731 research outputs found

    New versions of the all-ones problem

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    We study three new versions of the All-Ones Problem and the Minimum All-Ones Problem. The original All-Ones Problem is simply called the Vertex-Vertex Problem, and the three new versions are called the Vertex-Edge Problem, the Edge-Vertex Problem and the Edge-Edge Problem, respectively. The Vertex-Vertex Problem has been studied extensively. For example, existence of solutions and efficient algorithms for finding solutions were obtained, and the Minimum Vertex-Vertex Problem for general graphs was shown to be NP-complete and for trees it can be solved in linear time, etc. In this paper, for the Vertex-Edge Problem, we show that a graph has a solution if and only if it is bipartite, and therefore it has only two possible solutions and optimal solutions. A linear program version is also given. For the Edge-Vertex Problem, we show that a graph has a solution if and only if it contains even number of vertices. By showing that the Minimum Edge-Vertex Problem can be polynomially transformed into the Minimum Weight Perfect Matching Problem, we obtain that the Minimum Edge-Vertex Problem can be solved in polynomial time in general. The Edge-Edge Problem is reduced to the Vertex-Vertex Problem for the line graph of a graph.Comment: 12 page

    Good upper bounds for the total rainbow connection of graphs

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    A total-colored graph is a graph GG such that both all edges and all vertices of GG are colored. A path in a total-colored graph GG is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph GG is total-rainbow connected if any two vertices of GG are connected by a total rainbow path of GG. The total rainbow connection number of GG, denoted by trc(G)trc(G), is defined as the smallest number of colors that are needed to make GG total-rainbow connected. These concepts were introduced by Liu et al. Notice that for a connected graph GG, 2diam(G)βˆ’1≀trc(G)≀2nβˆ’32diam(G)-1\leq trc(G)\leq 2n-3, where diam(G)diam(G) denotes the diameter of GG and nn is the order of GG. In this paper we show, for a connected graph GG of order nn with minimum degree Ξ΄\delta, that trc(G)≀6n/(Ξ΄+1)+28trc(G)\leq6n/{(\delta+1)}+28 for Ξ΄β‰₯nβˆ’2βˆ’1\delta\geq\sqrt{n-2}-1 and nβ‰₯291n\geq 291, while trc(G)≀7n/(Ξ΄+1)+32trc(G)\leq7n/{(\delta+1)}+32 for 16≀δ≀nβˆ’2βˆ’216\leq\delta\leq\sqrt{n-2}-2 and trc(G)≀7n/(Ξ΄+1)+4C(Ξ΄)+12trc(G)\leq7n/{(\delta+1)}+4C(\delta)+12 for 6≀δ≀156\leq\delta\leq15, where C(Ξ΄)=e3log⁑(Ξ΄3+2Ξ΄2+3)βˆ’3(log⁑3βˆ’1)Ξ΄βˆ’3βˆ’2C(\delta)=e^{\frac{3\log({\delta}^3+2{\delta}^2+3)-3(\log3-1)}{\delta-3}}-2. This implies that when Ξ΄\delta is in linear with nn, then the total rainbow number trc(G)trc(G) is a constant. We also show that trc(G)≀7n/4βˆ’3trc(G)\leq 7n/4-3 for Ξ΄=3\delta=3, trc(G)≀8n/5βˆ’13/5trc(G)\leq8n/5-13/5 for Ξ΄=4\delta=4 and trc(G)≀3n/2βˆ’3trc(G)\leq3n/2-3 for Ξ΄=5\delta=5. Furthermore, an example shows that our bound can be seen tight up to additive factors when Ξ΄β‰₯nβˆ’2βˆ’1\delta\geq\sqrt{n-2}-1.Comment: 8 page

    Investigation of Monaural Front-End Processing for Robust ASR without Retraining or Joint-Training

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    In recent years, monaural speech separation has been formulated as a supervised learning problem, which has been systematically researched and shown the dramatical improvement of speech intelligibility and quality for human listeners. However, it has not been well investigated whether the methods can be employed as the front-end processing and directly improve the performance of a machine listener, i.e., an automatic speech recognizer, without retraining or joint-training the acoustic model. In this paper, we explore the effectiveness of the independent front-end processing for the multi-conditional trained ASR on the CHiME-3 challenge. We find that directly feeding the enhanced features to ASR can make 36.40% and 11.78% relative WER reduction for the GMM-based and DNN-based ASR respectively. We also investigate the affect of noisy phase and generalization ability under unmatched noise condition.Comment: 5 pages, 0 figures, 4 tables, conferenc

    Rainbow vertex-connection and forbidden subgraphs

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    A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph GG is \emph{rainbow vertex-connected} if for any two distinct vertices of GG, there is a vertex-rainbow path connecting them. For a connected graph GG, the \emph{rainbow vertex-connection number} of GG, denoted by rvc(G)rvc(G), is defined as the minimum number of colors that are required to make GG rainbow vertex-connected. In this paper, we find all the families F\mathcal{F} of connected graphs with ∣F∣∈{1,2}|\mathcal{F}|\in\{1,2\}, for which there is a constant kFk_\mathcal{F} such that, for every connected F\mathcal{F}-free graph GG, rvc(G)≀diam(G)+kFrvc(G)\leq diam(G)+k_\mathcal{F}, where diam(G)diam(G) is the diameter of GG.Comment: 11 page

    Total monochromatic connection of graphs

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    A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same color. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph. For a connected graph GG, the {\it total monochromatic connection number}, denoted by tmc(G)tmc(G), is defined as the maximum number of colors used in a TMC-coloring of GG. These concepts are inspired by the concepts of monochromatic connection number mc(G)mc(G), monochromatic vertex connection number mvc(G)mvc(G) and total rainbow connection number trc(G)trc(G) of a connected graph GG. Let l(T)l(T) denote the number of leaves of a tree TT, and let l(G)=max⁑{l(T)∣l(G)=\max\{ l(T) | TT is a spanning tree of GG }\} for a connected graph GG. In this paper, we show that there are many graphs GG such that tmc(G)=mβˆ’n+2+l(G)tmc(G)=m-n+2+l(G), and moreover, we prove that for almost all graphs GG, tmc(G)=mβˆ’n+2+l(G)tmc(G)=m-n+2+l(G) holds. Furthermore, we compare tmc(G)tmc(G) with mvc(G)mvc(G) and mc(G)mc(G), respectively, and obtain that there exist graphs GG such that tmc(G)tmc(G) is not less than mvc(G)mvc(G) and vice versa, and that tmc(G)=mc(G)+l(G)tmc(G)=mc(G)+l(G) holds for almost all graphs. Finally, we prove that tmc(G)≀mc(G)+mvc(G)tmc(G)\leq mc(G)+mvc(G), and the equality holds if and only if GG is a complete graph.Comment: 12 page

    Total proper connection of graphs

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    A graph is said to be {\it total-colored} if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a {\it total proper path} if (i)(i) any two adjacent edges on the path differ in color, (ii)(ii) any two internal adjacent vertices on the path differ in color, and (iii)(iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called {\it total-proper connected} if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph GG, the {\it total proper connection number} of GG, denoted by tpc(G)tpc(G), is defined as the smallest number of colors required to make GG total-proper connected. These concepts are inspired by the concepts of proper connection number pc(G)pc(G), proper vertex connection number pvc(G)pvc(G) and total rainbow connection number trc(G)trc(G) of a connected graph GG. In this paper, we first determine the value of the total proper connection number tpc(G)tpc(G) for some special graphs GG. Secondly, we obtain that tpc(G)≀4tpc(G)\leq 4 for any 22-connected graph GG and give examples to show that the upper bound 44 is sharp. For general graphs, we also obtain an upper bound for tpc(G)tpc(G). Furthermore, we prove that tpc(G)≀3nΞ΄+1+1tpc(G)\leq \frac{3n}{\delta+1}+1 for a connected graph GG with order nn and minimum degree Ξ΄\delta. Finally, we compare tpc(G)tpc(G) with pvc(G)pvc(G) and pc(G)pc(G), respectively, and obtain that tpc(G)>pvc(G)tpc(G)>pvc(G) for any nontrivial connected graph GG, and that tpc(G)tpc(G) and pc(G)pc(G) can differ by tt for 0≀t≀20\leq t\leq 2.Comment: 15 page

    Integrated Speech Enhancement Method Based on Weighted Prediction Error and DNN for Dereverberation and Denoising

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    Both reverberation and additive noises degrade the speech quality and intelligibility. Weighted prediction error (WPE) method performs well on the dereverberation but with limitations. First, WPE doesn't consider the influence of the additive noise which degrades the performance of dereverberation. Second, it relies on a time-consuming iterative process, and there is no guarantee or a widely accepted criterion on its convergence. In this paper, we integrate deep neural network (DNN) into WPE for dereverberation and denoising. DNN is used to suppress the background noise to meet the noise-free assumption of WPE. Meanwhile, DNN is applied to directly predict spectral variance of the target speech to make the WPE work without iteration. The experimental results show that the proposed method has a significant improvement in speech quality and runs fast

    Using Optimal Ratio Mask as Training Target for Supervised Speech Separation

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    Supervised speech separation uses supervised learning algorithms to learn a mapping from an input noisy signal to an output target. With the fast development of deep learning, supervised separation has become the most important direction in speech separation area in recent years. For the supervised algorithm, training target has a significant impact on the performance. Ideal ratio mask is a commonly used training target, which can improve the speech intelligibility and quality of the separated speech. However, it does not take into account the correlation between noise and clean speech. In this paper, we use the optimal ratio mask as the training target of the deep neural network (DNN) for speech separation. The experiments are carried out under various noise environments and signal to noise ratio (SNR) conditions. The results show that the optimal ratio mask outperforms other training targets in general

    Tight Nordhaus-Gaddum-type upper bound for total-rainbow connection number of graphs

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    A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph GG, the \emph{total-rainbow connection number} of GG, denoted by trc(G)trc(G), is the minimum number of colors required in a total-coloring of GG to make GG total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus-Gaddum-type upper bound for the total-rainbow connection number. We prove that if GG and Gβ€Ύ\overline{G} are connected complementary graphs on nn vertices, then trc(G)+trc(Gβ€Ύ)≀2ntrc(G)+trc(\overline{G})\leq 2n when nβ‰₯6n\geq 6 and trc(G)+trc(Gβ€Ύ)≀2n+1trc(G)+trc(\overline{G})\leq 2n+1 when n=5n=5. Examples are given to show that the upper bounds are sharp for nβ‰₯5n\geq 5. This completely solves a conjecture in [Y. Ma, Total rainbow connection number and complementary graph, Results in Mathematics 70(1-2)(2016), 173-182].Comment: 20 page

    On (strong) proper vertex-connection of graphs

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    A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex kk-connected} if any two vertices of the graph are connected by kk disjoint vertex-proper paths of the graph. For a kk-connected graph GG, the {\it proper vertex kk-connection number} of GG, denoted by pvck(G)pvc_{k}(G), is defined as the smallest number of colors required to make GG proper vertex kk-connected. A vertex-colored graph is {\it strong proper vertex-connected}, if for any two vertices u,vu,v of the graph, there exists a vertex-proper uu-vv geodesic. For a connected graph GG, the {\it strong proper vertex-connection number} of GG, denoted by spvc(G)spvc(G), is the smallest number of colors required to make GG strong proper vertex-connected. These concepts are inspired by the concepts of rainbow vertex kk-connection number rvck(G)rvc_k(G), strong rainbow vertex-connection number srvc(G)srvc(G), and proper kk-connection number pck(G)pc_k(G) of a kk-connected graph GG. Firstly, we determine the value of pvc(G)pvc(G) for general graphs and pvck(G)pvc_k(G) for some specific graphs. We also compare the values of pvck(G)pvc_k(G) and pck(G)pc_k(G). Then, sharp bounds of spvc(G)spvc(G) are given for a connected graph GG of order nn, that is, 0≀spvc(G)≀nβˆ’20\leq spvc(G)\leq n-2. Moreover, we characterize the graphs of order nn such that spvc(G)=nβˆ’2,nβˆ’3spvc(G)=n-2,n-3, respectively. Finally, we study the relationship among the three vertex-coloring parameters, namely, $spvc(G), \ srvc(G)andthechromaticnumber and the chromatic number \chi(G)ofaconnectedgraph of a connected graph G$.Comment: 12 page
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