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Cycle Extensions in BIBD Block-Intersection Graphs

JOURNAL OF COMBINATORIAL DESIGNS 2013年第7期21卷 303-310页

作者： Abueida, Atif A. Pike, David A. Univ Dayton Dept Math Dayton OH 45469 USA Mem Univ Newfoundland Dept Math & Stat St John NF A1C 5S7 Canada

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non-Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD(v,k,), consists of a set V of v elements and a block set B of k-subsets of V such that each 2-subset of V occurs in exactly of the blocks of B. The block-intersection graph of a design D=(V,B) is the graph GD having B as its vertex set and such that two vertices of GD are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block-intersection graph of any BIBD(v,k,) is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block-intersection graph. (C) 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 303-310, 2013

关键词： cycle extension block design intersection graph polynomial time algorithm

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Near optimal colourability on hereditary graph families

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THEORETICAL COMPUTER SCIENCE 2024年 993卷

作者： Ju, Yiao Huang, Shenwei Nankai Univ Coll Comp Sci Tianjin 300350 Peoples R China Nankai Univ Tianjin Key Lab Network & Data Secur Technol Tianjin 300071 Peoples R China

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lovasz theta function. A graph family G is near optimal colourable if there is a constant number c such that every graph G is an element of G satisfies chi(G) <= max{c, omega(G)}, where chi (G) and omega(G) are the chromatic number and clique number of G, respectively. The near optimal colourable graph families together with the Lovasz theta function are useful for the study of the colouring problems for hereditary graph families. We investigate the near optimal colourability for (H, H-2)-free graphs. Our main result is an almost complete characterisation for the near optimal colourability for (H, H-2)-free graphs with two exceptional cases, one of which is the celebrated Gyarfas conjecture. As an application of our results, we show that the colouring problem for (2K(2), P-4 v K-n)-free graphs is polynomial time solvable, which solves an open problem in Dabrowski and Paulusma (2018) [6].

关键词： Graph colouring Lovasz theta function polynomial time algorithm Forbidden induced subgraphs

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Constrained flows in networks

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THEORETICAL COMPUTER SCIENCE 2024年 1010卷

作者： Bang-Jensen, J. Bessy, S. Picasarri-Arrieta, L. Univ Southern Denmark Dept Math & Comp Sci Odense Denmark Univ Montpellier LIRMM CNRS Montpellier France Univ Cote Azur CNRS I3S Inria Sophia Antipolis France

The support of a flow x in a network is the subdigraph induced by the arcs uv for which x(uv) > 0. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding whether a network N = (D. s, t, c) has a maximum flow x such that the maximum out-degree of the support D-x of x is at most 2 is NP-complete as it contains the 2-linkage problem as a very special case. Another problem which is NP-complete for the same reason is that of computing the maximum flow we can send from s to t along p paths (called a maximum p-path-flow) in N. Baier et al. (2005) gave a polynomial time algorithm which finds a-path-flow x whose value is at least 2/3 of the value of a optimum p-path-flow when p is an element of{2, 3}, and at least 1/2 when p >= 4. When p = 2, they show that this is best possible unless P=NP. We show for each p >= 2 that the value of a maximum p-path-flow cannot be approximated by any ratio larger than 9/11, unless P=NP. We also consider a variant of the problem where the p paths must be disjoint. For this problem, we give an algorithm which gets within a factor 1/H(p) of the optimum solution, where H(p) is the p'th harmonic number (H(p) similar to ln(p)). We show that in the case where the network is acyclic, we can find such a maximum p-path-flow in polynomial time for every p. We determine the complexity of a number of related problems concerning the structure of flows. For the special case of acyclic digraphs, some of the results we obtain are in some sense best possible.

关键词： Flows (Arc-)disjoint paths with prescribed end vertices Acyclic digraph polynomial time algorithm NP-complete problem Approximation algorithm Parameterised complexity

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How can we solve a linear Diophantine equation by the basis reduction algorithm

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2005年第10期82卷 1227-1234页

作者： Esmaeili, H Bu Ali Sina Univ Hamadan Iran

The main difficulty in solving the systems of linear Diophantine equations is the very rapid growth of the intermediate results. Hence, it is important to design algorithms that restraint the growth of intermediate results. The use of the basis reduction algorithm for solving a linear Diophantine equation is examined in this paper. The motivation behind choosing the basis reduction is twofold. First the basis reduction allows us to work only with integers, which avoids the round-off problems. Second, the basis reduction finds short and nearly orthogonal vectors belonging to the lattice described by the basis. Then, we expect the general solution obtained by this basis will also be short. It is important to note that the basis reduction does not change the lattice, it only derives an alternative way of spanning it. Once we have obtained the vectors given by the reduced basis, we use them to find the general solution of the linear Diophantine equation.

关键词： linear Diophantine equation polynomial time algorithm unimodular matrices basis reduction algorithm

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An algorithm for node-to-node disjoint paths problem in burnt pancake graphs

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2007年第1期E90D卷 306-313页

作者： Kaneko, Keiichi Sawada, Naoki Tokyo Univ Agr & Technol Grad Sch Engn Koganei Tokyo 1848588 Japan

In this paper, we propose an algorithm that solves the node-to-node disjoint paths problem in n-burnt pancake graphs in polynomial-order time of n. We also give a proof of its correctness as well as the estimates of time complexity O(n(3)) and the maximum path length 3n + 4. We conducted a computer experiment for n = 2 to 100 to measure the average performance of our algorithm. The results show that the average time complexity is O(n(3.0)) and the maximum path length is 3n + 4.

关键词： burnt pancake graphs node-to-node disjoint paths problem container problem internally disjoint paths polynomial time algorithm

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ON THE EDGE IMPORTANCE USING ITS TRAFFIC BASED ON A DISTRIBUTION FUNCTION ALONG SHORTEST PATHS IN A NETWORK

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 1995年第3期E78A卷 440-443页

作者： CHENG, P MASUYAMA, S Coll of Information Science and Cultural Studies Nisshin-shi Japan

We model a road network as a directed graph G = (V, E) with a source s and a sink t, where each edge e has a positive length l(e) and each vertex v has a distribution function alpha(v) with respect to the traffic entering and leaving v. This paper proposes a polynomial time algorithm for evaluating the importance of each edge e is an element of E which is defined to be the traffic f(e) passing through e in order to assign the required traffic F-st(> 0) from a to t along only shortest s-t paths in accordance with the distribution function alpha(v) at each vertex v.

关键词： GRAPH THEORY SHORTEST PATH EDGE IMPORTANCE WITH RESPECT TO TRAFFIC DISTRIBUTION FUNCTION polynomial time algorithm

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Combining decomposition approaches for the Maximum Weight Stable Set problem

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THEORETICAL COMPUTER SCIENCE 2023年 960卷

作者： Brandstaedt, Andreas Mosca, Raffaele Univ Rostock Inst Informat D-18051 Rostock Germany Univ G DAnnunzio Dipartimento Econ I-65121 Pescara Italy

The Maximum Weight Stable Set Problem (MWS) is a well-known NP-hard problem. A popular way to study MWS is to detect graph classes for which MWS can be solved in polynomial time. In this context some decomposition approaches have been introduced in the literature, in particular decomposition by homogeneous sets and decomposition by clique separators, which had several applications in the literature. Brandstadt and Hoang (2007) [6] showed how to combine such two decomposition approaches and stated the following result: If the MWS problem can be solved in polynomial time for every subgraph of a graph G which has no homogeneous set and has no clique separators then so can the problem for G. This result, namely Corollary 9 of [6], was used in various papers. Unfortunately, it was stated in a successive paper by Brandstadt and Giakoumakis (2015) [5] that "Actually, [6] does not give a proof for this, and it seems that Corollary 9 of [6] promises too much;later attempts for proving it failed, and thus, it is unproven and its use has to be avoided." This manuscript introduces a proof of Corollary 9 of [6]. Furthermore a third decomposition approach, namely decomposition by anti-neighborhoods of vertices, is combined together with such two decomposition approaches. Then the various papers in which Corollary 9 of [6] was used would be fixed.(c) 2023 Elsevier B.V. All rights reserved.

关键词： Graph algorithms Maximum Weight Independent Set problem polynomial time algorithm Decomposition by homogeneous sets Decomposition by clique separators Decomposition by anti-neighborhood of vertices

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Optimal resilient sensor placement problem for secure state estimation

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AUTOMATICA 2024年 160卷

作者： Shinohara, Takumi Namerikawa, Toru Keio Univ Grad Sch Sci & Technol 3-14-1 HiyoshiKohoku Ku Yokohama Kanagawa 2238522 Japan Keio Univ Dept Syst Design Engn 3-14-1 HiyoshiKohoku Ku Yokohama Kanagawa 2238522 Japan

In this paper, we consider a sensor placement problem for secure state estimation in an adversarial environment. Specifically, this paper deals with a system consisting of n agents where up to l sensor measurements are potentially compromised by malicious adversaries and tackles the problem of sensor placement such that the system state can be reconstructed even if the measurements are compromised. However, since it is costly to deploy sensors for all agents, we also aim to place sensors at the lowest possible cost. We call this problem the optimal resilient sensor placement problem (ORSPP) and show that this problem is coNP-hard. Therefore, it is, in general, impossible to compute the ORSPP in polynomial time unless P = coNP. On the positive side, however, we also provide that this problem can be solved in polynomial time if the network structure of the agents satisfies certain conditions. The concrete algorithm for the ORSPP is provided, and further, in the process of providing the coNP-hardness, we also derive a new form of a necessary and sufficient condition for secure state estimation. Finally, we show the results of numerical simulations using a random graph and a diffusion process.(c) 2023 Elsevier Ltd. All rights reserved.

关键词： Secure state estimation Optimal sensor placement Eigenvalue and sparse observability polynomial time algorithm

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Completing orientations of partially oriented graphs

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JOURNAL OF GRAPH THEORY 2018年第3期87卷 285-304页

作者： Bang-Jensen, J. Huang, J. Zhu, X. Univ Southern Denmark Dept Math & Comp Sci DK-5230 Odense Denmark Univ Victoria Dept Math & Stat Victoria BC V8W 3R4 Canada Zhejiang Normal Univ Dept Math Jinhua Peoples R China

We initiate a general study of what we call orientation completion problems. For a fixed class C of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in. by orienting the (nonoriented) edges in P. Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs. We study orientation completion problems for various classes of oriented graphs, including k-arc-strong oriented graphs, k-strong oriented graphs, quasi-transitive-oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in-tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP-complete. We also show that some of the NP-complete problems become polynomial time solvable when the input-oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.

关键词： friendly partial oriented graph in-tournament local tournament locally transitive local tournament NP-complete orientation completion problem partially oriented graph polynomial time algorithm proper circular arc graph proper interval graph recognition representation extension

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THE COMPLEXITY OF COMPUTING THE AUTOMORPHISM GROUP OF AUTOMATA AND RELATED PROBLEMS

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 1988年第1期25卷 17-31页

作者： ZHANG, SH LI, LA Department of Mathematics Lanzhou University Lanzhou Gansu China

A polynomial time algorithm for computing the automorphism group of cyclic automata is given. The problems of computing the automorphism group of automata, testing either two graphs, two strongly connected automata for isomorphism, or two automata for strong-isomorphism are shown to be polynomially equivalent. © 1988, Taylor & Francis Group, LLC. All rights reserved.

关键词： automorphism group Classification of automata graph isomorphism polynomial time algorithm polynomially equivalent problems strong-isomorphism

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