Constrained flows in networks
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 proble...
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| Vydané v: | Theoretical computer science Ročník 1010; s. 114702 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier B.V
27.09.2024
Elsevier |
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | 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 Dx 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 p-path-flow x whose value is at least 23 of the value of a optimum p-path-flow when p∈{2,3}, and at least 12 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 911, 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 1H(p) of the optimum solution, where H(p) is the p'th harmonic number (H(p)∼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. |
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| AbstractList | 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 Dx 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 p-path-flow x whose value is at least 23 of the value of a optimum p-path-flow when p∈{2,3}, and at least 12 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 911, 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 1H(p) of the optimum solution, where H(p) is the p'th harmonic number (H(p)∼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. The support of a flow $x$ in a network is the subdigraph induced by the arcs $ij$ for which $x_{ij}>0$. 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 ${\cal N}=(D,s,t,c)$ 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 deciding the maximum flow we can send from $s$ to $t$ along 2 paths (called a maximum 2-path-flow) in ${\cal N}$. Baier et al. (2005) gave a polynomial algorithm which finds a 2-path-flow $x$ whose value is at least $\frac{2}{3}$ of the value of a optimum 2-path-flow. This is best possible unless P=NP. They also obtained a $\frac{2}{p}$-approximation for the maximum value of a $p$-path-flow for every $p\geq 2$. In this paper we give an algorithm which gets within a factor $\frac{1}{H(p)}$ of the optimum solution, where $H(p)$ is the $p$'th harmonic number ($H(p) \sim \ln(p)$). This improves the approximation bound due to Baier et al. when $p\geq 5$. We show that in the case where the network is acyclic, we can find 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. |
| ArticleNumber | 114702 |
| Author | Picasarri-Arrieta, L. Bessy, S. Bang-Jensen, J. |
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| Cites_doi | 10.1016/j.tcs.2015.06.026 10.1016/j.dam.2023.11.026 10.1007/s101070100260 10.1016/j.tcs.2014.01.011 10.1016/0304-3975(80)90009-2 10.1137/070697781 10.1016/j.dam.2015.10.012 10.1016/j.tcs.2012.03.003 10.1007/s00453-005-1167-9 10.1016/j.orl.2019.10.012 10.1137/0205048 |
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| Keywords | (Arc-)disjoint paths with prescribed end vertices Acyclic digraph Polynomial time algorithm Parameterised complexity Flows NP-complete problem Approximation algorithm |
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| References | Even, Itai, Shamir (br0120) 1976; 5 Fortune, Hopcroft, Wyllie (br0130) 1980; 10 Baier, Köhler, Skutella (br0020) 2005; 42 Bang-Jensen, Gutin (br0040) 2009 Bessy, Hörsch, Maia, Rautenbach, Sau (br0090) 2024; 346 Skutella (br0150) 2002; 91 Disser, Matuschke (br0110) 2020; 48 Slivkins (br0160) 2010; 24 Bang-Jensen, Havet, Yeo (br0050) 2016; 209 Fürer, Raghavachari (br0140) 1992 Bang-Jensen, Bessy (br0030) 2014; 526 Bang-Jensen, Yeo (br0060) 2012; 438 Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Pilipczuk, Saurabh (br0100) 2015 Berman, Karpinski, Scott (br0080) 2003; TR03 Ahuja, Magnanti, Orlin (br0010) 1993 Bang-Jensen, Yeo (br0070) 2015; 595 Bang-Jensen (10.1016/j.tcs.2024.114702_br0040) 2009 Bang-Jensen (10.1016/j.tcs.2024.114702_br0060) 2012; 438 Berman (10.1016/j.tcs.2024.114702_br0080) 2003; TR03 Ahuja (10.1016/j.tcs.2024.114702_br0010) 1993 Disser (10.1016/j.tcs.2024.114702_br0110) 2020; 48 Skutella (10.1016/j.tcs.2024.114702_br0150) 2002; 91 Slivkins (10.1016/j.tcs.2024.114702_br0160) 2010; 24 Bang-Jensen (10.1016/j.tcs.2024.114702_br0030) 2014; 526 Bessy (10.1016/j.tcs.2024.114702_br0090) 2024; 346 Bang-Jensen (10.1016/j.tcs.2024.114702_br0070) 2015; 595 Fürer (10.1016/j.tcs.2024.114702_br0140) 1992 Baier (10.1016/j.tcs.2024.114702_br0020) 2005; 42 Bang-Jensen (10.1016/j.tcs.2024.114702_br0050) 2016; 209 Even (10.1016/j.tcs.2024.114702_br0120) 1976; 5 Fortune (10.1016/j.tcs.2024.114702_br0130) 1980; 10 Cygan (10.1016/j.tcs.2024.114702_br0100) 2015 |
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| Snippet | 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... The support of a flow $x$ in a network is the subdigraph induced by the arcs $ij$ for which $x_{ij}>0$. We discuss a number of results on flows in networks... |
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| SubjectTerms | (Arc-)disjoint paths with prescribed end vertices Acyclic digraph Approximation algorithm Combinatorics Flows Mathematics NP-complete problem Parameterised complexity Polynomial time algorithm |
| Title | Constrained flows in networks |
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