1. Introduction
Belonging to R. M. Karp’s famous list of 21 NP–complete problems (Karp:1972), the FVSP & FASP are of central interest in theoretical computer science and applications beyond. While the undirected version of the FASP can be solved efficiently by computing a maximal spanning tree the undirected FVSP remains NP–complete. The only known (directed) instance classes possessing polynomial time solutions are planar or more general weakly acyclic graphs (Groetschel:1985). Parameter tractable algorithms are given in (Chen; FASP). Further, by –reduction of the minimum vertex cover problem (MVCP) both problems are known to be APX–hard (Karp:1972) and inapproximable beneath a ratio of (Dinur), unless P=NP. The undirected FVSP can be approximated within ratio (bafna; becker) and is thereby APX–complete. The FASP on tournaments possesses a PTAS (tour). We recommend (bang) for further studies. That the directed FVSP & FASP are approximation preserving reducible to each other is known due to (ausiello; crescenzi1995compendium; Even:1998; kann1992). In our previous work (FASP) we compactified these constructions. However, we missed the crucial difference between the directed and undirected FVSP and therefore misleadingly claimed the APX–completeness of the FASP. In addition to its new contributions, we want to take this article as a chance to correct our misunderstanding.
The complementary problem of finding the maximum acyclic subgraph is known to be MAX SNP complete and thereby approximable (hassin). However, this fact is not sufficient to approximate the directed FVSP & FASP. Though approximations of constrained versions of the problems (Even:1998) were delivered, these approximations depend logarithmically on the number of cycles a graph possesses. By reduction from the Hamiltonian cycle problem, counting all cycles is already a #P–hard counting problem (Arora) and thereby the proposed approximation is not bounded constantly.
To give a by no means exhaustive list of related problems and applications we here cite publications regarding minimum transversals of directed cuts (Lucchesi:1978) and general minimum multi cuts (Even:1998), circuit testing (leiserson; gupta; kunzmann) and efficient deadlock resolution (logic), computational biology and neuroscience (bao; greedy), (logical) network analysis and operating systems (Silber; unger). We recommend (marti) for further considerations. It is notable that in most of the applications the appearing graphs are of large and sparse nature.
In (Eades1993; Saab) an excellent overview of heuristical solutions is given. Further,(Eades1993) proposes the most common state of the art heuristic termed Greedy Removal (GR). Exact methods use ILP–solvers with modern formulations given in (exact; sagemath) based on the results of (Groetschel:1985; younger). However, for dense or large sparse graphs the ILP–approaches run into time out while the heuristical approach GR performs inaccurately. As we present in this article, our proposed heuristical solution can fill this gap and produce reasonable results.
2. Theoretical Considerations
In this section we provide the main graph theoretical concepts, which are required to formulate our heuristical solution.
2.1. Preliminaries
For rendering the introduced concepts of this article consistent, we introduce a non–classical definition of graphs as follows.
Definition 2.1 ().
Let be a –tuple, where are finite sets and are some maps. We call the elements vertices and the elements arcs of , while are called head and tail of the arc . An arc with is called a loop. In general, we call a multi–digraph. The following cases are often relevant:

is called a digraph iff the map , with is injective.

is called an undirected graph iff is a digraph and for every there is with , . In this case, we slightly simplify notation by shortly writing for the pair , which is then called an edge. The notion of can thereby be replaced by with .

In the special case, were the maps are assumed to be canonically given by the relation of , i.e., , for all .
One readily observes that in the cases our definition coincides with the common understanding of graphs. In the general case of multi–digraphs, our definition has the advantage that though multiple arcs with , are allowed are distinguished. Thus, is no multi–set, as it is assumed usually, but a simple set, simplifying our considerations. For we denote with , , , the set of all parallel and anti–parallel arcs and their union, respectively.
Further, two arcs and are called consecutive if and are called connected if . A directed path of length from a vertex to a vertex is a list of consecutive arcs , such that and . Thereby, repetition is allowed, i.e., , is possible.
A directed cycle is a directed path from some vertex to itself, which can also be a loop. shall denote the sets of all directed cycles of . A cycle is called simple or elementary if every arc or vertex it contains is passed exactly once, respectively. Certainly, every cycle is given by passing through several elementary cycles. We denote with , the set of all directed elementary cycles. While non–elementary cycles can coincide in their arc sets but differ in their orderings the only reorderings of elementary cycles are of cyclic nature. Thus, an elementary cycle is uniquely determined up to cyclic reorderings by its arcs.
Example 2.2 ().
Consider the graph in Figure 1. The cycle is a simple and nonelementary cycle, while the cycles and are elementary. Certainly, is given by passing through and . Further, the cycle while . The only reorderings of keeping the arcs consecutive are cyclic reorderings.
With , we denote the graphs obtained by deleting the arc or the vertex and all its connected arcs. Further, , , denote the graph, the set of all arcs, and the set of all vertices induced by a set of graphs, arcs and vertices. By we denote the power set of a given set of finite cardinality .
2.2. Problem Formulation
In the following we formulate the classical optimization problems considered in this article.
Problem 1 (FASP & FVSP).
Let be a multi–digraph and be an arc weight function. Then the weighted FASP is to find a set of arcs such that is acyclic, i.e., and
(1) 
is minimized. The weighted minimum feedback vertex set problem (FVSP) is given by considering a vertex weight function and ask for a set of vertices such that is acyclic, i.e., and
(2) 
is minimized. We denote the set of solutions of the FASP & FVSP with , respectively.
Further, we call a minimum feedback arc set and a minimum feedback vertex set and denote with , the minimum feedback arc/vertex length. If or are constant functions then we derive the unweighted versions of the FASP & FVSP, respectively.
Remark 2.1 ().
Note that, checking whether a graph is acyclic or not can be done by topological sorting in –time (cormen; kahn; tarjan_sort). Further, every directed cycle is given by passing through several elementary cycles. Thus, the conditions and are equivalent. The FVSP & FASP can be also formulated in terms of the maximum linear ordering problem see for instance (exact; younger).
2.3. Isolated Cycles
The complexity of an instance for the FVSP or FASP is certainly correlated to the structure of its cycles. However, by reducing to the Hamiltonian cycle problem, already counting all directed elementary cycles turns out to be a #Phard problem. This makes it hard to study the structure of . Here, we propose to use a technique developed in our previous article (FASP) to overcome this issue.
Definition 2.3 (cycle cover & isolated cycles).
Let be a multi–digraph and . We call the subgraph
(3) 
induced by all elementary cycles passing through or a parallel arcs the cycle cover of . Further, we denote with
the subgraph induced by all isolated cycles passing through , i.e., if is isolated then intersects with no cycle passing not through or some parallel arc of .
Remark 2.2 ().
Note, that the sets of isolated cycles possess a flat hierarchy in the following sense. If , with then . Vice versa implies and further .
Example 2.4 ().
Proposition 2.5 ().
Let be a multi–digraph with arc weight .

There exist algorithms computing the subgraph in and in .

If is a minimum feedback arc set with then .

If and be a minimum–––cut with , w.r.t. such that:
(4) Then there is with .

Checking whether and (4) holds can be done in .
Proof.
For better readability we recaptured this statement from our previous article (FASP) in Appendix D. ∎
3. The Algorithm
Proposition 2.5 allows to localize optimal arc cuts even in the weighted case, whenever isolated cycles with property exist. However, isolated cycles do not need to exist at all as the Example 2.4 shows. Dealing with that issue is the challenge we take on in this section.
3.1. Building Block
Given a multi–digraph we formulate an algorithm termed ISO–CUT, which searches for arcs which satisfy the assumption of Proposition 2.5. If such an arc is located we store in a list , consider and continue the search until either the resulting graph is acyclic or no desired arc can be localized. In any case, the stored arcs are an optimal subsolution for the FASP on . A formal pseudo–code for the algorithm is given in Algorithm 1.
Lemma 3.1 ().
Let be a multi–digraph with arc weight .

The algorithm ISO–CUT requires runtime to return an optimal subsolution of the FASP on and the remaining graph in the unweighted case.

The analogous return in the weighted case requires runtime.

If is acyclic then is a minimum feedback arc set.
Proof.
As one can verify readily Algorithm 1 contains 2 recursion over
with line 6 being the bottleneck for each recursion. The runtime estimation thereby follow directly from Proposition
2.5 and . Statement follows from Proposition 2.5 . ∎3.2. A Good Guess
If algorithm ISO–CUT does not return an acyclic graph we have to develop a concept of a good guess for cutting in a pseudo–optimal way until it possesses isolated cycles and thereby ISO–CUT can proceed. Our idea is based on the following fact.
Proposition 3.2 ().
Let be a multi–digraph with arc weight , , with and . Denote with a minimum–––cut and with a minimum feedback arc set, while shall denote its restriction to . If
(5) 
then .
Proof.
Assume . Since is acyclic . Hence,
yields a contradiction proving the claim. ∎
Certainly, the right hand side of (5) is hard to compute or even to estimate. Intuitively, one could guess that the larger the left hand side becomes the more likely it is that the inequality in (5) holds. This intuition is the basic idea of our concept of a good guess.
However, maximizing the left hand side of (5) is too costly for a heuristic guess. Therefore, we restrict our considerations to cycles and find the arc maximizing the left hand side of (5) on . Algorithm 2 formalizes this procedure termed GOOD–GUESS.
Lemma 3.3 ().
Let be a multi–digraph with arc weight and be a constant. Then the algorithm GOOD–GUESS requires runtime to propose an arc .
Proof.
Certainly, for all cycles . Thereby, the two loops in Algorithm 2 are called at most times. Computing minimum ––cuts requires by combining (KRT) and (orlin) yielding the runtime estimation. ∎
3.3. The Global Approach
Now we combine the algorithms ISO–CUT and GOOD–GUESS to yield an algorithm termed TIGHTCUT computing feedback arc sets formalized in Algorithm 3.
Proposition 3.4 ().
Let be a multi–digraph with arc weight and vertex weight .

The algorithm TIGHT–CUT proposes a feedback arc set in in the unweighted case.

In the weighted case the analogous return requires runtime.

The algorithm TIGHT–CUT can be adapted to proposes a feedback vertex set in in the unweighted case and in the weighted case, where denotes the maximum degree of .
Proof.
Obviously TIGHT–CUT runs once through all arcs in the worst case. The bottlenecks are thereby ISO–CUT and GOOD–GUESS. Due to Lemmas 3.1,3.3 we obtain , . Now is a consequence of an existing approximation preserving –reduction from the FVSP to the FASP relying on Definition C.3 and Proposition C.4. ∎
Next we bound the ratio for any feedback arc set proposed by TIGHT–CUT.
Theorem 3.5 ().
Let be a multi–digraph with arc weight and and – be a feedback arc set proposed by –. Then
(6) 
If in particular, then is a minimum feedback arc set.
Proof.
Let be a minimum feedback arc set. We consider the subgraph . Let be the list of arcs belonging to and , w.r.t. order of appearance in TIGHT–CUT, respectively. Let for some , and be the partial list of all arcs appearing prior and be the list of all arcs appearing posterior . Then by considering Algorithm 3 one can readily observe that and the assumption of Proposition 2.5 is fulfilled w.r.t. while . Hence, ISO–CUT is a possible return of ISO–CUT. Due to Lemma 3.1 this implies that is an optimal subsolution, i.e., . Therefore, and together with imply (6) and whenever . ∎
4. Validation & Benchmarking
To speed up the heuristic TIGHT–CUT we formulated a relaxed version, which we implemented in C++. The relaxation relies on weakening condition in Proposition 2.5 by a notion of almost isolated cycles. Further explanations and a pseudo–code are given in Appendix A and Algorithm 4. All benchmarks were run on a single CPU core on a machine with CPUs: 2x Intel(R) Xeon(R) E52660 v3 @ 2.60GHz Memory: 128GB with OS: Ubuntu 16.04.6 LTS using compiler: GCC/G++ 9.2.1.
The following implementations were used for the experiments:

An exact integer linear programming based approach implemented as the
feedback_edge_set function from SageMath 8.9 (sagemath) with iterative constraint generation termed EM. 
The greedy removal approach from (Eades1993) termed GR imported from the igraph library (IGR).

The relaxed version of TIGHT–CUT termed TIGHT–CUT* presented in Appendix A with settings , , .
EM is similar to the approach from (exact) and iteratively increases the cycle matrix required for the optimization. Thereby, a sequence , of optimal subsolutions is generated with , being a global solution for . Indeed, the method can not handle the weighted case. We chose the GLPK back end for SageMath’s integer programming solver, which we found to perform significantly better than COINOR’s CBC or Gurobi. In order to compare approximation ratios and runtimes we generated the following instance classes:

We used the Erdős–Rényi model in order to generate random digraphs (model).

We uniform randomly chose a direction for every edge in a complete undirected graph in order to generate tournaments of size .

We generated random maximal planar digraphs, then considered uniform perturbations of planarity by randomly rewiring, i.e. removing and reinserting, a fraction of arcs. This construction is similar to the Watts–Strogatz smallworld model (Watts1998).

We followed (Saab) in order to generate large digraphs of known feedback arc length. Adaptions to treat the weighted case were made.
All implementations and benchmark sets will be made available on GitHub.
Experiment 4.1 ().
In total we generated 1869 random digraphs. Figure 3 shows the approximation ratios obtained by TIGHT–CUT* and GR on 967 out of these 1869 graphs plotted once against and once against the exact minimum feedback arc length. The exact feedback length was determined by EM whose runtime ratio w.r.t. TIGHT–CUT* is plotted in Figure 4. Thereby, the empty region in the left panel reflects the 902 instances which EM could not process within EM–time–out min.
Figure 3 validates that TIGHT–CUT* approximates the FASP beneath a ratio of at most by being much tighter in most of the cases. On the other hand, GR reaches ratios up to . The parameter tractable algorithm of (Chen) indicates that the feedback length reflects the complexity of a given instance. However, the accuracy of GR decreases quickly with increasing graph size and regardless of the feedback length. In contrast, the accuracy behavior of TIGHT–CUT* reflects that circumstance. Whatsoever, TIGHT–CUT* performs significantly better than GR. Especially, when approaching the time–out–region of EM, the approximation ratios of TIGHT–CUT* remain small. Thus, for digraphs located above the red region in Figure 4 the plot in Figure 6 shows that TIGHT–CUT* is up to –times faster than EM. Thus, even though TIGHT–CUT* requires up to min and GR runs beneath sec on these graphs TIGHT–CUT* is the only approach producing reasonable results.
5. Conclusion
We presented a new –heuristic termed TIGHT–CUT of the FASP which is adaptable for the FVSP in processing even weighted versions of the problems. At least by validation the ratio of the implemented relaxation TIGHT–CUT* is shown to be bounded by and is much smaller for most of the considered instances. Though we followed several ideas we can not deliver a proof of the APX–completeness for the directed FVSP & FASP at this time. Nevertheless, we are optimistic that Theorem 3.5 can be generalized in the sense that the estimation in (6) becomes independent of the considered instance. Currently, Theorem 3.5 also does not extend to the relaxed version TIGHT–CUT*. From our perspective, a deeper understanding of isolated cycles is necessary in order to provide such an extension and generalisation. In any case, by Proposition C.4 the directed FVSP & FASP can be reduced to each other. Hence, either both problems are APX–complete or none of them.
Regardless of these theoretical questions, validation and benchmarking with the heuristic GR (Eades1993) and the ILP–method EM from (sagemath) demonstrated the high–quality performance of TIGHT–CUT* even in the weighted case. Though of runtime complexity , especially the most relevant instance class of large sparse graphs can be solved efficiently within tight error bounds for the first time. Runtime improvements of TIGHT–CUT* are certainly possible, for instance by using the improved minimum–––cut algorithms from (Goldberg2014) and parallelizing the algorithm. Executing faster than EM for feedback lengths thereby might reachable, see Figure 6. A fast implementation of the –reduction from the FVSP to the FASP is in progress allowing to solve the FVSP by TIGHT–CUT* with the same accuracy in similar time.
We hope that many of the applications, even those which kept unmentioned here, will benefit from our approach.
Acknowledgements.
We thank Ivo F. Sbalzarini and Florian Jug for proofreading and inspiring discussions. Further, we are grateful for suggestions and hints from Christian L. Müller.Appendix A Relaxed Version of TIGHT–CUT
In order to make the algorithm ISO–CUT faster and more effective we propose the following relaxation within TIGHT–CUT.
Definition A.1 (almost isolated cycles).
Let be a multi–digraph and . If there is a set of arcs, i.e., such that
then we call the cycles almost isolated cycles. If then we obtain the notion of Definition 2.3.
As long as there are isolated cycles for small one can hope that the accuracy of TIGHT–CUT remains high. We take this relaxed notion into account as follows. If no isolated cycles were found then we generate graphs by randomly deleting arcs , , , and ask for the existence of almost isolated cycles,i.e., search for arcs in with . The arc appearing most in all the explored graphs is assumed to be a good choice for cutting it in the original graph. If no such arc can be found then we use GOOD–GUESS for making a choice in any case. The relaxation is formalized in Algorithm 4.
Remark A.1 ().
From our perspective the relaxation seems to follow a good intuition. However, Theorem 3.5 does not extend to this relaxed version of TIGHT–CUT.
Appendix B Additional Experiments
Experiment B.1 ().
We focus our considerations to tournaments. In Figure 5 the results for with 10 instances for each size are shown. is thereby the maximum size for EM not running into time–out min. GR seems to perform only slightly worse than TIGHT–CUT*. However, the feedback arc length for tournaments averages about of its arc count. Thus, the improvement in accuracy TIGHT–CUT* gains compared to GR is as significant. Again the NPhardness of the FASP becomes visible for the runtime ratios in Figure 5 and Figure 6 (left). As expected the feedback arc length is correlated to the complexity of the cycle structure of . Regardless of the type of the graphs we thereby reach intractable instances for EM beyond a feedback arc length of . Thus, for instances allowing EM might run into time out while TIGHT–CUT* processes efficiently.
Experiment B.2 ().
Since EM can not handle the weighted FASP we adapted the method of (Saab) to generate 77700 weighted multi–digraphs of integer weights with known feedback arc length and sizes from and . Figure 6 (right) illustrates the results. To merge the ratio distributions of GR and TIGHT–CUT* on one plot we chose a logarithmic scaling for the –axis. Indeed, TIGHTCUT* approximates the FASP beneath a ratio of and solves more than exactly and beneath a ratio of . In contrast GR is spread over ratios from to producing exact solutions only for and beneath a ratio of . Thus, though GR runs beneath sec and the runtimes of TIGHT–CUT* vary from seconds to minutes, this accuracy improvement justifies the larger amount of time.
Experiment B.3 ().
In Figure 7 the EM runtimes and TIGHT–CUT* approximation ratios for 541 small–world (perturbed planar digraphs) are plotted. As one can observe already for small perturbations a similar behavior as for random graphs occurs. In applications one can rarely guarantee planarity. At best, one can hope for planar–like instances. Consequently, realworld instances, hinder the efficiency of ILP–Solvers on planar graphs to come into effect. Therefore, TIGHT–CUT* is an alternative to EM worth considering even for planar–like graphs.
Experiment B.4 ().
We measured the accuracy behavior of GR and TIGHT–CUT* in the time–out region of EM. Therefore, we generated very large unweighted digraphs with known feedback arc length and varying vertex size , instances for each size, with density, i.e., . In Figure 8
the computed feedback lengths of both approaches are plotted with error bars indicating the standard deviation. While TIGHTCUT* delivers almost exact solutions GR is infeasible for these large graphs.
Experiment B.5 ().
We generated a few very large and dense unweighted graphs with large feedback length . The results are listed in Table 1 and validate that TIGHT–CUT* approximates the FASP beneath a ratio of .
Appendix C The Dualism of the FVSP & FASP
Though the dualism of the FVSP and the FASP is a known fact, its treatment is spread over the following publications (ausiello; crescenzi1995compendium; Even:1998; FASP; kann1992). Here, we summarize and simplify the known results into one compact presentation allowing also to consider weighted versions. We recommend (kann1992) for a modern introduction into approximation theory. The following additional notions and definitions are required.
For a given vertex , , , shall denote the set of all incoming or outgoing arcs of , and their unions. The indegree (respectively outdegree) of a vertex is given by , and the degree of a vertex is . The maximum degree of a graph is denoted by .
Definition C.1 (directed line graph).
The directed line graph of a multi–digraph is a digraph where each vertex represents one of the arcs of , i.e., . Two vertices are connected by an arc if and only if the corresponding arcs are consecutive, i.e., , with
If there is an arc weight on then we consider the induced vertex weight given by , for all .
vertices  arcs  TIGHT–CUT*  approx. ratio  

100  990  200  321  1.61 
100  990  200  279  1.40 
200  3980  200  280  1.40 
500  1500  200  334  1.67 
501  1501  200  345  1.73 
Remark C.1 ().
Note that the line graph has no multiple arcs and can be constructed in .
The dual concept is to derive the natural hyper–graph of a multi–digraph such that becomes the line graph of , i.e., . More precisely:
Definition C.2 (natural hyper–graph).
Let be a multi–digraph. We set and introduce hyper–arcs with , . The natural hyper–graph is then given by . See Figure 9 for an example. If there is a vertex weight on then we consider the hyper–arc weight given by for all .
Remark C.2 ().
Note that for any multidigraph the natural hyper–graph contains no multiple hyper–arcs and can be constructed in . Further, , is allowed. While a loop with in is represented in by the property .
Definition C.3 (dual digraph).
Let be a multi–digraph and its natural hyper–graph. For every hyper–arc we consider the bipartite graph given by
Further, we set and
Finally, we consider the sets
define the maps as the continuation of onto and denote the dual multi–digraph of by . If there is a vertex weight on then we consider the arc weight given by
Figure 9 illustrates an example. The thin arcs of are weighted with and the non–filled vertices correspond to the artificially introduced vertices .
Remark C.3 ().
Again we observe that the dual digraph possesses no multiple arcs and can be constructed from in .
Combining the definitions above we obtain maps
(7) 
Indeed and allow to show that weighted FVSP and the the weighted FASP are approximation preservable reducible to each other.
Proposition C.4 ().
Let be a weighted multi–digraph and be a minimum feedback arc set and be a minimum feedback vertex set of .

is a minimum feedback arc set of with .

is a minimum feedback vertex set of with .
Proof.
We show that the cycles , and are in correspondence. Indeed, the vertex set of any cycle induces exactly one cycle . Vice versa since is a digraph without multiple arcs we observe that the arc set of any cycle induces exactly one cycle with . Analogously, we note that a cycle is uniquely determined by knowing all its arcs see Definition C.3. Since possesses no multiple arcs this implies again that the vertex set of any cycle
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