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    <publishedDate>2017-06-28</publishedDate>
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    <title language="eng">The Cone of Flow Matrices: Approximation Hierarchies and Applications</title>
    <abstract language="eng">Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone&#13;
generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where &#13;
$\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P.&#13;
We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation&#13;
of a flow, reduce to a linear optimization problem over $\mathcal{K}$.&#13;
This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$&#13;
is NP-complete. However, the affine hull of this cone admits a nice description,&#13;
and we give an algorithm which computes in polynomial-time the decomposition of a matrix&#13;
$X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s.&#13;
Then, we provide two convergent approximation hierarchies, one of them based on a&#13;
completely positive representation of~K.&#13;
We illustrate this approach by computing bounds for &#13;
the quadratic shortest path problem, as well as&#13;
a maximum flow problem with pairwise arc-capacities.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-64399</identifier>
    <identifier type="doi">10.1002/net.21820</identifier>
    <enrichment key="SourceTitle">Appeared in: Networks 72(1): 128-150</enrichment>
    <author>Guillaume Sagnol</author>
    <submitter>Guillaume Sagnol</submitter>
    <author>Marco Blanco</author>
    <author>Thibaut Sauvage</author>
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      <title>ZIB-Report</title>
      <number>17-32</number>
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    <subject>
      <language>eng</language>
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      <value>Flows in graphs</value>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Approximation Hierarchies</value>
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    <subject>
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      <type>uncontrolled</type>
      <value>Copositive Programming</value>
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    <subject>
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      <value>Semidefinite Programming</value>
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    <publishedYear>2018</publishedYear>
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    <title language="eng">The Cone of Flow Matrices: Approximation Hierarchies and Applications</title>
    <abstract language="eng">Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone&#13;
generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where &#13;
$\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P.&#13;
We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation&#13;
of a flow, reduce to a linear optimization problem over $\mathcal{K}$.&#13;
This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$&#13;
is NP-complete. However, the affine hull of this cone admits a nice description,&#13;
and we give an algorithm which computes in polynomial-time the decomposition of a matrix&#13;
$X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s.&#13;
Then, we provide two convergent approximation hierarchies, one of them based on a&#13;
completely positive representation of~K.&#13;
We illustrate this approach by computing bounds for &#13;
the quadratic shortest path problem, as well as&#13;
a maximum flow problem with pairwise arc-capacities.</abstract>
    <parentTitle language="eng">Networks</parentTitle>
    <identifier type="doi">10.1002/net.21820</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">March 2018</enrichment>
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    <author>Guillaume Sagnol</author>
    <submitter>Guillaume Sagnol</submitter>
    <author>Marco Blanco</author>
    <author>Thibaut Sauvage</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
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    <title language="eng">Approximation Hierarchies for the cone of flow matrices</title>
    <abstract language="eng">Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone&#13;
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where &#13;
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.&#13;
Several combinatorial problems reduce to a linear optimization problem over $K$.&#13;
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a&#13;
completely positive representation of $K$.&#13;
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-68424</identifier>
    <identifier type="doi">10.1016/j.endm.2018.02.002</identifier>
    <enrichment key="SourceTitle">Appeared in: Electronic Notes in Discrete Mathematics  Volume 64, February 2018, Pages 275–284  INOC 2017 – 8th International Network Optimization Conference</enrichment>
    <author>Guillaume Sagnol</author>
    <submitter>Guillaume Sagnol</submitter>
    <author>Marco Blanco</author>
    <author>Thibaut Sauvage</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-20</number>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Flows in graphs</value>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Approximation hierarchies</value>
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    <subject>
      <language>eng</language>
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      <value>Copositive programming</value>
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    <publishedYear>2018</publishedYear>
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    <language>eng</language>
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    <title language="eng">Approximation Hierarchies for the cone of flow matrices</title>
    <abstract language="eng">Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone&#13;
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where &#13;
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.&#13;
Several combinatorial problems reduce to a linear optimization problem over $K$.&#13;
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a&#13;
completely positive representation of $K$.&#13;
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.</abstract>
    <parentTitle language="eng">INOC 2017 – 8th International Network Optimization Conference</parentTitle>
    <identifier type="doi">10.1016/j.endm.2018.02.002</identifier>
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    <author>Guillaume Sagnol</author>
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    <author>Thibaut Sauvage</author>
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