## Some Parameterized Problems on Digraphs

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Venue: | The Computer Journal |

Citations: | 6 - 2 self |

### BibTeX

@ARTICLE{Gutin_someparameterized,

author = {Gregory Gutin and Anders Yeo},

title = {Some Parameterized Problems on Digraphs},

journal = {The Computer Journal},

year = {},

volume = {51},

pages = {363--371}

}

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### Abstract

We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arrangement problem for undirected graphs. We state several open problems. 1

### Citations

871 | Parametrized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...ating some open parameterized problems for digraphs. We provide only very basic terminology of parameterized complexity, for an in-depth treatment of the topic we refer the reader to the classic text =-=[12]-=- and the new monographs [15, 36]. A parameterized problem Π can be considered as a set of pairs (I, k) where I is the problem instance and k (usually an integer) is the parameter. Π is called fixed-pa... |

466 |
Reducibility among combinatorial problems, in Complexity of Computer Computations
- Karp
- 1972
(Show Context)
Citation Context ...ace the arcs of D by the following: arc x − x + for each vertex x ∈ V (D) and x + y − for each arc xy ∈ A(D). It is not difficult to verify that the equalities of this proposition indeed hold. ⋄ Karp =-=[29]-=- was the first to prove that FAS is N P-complete. Proposition 2.1 and Karp’s result imply immediately that FVS is also N P-complete. For a digraph D, let ∆ 0 (D) be the minimum integer such that every... |

359 |
Parameterized Complexity Theory
- Flum, Grohe
- 2006
(Show Context)
Citation Context ...d problems for digraphs. We provide only very basic terminology of parameterized complexity, for an in-depth treatment of the topic we refer the reader to the classic text [12] and the new monographs =-=[15, 36]-=-. A parameterized problem Π can be considered as a set of pairs (I, k) where I is the problem instance and k (usually an integer) is the parameter. Π is called fixed-parameter tractable (FPT) if membe... |

325 | Retiming synchronous circuitry - Leiserson, Saxe - 1991 |

284 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
(Show Context)
Citation Context ...d problems for digraphs. We provide only very basic terminology of parameterized complexity, for an in-depth treatment of the topic we refer the reader to the classic text [12] and the new monographs =-=[15, 36]-=-. A parameterized problem Π can be considered as a set of pairs (I, k) where I is the problem instance and k (usually an integer) is the parameter. Π is called fixed-parameter tractable (FPT) if membe... |

234 | Digraphs: Theory, Algorithms and Applications
- Bang-Jensen, Gutin
- 2002
(Show Context)
Citation Context ... digraph H with V (H) = A(D) and in which there is an arc from xy ∈ V (H) to uv ∈ V (H) \ {xy} in H if y = u. A digraph H is a line digraph if there is a digraph D such that H = L(D). Proposition 2.1 =-=[4]-=- For every digraph D there exist digraphs D ′ and D ′′ such that τ0(D) = τ1(D ′ ) and τ1(D) = τ0(D ′′ ). The digraphs D ′ and D ′′ can be constructed from D in polynomial time. Proof: The digraph D ′′... |

174 | Representing Web Graphs
- Raghavan, Molina
- 2003
(Show Context)
Citation Context ... with at least k leaves? A bipartite digraph H = (V, A) with partite sets U and W is called a (k, ℓ)-dominator if |U| = k, |W | = ℓ and A = {uw : u ∈ U, w ∈ W }. 13sThe following problem was posed in =-=[32]-=-, a paper on data-mining the internet to identify online communities. The problem seems to be an obvious and perhaps easy candidate for W [1]-hardness, but in fact has resisted much effort. (k, l)-Dom... |

82 | Finding and counting given length cycles - Alon, Yuster, et al. - 1997 |

81 |
Finding a minimum circuit in a graph
- Itai, Rodeh
- 1978
(Show Context)
Citation Context ...f length 3 or the minimum degree of G is a least 3. In the last case, find a cycle C of length at most 2 lg |V | + 1 using a shortest cycle algorithm (there is such an algorithm of complexity O(n ω ) =-=[25]-=-, the complexity of multiplying two boolean matrices of size n × n). For each v ∈ V (C), run the algorithm recursively with input G − v and parameter k − 1 (rather than k). Observe that the running ti... |

72 | Ranking tournaments
- Alon
(Show Context)
Citation Context ...en for undirected graphs [16]. Bang-Jensen and Thomassen [5] conjectured that FAS is N P-complete even for tournaments. This conjecture was proved independently by at least four groups of researchers =-=[1, 2, 8, 9]-=-. Interestingly, FAS is polynomial time solvable for planar digraphs [4, 35] and trivially polynomial time solvable for undirected graphs. 3 Parameterized FVS and FAS The natural parameterizations of ... |

42 | Computing Slater rankings using similarities among candidates
- Conitzer
- 2006
(Show Context)
Citation Context ...en for undirected graphs [16]. Bang-Jensen and Thomassen [5] conjectured that FAS is N P-complete even for tournaments. This conjecture was proved independently by at least four groups of researchers =-=[1, 2, 8, 9]-=-. Interestingly, FAS is polynomial time solvable for planar digraphs [4, 35] and trivially polynomial time solvable for undirected graphs. 3 Parameterized FVS and FAS The natural parameterizations of ... |

36 | On the maximal number of disjoint circuits of a graph, Publ
- Erdos, Posa
- 1962
(Show Context)
Citation Context ...can replace the edges yx, zx incident to x by yz without creating parallel edges and without changing τ0(G). If the minimum degree of a vertex in G is at least 3, then, by a theorem of Erdős and Posa =-=[24]-=-, G has a cycle of length at most 2 lg |V | + 1. This suggests the following algorithm first considered in [38]: transform an input graph G = (V, E) such that either G has a cycle C of length 3 or the... |

34 |
On well-partial-order theory and its applications to combinatorial problems of VLSI design
- Fellows, Langston
- 1992
(Show Context)
Citation Context ...raphs it is equivalent to ask whether a graph G has a tree with at least k leaves or a spanning tree with at least k leaves. The last problem is well known to be FPT for undirected graphs, see, e.g., =-=[10, 11, 19]-=-. Let L is the set of digraphs D for which either ℓs(D) = 0 or ℓ(D) = ℓs(D) holds. Note that many digraphs do not belong to L. Let Hn be a digraph with vertex set {1, 2, . . . , n} and arc set {(i, i ... |

33 |
Some NP-complete problems on graphs
- Gavril
- 1977
(Show Context)
Citation Context ...d Karp’s result imply immediately that FVS is also N P-complete. For a digraph D, let ∆ 0 (D) be the minimum integer such that every vertex x of D has in-degree and out-degree at most ∆ 0 (D). Gavril =-=[17]-=- proved that FAS remains N P-complete even for digraphs D with ∆ 0 (D) ≤ 3 and for line digraphs. Bang-Jensen and Thomassen [5] proved that FVS is N P-complete even for tournaments. FVS remains N P-co... |

33 | The Creative Compute - Michie, Johnston - 1984 |

29 |
An o(2 O(k) n 3 ) fpt algorithm for the undirected feedback vertex set problem
- Dehne, Fellows, et al.
(Show Context)
Citation Context ...time of this algorithm is O((2 lg |V | + 1) k n ω ). Since (lg |V |) k ≤ |V | + (3k lg k) k , FVS is FPT for undirected graphs. Note that there are faster FPT algorithms for FVS for undirected graphs =-=[10, 19]-=- that run in time O(c k n O(1) ) for some constant c. The shortest cycle approach can be used for tournaments, i.e., digraphs in which there is exactly one arc between every pair of distinct vertices.... |

27 | private communications
- Kim
(Show Context)
Citation Context ...ghbors of x are linked by at least one arc. Locally semicomplete digraphs constitute another class of digraphs for which it is easy to prove that FVS is FPT. An easy way to see this (suggested by Kim =-=[30]-=-) is to apply the following theorem by Bang-Jensen [3]. A digraph on n vertices is round if we can label its vertices v1, v2, . . . , vn so that for each i, the out-neighbors of vi are vi+1, vi+2 . . ... |

26 |
A polynomial algorithm for 2-path problem in semicomplete graphs
- Bang-Jensen, Thomassen
- 1992
(Show Context)
Citation Context ...very vertex x of D has in-degree and out-degree at most ∆ 0 (D). Gavril [17] proved that FAS remains N P-complete even for digraphs D with ∆ 0 (D) ≤ 3 and for line digraphs. Bang-Jensen and Thomassen =-=[5]-=- proved that FVS is N P-complete even for tournaments. FVS remains N P-complete for digraphs D with ∆ 0 (D) ≤ 2 and for planar digraphs D with ∆ 0 (D) ≤ 3 [16]. This problem, unlike FAS, is N P-comple... |

24 | An Analytical Approach to the Partial Scan Problem
- Kunzmann, Wunderlich
- 1990
(Show Context)
Citation Context ... a small set of arcs whose removal makes the resulting digraph acyclic can help reduce the hardware overhead needed for testing the circuit using so-called scan registers (see Kunzmann and Wunderlich =-=[33]-=-). The classical decision problems Feedback Vertex Set (FVS) and Feedback Arc Set 2s(FAS) can be stated as follows. FVS Instance: A digraph D and a positive integer k. Question: Is τ0(D) ≤ k? FAS Inst... |

22 | A structural view on parameterizing problems: Distance from triviality
- Guo, Hüffner, et al.
(Show Context)
Citation Context ...se to acyclic. Indeed, k can be considered as an indicator of the ’distance’ from a given PCD to the closest acyclic digraph. So, k can be viewed as a ’distance from triviality’ in the terminology of =-=[20]-=-. We would rather call it ’distance from tractability.’ In this section, we consider another distance from tractability, which is of interest for PCDs that may be far from acyclic digraphs. Let D = (V... |

21 | The minimum feedback arc set problem is NP-hard for tournaments
- Charbit, Thomassé, et al.
- 2007
(Show Context)
Citation Context ...en for undirected graphs [16]. Bang-Jensen and Thomassen [5] conjectured that FAS is N P-complete even for tournaments. This conjecture was proved independently by at least four groups of researchers =-=[1, 2, 8, 9]-=-. Interestingly, FAS is polynomial time solvable for planar digraphs [4, 35] and trivially polynomial time solvable for undirected graphs. 3 Parameterized FVS and FAS The natural parameterizations of ... |

21 |
Parameterized Algorithmics: A Graph-Theoretic Approach. Habilitationsschrift. WilhelmSchickard-Institut für Informatik, Universität
- Fernau
- 2005
(Show Context)
Citation Context ... which gives an O(4 k |V | 3 )-algorithm. Another approach brings the complexity down. In particular, for bipartite tournaments, Truß [44] noted that a result on hitting sets in hypergraphs by Fernau =-=[14]-=- implies an O(3.12 k + |V | O(1) )-algorithm. We will give a very brief outline of the O(3.12 k + |V | O(1) )-algorithm for bipartite tournaments. Let T be any bipartite tournament and let HT be the h... |

21 | Color-coding: a new method for finding simple paths, cycles and other small subgraphs within large graphs
- Alon, Yuster, et al.
- 1994
(Show Context)
Citation Context ...-Leaf is FPT (for all digraphs) remains an open problem. 10 Other parameterized problems on digraphs In this section we give a short overview of some other papers on the topic. Alon, Yuster and Zwick =-=[4, 5]-=- introduced a new powerful method called color-coding that allows one to prove that the following two problems are FPT: given a digraph D and a parameter k check whether D has a directed path (cycle) ... |

20 |
Locally semicomplete digraphs: a generalization of tournaments
- Bang-Jensen
- 1990
(Show Context)
Citation Context ...icomplete digraphs constitute another class of digraphs for which it is easy to prove that FVS is FPT. An easy way to see this (suggested by Kim [30]) is to apply the following theorem by Bang-Jensen =-=[3]-=-. A digraph on n vertices is round if we can label its vertices v1, v2, . . . , vn so that for each i, the out-neighbors of vi are vi+1, vi+2 . . . , v i+d + (vi) and the in-neighbors of vi are v i−d ... |

19 | Graph-modeled data clustering: Fixedparameter algorithms for clique generation
- Gramm, Guo, et al.
(Show Context)
Citation Context ...if and only if they are not adjacent in D. Observe that cn(D) equals qd(G), which is the minimum number of edges that have to be deleted from G to obtain a collection of disjoint cliques. Gram et al. =-=[18]-=- obtained an O(n 3 + 1.77 k )-time algorithm to verify whether qd(G) ≤ k. We can use this algorithm to verify whether cn(D) ≤ k. Assume that cn(D) ≤ k. It is straightforward to use Theorems 6.1 and 6.... |

19 | Policy search using paired comparisons
- Strens, Moore
(Show Context)
Citation Context ...ngth can be different for different arcs depending on the ordering. We consider such a case in the next three sections. In the well-known method of paired comparisons, which has numerous applications =-=[31, 42, 43]-=-, we first compare objects of a set S pairwise (the comparison process is often performed by experts) and consequently produce an optimal ordering of S. In general, the outcome of the process of paire... |

18 |
A faster FPT algorithm for finding spanning trees with many leaves
- Bonsma, Brüggemann, et al.
- 2003
(Show Context)
Citation Context ...raphs it is equivalent to ask whether a graph G has a tree with at least k leaves or a spanning tree with at least k leaves. The last problem is well known to be FPT for undirected graphs, see, e.g., =-=[10, 11, 19]-=-. Let L is the set of digraphs D for which either ℓs(D) = 0 or ℓ(D) = ℓs(D) holds. Note that many digraphs do not belong to L. Let Hn be a digraph with vertex set {1, 2, . . . , n} and arc set {(i, i ... |

17 |
Finding and counting given length cycles. Algorithmica 17(3):209–223
- Alon, Yuster, et al.
- 1997
(Show Context)
Citation Context ...-Leaf is FPT (for all digraphs) remains an open problem. 10 Other parameterized problems on digraphs In this section we give a short overview of some other papers on the topic. Alon, Yuster and Zwick =-=[4, 5]-=- introduced a new powerful method called color-coding that allows one to prove that the following two problems are FPT: given a digraph D and a parameter k check whether D has a directed path (cycle) ... |

16 | Fixed-parameter tractability results for feedback set problems in tournaments
- Dom, Guo, et al.
- 2006
(Show Context)
Citation Context ... z ∈ V − {x} (overall O(|V | 2 ) time). This implies that FVS has an FPT algorithm of complexity O(3 k |V | 2 ). A faster algorithm was designed in [39]. The current fastest FPT algorithm is given in =-=[11]-=- and its time complexity is O(2 k |V | 2 (lg |V | + k)). Certainly, the shortest cycle approach can be used for the whole class of multipartite tournaments, i.e., directed graphs obtained from complet... |

16 | Improved fixedparameter algorithms for two feeback set problems
- Guo, Gramm, et al.
- 2004
(Show Context)
Citation Context ...time of this algorithm is O((2 lg |V | + 1) k n ω ). Since (lg |V |) k ≤ |V | + (3k lg k) k , FVS is FPT for undirected graphs. Note that there are faster FPT algorithms for FVS for undirected graphs =-=[10, 19]-=- that run in time O(c k n O(1) ) for some constant c. The shortest cycle approach can be used for tournaments, i.e., digraphs in which there is exactly one arc between every pair of distinct vertices.... |

15 | Analogs & duals of the MAST problem for sequences & trees
- Fellows, Hallett, et al.
- 1998
(Show Context)
Citation Context ...these problems are either both FPT, or are both not. So far, the parameterized complexity remains a (well-known) open problem [12]. Many researchers believe that both problems are FPT. Fellows et al. =-=[13]-=- introduce a parameterized version of the shortest common sequence problem such that if FVS is not FPT, then the problem in [13] is not FPT either. However, FVS was proved to be FPT for some special c... |

15 | Kernels in planar digraphs
- Gutin, Kloks, et al.
- 2005
(Show Context)
Citation Context ...h undirected and directed graphs. 9 Other parameterized problems on digraphs We have already observed that there are not so many papers on parameterized problems on digraphs. We would like to mention =-=[21]-=-, where an FPT algorithm is obtained for the kernel problem on planar digraphs, and [14], where an FPT algorithm is derived for the Directed LAP Parameterized Above Guaranteed Value (DLAPAGV). A kerne... |

14 |
On linear time minor tests and depth-first search
- Bodlaender
- 1993
(Show Context)
Citation Context ...raphs it is equivalent to ask whether a graph G has a tree with at least k leaves or a spanning tree with at least k leaves. The last problem is well known to be FPT for undirected graphs, see, e.g., =-=[10, 11, 19]-=-. Let L is the set of digraphs D for which either ℓs(D) = 0 or ℓ(D) = ℓs(D) holds. Note that many digraphs do not belong to L. Let Hn be a digraph with vertex set {1, 2, . . . , n} and arc set {(i, i ... |

12 | The linear arrangement problem parameterized above guaranteed value. Theory Comput
- Gutin, Rafiey, et al.
(Show Context)
Citation Context ...P can naturally be parameterized by asking whether ola(G) ≤ k. It is easy to see that this problem is FPT. It is significantly harder to prove that the problem ola(G) − |E| ≤ k is FPT; the authors of =-=[22]-=- obtained an O(|V | + |E| + 5.88 k )-algorithm for this problem. Notice that the problem ola(G) − |E| ≤ |E| r is NP-complete for every fixed r > 0 [22]. It makes sense to consider ola(G) − |E| rather ... |

12 | Parameterized tractability of edge-disjoint paths on directed acyclic graphs, in
- Slivkins
(Show Context)
Citation Context ...x Disjoint Cycle Problem (VDCP): given a digraph D and a parameter k, check whether D has at least k vertex-disjoint cycles. The fact that VDCP is W[1]-hard easily follows from the results of Slivkis =-=[51]-=-. For more details, see [25]. 4 Methods of Paired Comparison Proposition 3.2 shows that the minimal size of a feedback arc set in a digraph D equals the minimal number of backward arcs in an ordering ... |

11 | S.: Parameterized algorithms for directed maximum leaf problems - Alon, Fomin, et al. |

10 |
Ranking the vertices of a paired comparison digraph
- Kano, Sakamoto
- 1985
(Show Context)
Citation Context ...by (i) and (ii) the uniform PCD corresponding to D. The score of a vertex x ∈ V is σ + (x) = � xy∈A ɛ(xy). We describe two methods (forward and backward) of ordering V introduced by Kano and Sakamoto =-=[27, 28]-=-. Notice that, for semicomplete digraphs (digraphs that can be obtained from complete graphs by replacing every edge xy with the arc xy or the arc yx or both xy and yx), these methods agree with the s... |

10 | Short cycles in digraphs - Nishimura |

10 | Reducibility among combinatorial problems. In: Proceedings of a symposium on the Complexity of Computer Computations - Karp - 1972 |

8 |
Ranking the Vertices of a Weighted Digraph Using the Length of Forward Arcs
- Kano, Sakamoto
- 1983
(Show Context)
Citation Context ...by (i) and (ii) the uniform PCD corresponding to D. The score of a vertex x ∈ V is σ + (x) = � xy∈A ɛ(xy). We describe two methods (forward and backward) of ordering V introduced by Kano and Sakamoto =-=[27, 28]-=-. Notice that, for semicomplete digraphs (digraphs that can be obtained from complete graphs by replacing every edge xy with the arc xy or the arc yx or both xy and yx), these methods agree with the s... |

8 |
A minimax equality for directed graphs
- Lucchesi
- 1976
(Show Context)
Citation Context ... is N P-complete even for tournaments. This conjecture was proved independently by at least four groups of researchers [1, 2, 8, 9]. Interestingly, FAS is polynomial time solvable for planar digraphs =-=[4, 35]-=- and trivially polynomial time solvable for undirected graphs. 3 Parameterized FVS and FAS The natural parameterizations of FVS and FAS are formulated as follows. 3sParameterized FVS Instance: A digra... |

8 | Packing directed circuits - Reed, Robertson, et al. - 1996 |

8 |
Graphs with interlinked directed circuits
- Younger
- 1973
(Show Context)
Citation Context ...an in-neighbor of a vertex xy of L(D) if v = x). We finish this section with the following graph-theoretical result potentially useful in establishing parameterized complexity of FVS and FAS. Younger =-=[45]-=- conjectured that for every k, there exists a (least) natural number t0(k) (t1(k), respectively) such that for every digraph D the following holds: either D contains k vertex-disjoint (arc-disjoint, r... |

5 |
Ranking the vertices of an r-partite paired comparison digraph
- Kano
- 1987
(Show Context)
Citation Context ...for a vertex x ∈ V , we define ψ(α, x) = σ + (x) + |{y ∈ U : α(y) > α(x)}|, where U is the partite set of D containing x. The following theorem is an important ingredient of our approach. Theorem 6.1 =-=[26]-=- Let D be a semicomplete multipartite PCD with n vertices. An ordering α with α(vi) = i, i = 1, 2, . . . , n is backward optimal if and only if the following two conditions hold: (a) ψ(α, v1) ≥ ψ(α, v... |

5 |
Parameterized complexity
- Raman
- 1997
(Show Context)
Citation Context ...either. However, FVS was proved to be FPT for some special classes of digraph as well as for undirected graphs. We describe the main idea behind the following FPT algorithm for undirected graphs from =-=[38]-=-. This idea, which we call the shortest cycle approach, is used by many FPT algorithms for FVS for both directed and undirected graphs. Let G = (V, E) be an undirected graph. Clearly, we may delete a ... |

5 | Parameterized approximability of the disjoint cycle problem, in - Grohe, Grüber |

4 | Ranking the vertices of a complete multipartite paired comparison digraph
- Gutin, Yeo
- 1996
(Show Context)
Citation Context ... ordering of D in time O(n 2 2 k ). ⋄ This result shows that BPCP with cn(D) as a parameter is FPT. We can obtain a similar result for FPCP with cn(D) using the polynomial time algorithm described in =-=[23]-=-. 7 Application of BPCP to the Linear Arrangement Problem For a graph G = (V, E), let α be an ordering of G. The length of an edge uv ∈ E relative to α is defined as λ ′ α(uv) = |α(u) − α(v)|. The cos... |

4 |
Experimental ergonomic evaluation with user trials: EEE product development procedures. Oulun yliopisto
- Kirvesoja
- 2001
(Show Context)
Citation Context ...ngth can be different for different arcs depending on the ordering. We consider such a case in the next three sections. In the well-known method of paired comparisons, which has numerous applications =-=[31, 42, 43]-=-, we first compare objects of a set S pairwise (the comparison process is often performed by experts) and consequently produce an optimal ordering of S. In general, the outcome of the process of paire... |

3 |
Applying a method of paired comparisons to measure economic values for multiple goods sets
- Rosenberger, Peterson, et al.
- 2002
(Show Context)
Citation Context ...ngth can be different for different arcs depending on the ordering. We consider such a case in the next three sections. In the well-known method of paired comparisons, which has numerous applications =-=[31, 42, 43]-=-, we first compare objects of a set S pairwise (the comparison process is often performed by experts) and consequently produce an optimal ordering of S. In general, the outcome of the process of paire... |

2 |
Cycles in Digraphs
- BERMOND, THOMASSEN
- 1981
(Show Context)
Citation Context ... the shortest cycle approach, we can prove the following: Theorem 3.3 FAS is FPT for digraphs in Sh. We provide only a scheme of the proof of Theorem 3.3 in [39]. We will use the following: Lemma 3.4 =-=[6]-=- Let l ≥ 2 be any integer and let H be a strongly connected digraph with n vertices and m arcs such that m ≥ n2 + (3 − 2l)n + (l2 − l) . 2 Then the length of a shortest cycle in H is at most l. Using ... |