{oi}(G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leaves we have gamma_t^{oi}(T) >= (2n-2l+2)/3,...
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Cost minimisation in multi-interface networks
PublikacjaPraca dotyczy problemu minimalizacji energii poprzez selektywne odłączanie urządzeń komunikacyjnych w wielointerfejsowych sieciach bezprzewodowych w taki sposób, by zapewnić realizację wymaganego grafu połączeń. Sformułowano problem optymalizacyjny, podano wyniki dotyczące jego trudności i zaproponowano algorytmy optymalizacyjne.
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Tighter bounds on the size of a maximum P3-matching in a cubic graph
PublikacjaW pracy pokazano, że największe P3-skojarzenie dla dowolnego grafu o n>16 wierzchołkach składa się z przynajmniej 117n/152 wierzchołków.
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Graph decomposition for improving memoryless periodic exploration
PublikacjaW ostatnich latach często badanym problem jest eksploracja anonimowych grafów z lokalnymi etykietami portów przy każdym wierzchołku. Niedawno pokazano [Czyzowicz et al., Proc. SIROCCO'09], że dla każdego grafu istnieje poetykietowanie prowadzące do eksploracji przez automat bezpamięciowy z okresem co najwyżej 13n/3. W niniejszej pracy poprawiamy to ograniczenie do 4n-2, stosując całkowicie nową technikę dekompozycji grafu.
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An upper bound on the 2-outer-independent domination number of a tree
PublikacjaA 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)D has a at least two neighbors in D, and the set V(G)D is independent. The 2-outer-independent domination number of a graph G, denoted by gamma_2^{oi}(G), is the minimum cardinality of a 2-outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leaves we have gamma_2^{oi}(T) <= (n+l)/2,...
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Distributed graph searching with a sense of direction
PublikacjaIn this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function w provides for each vertex v the minimum number of searchers required to guard v, i.e., the fugitive may not pass through v without being detected only if at least w(v) searchers are present at v. This problem is a generalization of the classical edge searching problem, in which one has...
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Lossless Compression of Binary Trees with Correlated Vertex Names
PublikacjaCompression schemes for advanced data structures have become the challenge of today. Information theory has traditionally dealt with conventional data such as text, image, or video. In contrast, most data available today is multitype and context-dependent. To meet this challenge, we have recently initiated a systematic study of advanced data structures such as unlabeled graphs [1]. In this paper, we continue this program by considering...
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New potential functions for greedy independence and coloring
PublikacjaA potential function $f_G$ of a finite, simple and undirected graph $G=(V,E)$ is an arbitrary function $f_G : V(G) \rightarrow \mathbb{N}_0$ that assigns a nonnegative integer to every vertex of a graph $G$. In this paper we define the iterative process of computing the step potential function $q_G$ such that $q_G(v)\leq d_G(v)$ for all $v\in V(G)$. We use this function in the development of new Caro-Wei-type and Brooks-type...
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Blurred quantum Darwinism across quantum reference frames
PublikacjaQuantum Darwinism describes objectivity of quantum systems via their correlations with their environment--information that hypothetical observers can recover by measuring the environments. However, observations are done with respect to a frame of reference. Here, we take the formalism of [Giacomini, Castro-Ruiz, & Brukner. Nat Commun 10, 494 (2019)], and consider the repercussions on objectivity when changing quantum reference...
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Modelling of dark fermentation of glucose and sour cabbage
PublikacjaIn the article, modified Anaerobic Digestion Models 1 (ADM-1) was tested for modelling dark fermentation for hydrogen production. The model refitting was done with the Euler method. The new model was based on sets of differential equations. The model was checked for hydrogen production from sour cabbage in batch and semi-batch in 5 g VSS (volatile solid suspension)/L and at the semi-batch process from glucose at 5 and 10 g VSS/L....
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An approximation algorithm for maximum P3-packing in subcubic graphs
PublikacjaW pracy podano algorytm 4/3-przyliżony dla trudnego obliczeniowo problemu umieszczania wierzchołkowo rozłącznych dwukrawędziowych ścieżek w grafach o stopniu maksymalnym 3 i stopniu minimalnym 2. Poprawiono tym samym wcześniejsze wyniki dla grafów kubicznych (A. Kelmans, D. Mubayi, Journal of Graph Theory 45, 2004).
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Comparing phylogenetic trees using a minimum weight perfect matching
PublikacjaA phylogenetic tree represents historical evolutionary relationshipbetween different species or organisms. There are various methods for reconstructing phylogenetic trees.Applying those techniques usually results in different treesfor the same input data. An important problem is to determinehow distant two trees reconstructed in such a wayare from each other. Comparing phylogenetic trees is alsouseful in mining phylogenetic information...
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Collaborative Delivery by Energy-Sharing Low-Power Mobile Robots
PublikacjaWe study two variants of delivery problems for mobile robots sharing energy. Each mobile robot can store at any given moment at most two units of energy, and whenever two robots are at the same location, they can transfer energy between each other, respecting the maximum capacity. The robots operate in a simple graph and initially each robot has two units of energy. A single edge traversal by an robot reduces its energy by one...
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Approximation Strategies for Generalized Binary Search in Weighted Trees
PublikacjaWe consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node t in a given tree T. Upon querying a node v of the tree, the strategy receives as a reply an indication of the connected component of T\{v} containing the target t. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total...
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2-bondage in graphs
PublikacjaA 2-dominating set of a graph G=(V,E) is a set D of vertices of G such that every vertex of V(G)D has at least two neighbors in D. The 2-domination number of a graph G, denoted by gamma_2(G), is the minimum cardinality of a 2-dominating set of G. The 2-bondage number of G, denoted by b_2(G), is the minimum cardinality among all sets of edges E' subseteq E such that gamma_2(G-E') > gamma_2(G). If for every E' subseteq E we have...
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Parity vertex colouring of graphs
PublikacjaA parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let Xp(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds X(G) <= Xp(G) <=|V(G)|− a(G)+1, where X(G) and a(G) are the chromatic number and the independence number of G, respectively. The bounds are improved for...
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Deterministic Rendezvous in Restricted Graphs
PublikacjaIn this paper we consider the problem of synchronous rendezvous in which two anonymous mobile entities (robots) A and B are expected to meet at the same time and point in a graph G = (V;E). Most of the work devoted to rendezvous in graphs assumes that robots have access to the same sets of nodes and edges, where the topology of connections may be initially known or unknown. In our work we assume the movement of robots is restricted...
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Device-independent quantum key distribution based on measurement inputs
PublikacjaWe provide an analysis of a family of device-independent quantum key distribution (QKD) protocols that has the following features. (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings. (b) Instead of a single security parameter (a violation of some Bell inequality) a set of them is used to estimate the level of trust in the secrecy of the key....
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Cops, a fast robber and defensive domination on interval graphs
PublikacjaThe game of Cops and ∞-fast Robber is played by two players, one controlling c cops, the other one robber. The players alternate in turns: all the cops move at once to distance at most one each, the robber moves along any cop-free path. Cops win by sharing a vertex with the robber, the robber by avoiding capture indefinitely. The game was proposed with bounded robber speed by Fomin et al. in “Pursuing a fast robber on a graph”,...
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Edge-coloring of 3-uniform hypergraphs
PublikacjaWe consider edge-colorings of 3-uniform hypergraphs which is a natural generalization of the problem of edge-colorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NP-complete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.
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Cost minimisation in unbounded multi-interface networks
PublikacjaW pracy badano problem odłączania niektórych urządzeń komunikacyjnych w wielointerfejsowych sieciach bezprzewodowych w taki sposób, by zapewnić realizację wymaganego grafu połączeń przy jednoczesnej minimalizacji zużycia energii. Sformułowano problem optymalizacyjny, podano wyniki dotyczące jego trudności i zaproponowano algorytmy optymalizacyjne dla wariantu, w którym liczba interfejsów komunikacyjnych jest potencjalnie nieograniczona...
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Derandomizing random walks in undirected graphs using locally fair exploration strategies
PublikacjaW pracy rozważono problem eksploracji anonimowego nieskierowanego grafu przez bezpamięciowego robota. Zaprojektowane strategie eksploracji cechują się własnością lokalnej sprawiedliwości, tj. kolejne krawędzie trawersowane przez robota wybierane są na podstawie lokalnych informacji tak, aby zapewnić równomierne wykorzystanie krawędzi w sensie pewnego kryterium. Okazuje się, że odpowiedni dobór kryterium jest kluczowy do zapewnienia...