Current technologies [1] like two-hybrid screening can find protein interactions, leading to protein-protein interaction (PPI) networks, but cannot decide the direction of the interaction. This can be complemented by gene knock-out experiments which constitute a way to determine causal relations in these networks, thus providing additional information on possible directions of information flow in them [2]. Given a list of so-called cause–effect pairs, the challenge consists in deducing an orientation of the PPI network which takes into account the causal relations of as many of these pairs as possible. Medvedovsky et al. [3] formalize this in terms of a graph theoretical problem as follows.

MTO was introduced by Medvedovsky et al. [3]; they showed that the problem is NP-complete even when the underlying tree is a star (that is, a diameter-two tree) or a tree with maximum vertex degree three. Moreover, they provided a cubic-time algorithm for MTO restricted to paths. Seeing MTO as the task to maximize the number of satisfied pairs, Medvedovsky et al. also provided polynomial-time approximation algorithms with approximation factor 1/ 4 in the case of stars and O(1/ log n) in the case of general n-vertex trees. The latter approximation factor was recently improved to O(log log n/ log n) by Gamzu et al. [8], who furthermore extended the studies of MTO to "mixed graphs" where some of the edges are already oriented based on causal relations known in advance. Besides these theoretical investigations, Medvedovsky et al. [3] also provided some experimental results based on a yeast PPI network and some synthetic data. Silverbush et al. [9] recently formulated a polynomial-size integer linear program for the generalization of mixed graphs and did some experiments with it. Also recently, Gitter et al. [10] considered graph orientation with the objective of maximizing the weight of all satisfied paths between sources and targets with length at most some constant k. They used approximation algorithms to discover pathways in biological networks. In an earlier work, Hakimi et al. [11] studied the special case of MTO where the list of pairs to be satisfied contains all possible pairs; they developed a quadratic-time algorithm for this case.

We mainly continue and complement previous work on MTO [3, 8] by starting a parameterized and multivariate complexity analysis of MTO. That is, we try to better understand the border between tractable and intractable cases of MTO while sticking to optimal (instead of approximate) solutions. In particular, our focus is on the "amount of signal flow" over vertices and edges, respectively, and how this influences the computational complexity of MTO.

For ease of presentation, for a W-MTO instance (T, P, ω), we always assume that ω([s, t]) = 0 for all pairs s, t ∈ V with [s, t] ∉ P. Moreover, subsequently mostly referring to MTO, the presented concepts and definitions clearly apply to W-MTO as well. Note that in a tree T = (V, E), for each ordered pair [a, b] of vertices, there exists a uniquely determined path connecting these vertices. We will therefore often write the path defined by the pair [a, b] when we refer to the unique path in the tree starting in vertex a and ending in vertex b, or talk about pairs and paths interchangeably. Sometimes, we also talk about paths in the tree which do not necessarily correspond to pairs. We denote the undirected path connecting vertices v and w in T by path _T (v, w). Moreover, P _v : = {[s, t] ∈ P | v ∈ V (path _T (s, t))} denotes the set of paths passing through a vertex v (note that this includes paths of which v is an endpoint). An MTO instance is called rooted if the underlying tree T is rooted. In a rooted tree T = (V, E), if vertex a ∈ V is an ancestor of vertex b ∈ V, then we use the notation a ≺ b. The subtree of T rooted at v ∈ V is denoted T _v .

Let (T = (V, E), P) be an MTO instance, and let x, y ∈ P be two pairs. We say that x conflicts with y if there exists no orientation of T for which both x and y are satisfied. From an n-vertex MTO instance, we build a so called conflict graph in which each vertex corresponds to an input pair of the MTO instance, and where there is an edge between two pairs if and only if they conflict with each other. More formally, given an MTO instance (T = (V, E), P), the corresponding conflict graph G _c (T, P) is defined G _c (T, P):= (P, E _c ) where E _c : = {{u, v} | u, v ∈ P ∧ u conflicts with v}.

The computation of the conflict graph can be done in Θ(n⁴) time. It clearly cannot be done faster, because up to O(n⁴) conflicts are possible. To achieve the desired bound, we thus need to decide in constant time whether two pairs conflict with each other. This is done using an appropriate data structure and two simple observations: First, in a rooted tree, least common ancestors (LCAs) can be calculated in constant time after some linear time preprocessing [12]. Second, two pairs are in conflict if and only if their paths run in different directions through an edge incident on the lower one of the two LCAs of the two pairs. Clearly, for an orientation of (T, P), in G _c there are no edges (that is, conflicts) between the vertices corresponding to the satisfied source–target pairs, and hence the vertices corresponding to the non-satisfied source–target pairs form a vertex cover for G _c , that is, a vertex set V' ⊆ P such that for every edge e ∈ E _c at least one endpoint of e is in V'. This yields the following useful observation.

Proposition 1. Finding a minimum-weight vertex cover in the conflict graph G _c (T, P) one-to-one corresponds to determining a minimum-weight set of pairs that cannot be satisfied in (T, P).

It is generally assumed that the fact that a problem is NP-hard implies that there is no algorithm that finds an optimal solution and has running time bounded by a polynomial of the size of the input. Parameterized complexity is a two-dimensional framework for the analysis of computational complexity [13–15]. One dimension is the input size n, and the other one is the parameter (usually a positive integer). A problem is called fixed-parameter tractable (fpt) with respect to a parameter x if it can be solved in f(x) ⋅ n ^{O (1)} time, where f is a computable function only depending on x. If a problem is fixed-parameter tractable with respect to x, we can hope for efficient optimal solutions as long as the parameter is not too large. Due to Proposition 1 we can immediately conclude that MTO and W-MTO are fixed-parameter tractable with respect to the parameter "number of pairs" p, since the conflict graph has p vertices and we can find a minimum-weight vertex cover by trying all possibilities in 2 ^p ⋅ n ^{O (1)} time. Further, since minimum-weight vertex covers can be found in O(1. 379 ^k + kn) time [16], we have fixed-parameter tractability with respect to the parameter "number of unsatisfied pairs", and if all weights are at least one, also with respect to the parameter "total weight of unsatisfied pairs".

Tree-decomposition-based algorithms have been successfully applied in the area of computational biology, for instance, in the context of structure–sequence alignment [17]. Informally speaking, the treewidth[15] measures the "tree-likeness" of a graph, and a tree decomposition is the "embedding" of a graph into a tree depicting the tree-like structure of the graph.

The width of $〈\left\{X_i|i\in I\right\},\mathcal{T}〉$, is max{|X _i | | i ∈ I} - 1. The treewidth of G is the minimum width over all tree decompositions of G.

WEIGHTED MAXIMUM TREE ORIENTATION on n-vertex paths can be solved in O(n²) time.

Next, we show that W-MTO is fixed-parameter tractable with respect to the parameter q _v by extending the dynamic programming algorithm for cross-pair-free instances. Formally, q _v is defined as follows. For a rooted W-MTO instance (T = (V, E), P) with root r, let Q denote the set of cross pairs. Moreover, for v ∈ V let Q _v := P _v ∩ Q be the set of cross pairs passing through v. With respect to the root r the maximum number q _v (r) of cross pairs passing through a vertex is given by max _v∈V |Q _v |. Then, q _v is the minimum value of q _v (r) over all possible choices r to root T.

Theorem 3. On n-vertex trees, WEIGHTED MAXIMUM TREE ORIENTATION with given root can be solved in O( 2 ^qv ⋅ q _v ⋅ n²) time, where q _v denotes the maximum number of cross pairs passing through a vertex.

Proof. The basic idea of the algorithm is to incorporate the cross pairs by trying for every vertex all possibilities to satisfy the cross pairs passing through this vertex. To this end, we extend the matrix S by an additional dimension. As a consequence, the dynamic programming update step becomes significantly more intricate.

Let (T = (V, E), P, ω) be a rooted W-MTO instance with root r. For the presentation of the algorithm we use the same notation as in the proof of Theorem 2. In addition, we employ the following definitions.

Let w ∈ V. A possibility to satisfy the cross pairs in Q _w is represented by a coloring c _w : Q _w → {0, 1}, meaning that a cross pair q ∈ Q _w must be satisfied iff c _w (q) = 1. Let C _w denote the set of all 0/1-colorings of Q _w . Note that $\left|C_w\right|=2^{\left|Q_w\right|}$. To incorporate the cross pairs, for every v, w ∈ V with v ≺ w or v = w and for every coloring c _w , the dynamic programming table S contains two entries S(v, w, c _w ) and S(w, v, c _w ). Informally speaking, S(v, w, c _w ) denotes the maximum weight of an orientation of $T_w^v$ under the assumption that all cross pairs q ∈ Q _v with c _w (q) = 1 are "satisfiable" and the edges in path _T (v, w) are oriented from v towards w. Entry S(w, v, c _w ) is defined analogously, but here we assume that the edges in path _T (v, w) are oriented from w towards v. For a precise description, we use the following notation.

For the case (1), a locally feasible coloring c _w is called feasible if for each cross pair [s, t] ∈ Q _w with c _w ([s, t]) = 1 orienting the edges on path _T (v, w) from v to w and the edges on path _T (s, t) from s to t is simultaneously possible. Analogously, a locally feasible coloring c _w is feasible for case (2) when orienting the edges on path _T (s, t) from s to t does not contradict the orientation of the edges on path _T (v, w) from w to v.

For a coloring c _w of Q _w , we must ensure that in the considered orientations of $\left(T_w^v,P_w^v\right)$ all cross pairs q ∈ Q _w with c _w (q) = 1 are satisfiable. Therefore, we call an orientation of $\left(T_w^v,P_w^v\right)$consistent with a cross pair [s, t] ∈ Q _w (note that $\left[s,t\right]\in P\backslash P_w^v$ is allowed) if the common edges of $T_w^v$ and path _T (s, t) are oriented from s to t. Finally, we call an orientation of $\left(T_w^v,P_w^v\right)$consistent with a coloring c _w of Q _w if the orientation is consistent with each cross pair q ∈ Q _w with c _w (q) = 1.

With these notations, we can formally define the meaning of the entries of S. For every two vertices v, w ∈ V with v ≺ w or v = w and for every 0/1-coloring c _w ∈ Q _w the entry S(v, w, c _w ) is -∞ if the coloring c _w is not feasible for case (1). Otherwise, S(v, w, c _w ) denotes the maximum weight of an orientation of $\left(T_w^v,P_w^v\right)$ among all orientations of $\left(T_w^v,P_w^v\right)$ fulfilling the following constraints:

This definition ensures that the orientation is not conflicting with the realization implied by the coloring c _w and the fixed orientation of path _T (v, w). The entry S(w, v, c _w ) is defined analogously with the difference that here we assume that the edges on path _T (v, w) are oriented from w to v. Note that the cross pairs having only one endpoint in $T_w^v$ are not contained in $P_w^v$, and hence do not contribute to the weight of an orientation of $\left(T_w^v,P_w^v\right)$. Observe that ${{\displaystyle max}}_{c_r\in C_r}S(r\mathrm{,}r\mathrm{,}{c}_r)$ is the maximum weight of an orientation of the whole instance (T, P, ω) since $T_r^r=T$ and we build the maximum over all colorings of the cross pairs in Q _r . Next, we provide a strategy to compute the entries of S in accordance with this definition. In the update step, we need to adjust the tables of a vertex w with the tables of its children. Doing so, we have to ensure that we only consider colorings that are not contradictory to each other. Let c _u denote a coloring of Q _u and c _w denote a coloring of Q _w . We use c _u |c _w to denote that c _u and c _w agree in the coloring of the cross pairs in Q _u ∩ Q _w , that is, for all q ∈ Q _u ∩ Q _w it holds that c _u (q) = c _w (q). Finally, let W_LCA(w, c _w ) denote the sum of the weights of the cross pairs [s, t] ∈ Q _w with c _w ([s, t]) = 1 that have w as their least common ancestor.

Herein, A is defined exactly as in the proof of Theorem 2.

Correctness. For the correctness, we argue that for v, w ∈ V with v ≺ w and for a coloring c _w ∈ C _w that is feasible for case (1), the maximum weight of an orientation of $\left(T_w^v,P_w^v\right)$ consistent with c _w that orients the edges on path _T (v, w) from v to w is $A\left(v,w\right)+W_{\mathsf{\text{LCA}}}\left(w,c_w\right)+{\sum }_{i=1}^{\ell }M\left(u_i,v,w,c_w\right)$, and, hence, S(v, w, c _w ) is computed correctly. To this end, first consider the case that w is a leaf. Then, $T_w^v$ is identical to path _T (v, w) and Q _w = ∅. Hence, in this case exactly the source–target pairs $\left[s,t\right]\in P\backslash P_w^v$ with s ≺ t are satisfied whose total weight per definition is A(v, w). Second, consider the case that w is an internal vertex with children u₁, …, u_ℓ. Moreover, let ${\overrightarrow{T}}_w^v$ be an optimal orientation consistent with c _w that orients the edges on path _T (v, w) from v to w. Assume that this orientation contains for a child u _i the arc (u _i , w). Then, with respect to w and u _i , the subgraph of ${\overrightarrow{T}}_w^v$ induced by the vertices in $V_{u_i}^w$ is an orientation of $T_{u_i}^w$ consistent with a coloring $c_{u_i}$ that clearly agrees with c _w . Thus, the contribution of u _i to the weight of ${\overrightarrow{T}}_w^v$ is the maximum $S\left(u_i,w,c_{u_i}\right)$ over all $c_{u_i}\in Q_{u_i}$ that agree with c _w . Similarly, if ${\overrightarrow{T}}_w^v$ contains the arc (w, u _i ) for a child u _i , the subgraph of ${\overrightarrow{T}}_w^v$ induced by the vertices in $V_{u_i}^w$ is an orientation of $T_{u_i}^w$ consistent with a coloring $c_{u_i}$ that clearly agrees with c _w . Thus, the contribution of the pairs in $P_w^v$ with at least one endpoint in $T_{u_i}$ is the maximum of $S\left(v,u_i,c_{u_i}\right)-A\left(v,w\right)$ over all $c_{u_i}\in {\mathcal{C}}_{u_i}$ that agree with c _w (here, we have to subtract the number of satisfied pairs with both endpoints on path _T (v, w) that are already accounted for by the term A(v, w) in (8)). Finally, observe that the contribution of the cross pairs q in $P_w^v$ with c _w (q) = 1 for which w is the least common ancestor are not taken into account in the contributions of the u _i 's. This is done by the term W_LCA in (8). The argumentation for the correctness of the computation of S(w, v, c _w ) follows analogously.

Running time. Next, we analyze the running time. We use the following notation and implementation details. For w ∈ V let $Q_w=\left\{p_1^w,\dots ,p_{n_w}^w\right\}$. A coloring c _w : Q _w → {0, 1} is realized by a tuple $\left(c_1,\dots ,c_{n_w}\right)\in {\left\{0,1\right\}}^{n_w}$ with $c_w\left(p_i^w\right)=c_i$ for all 1 ≤ i ≤ n _w . Moreover, the dynamic programming table S is realized by two tables $S_{v,w}^{up}$ and $S_{v,w}^{down}$ for every pair v, w ∈ V with v ≺ w or v = w with an entry for every coloring $c\in {\left\{0,1\right\}}^{n_w}$ where $S_{v,w}^{up}\left(c\right)=S\left(w,v,c\right)$ and $S_{v,w}^{down}\left(c\right)=S\left(v,w,c\right)$. The table A is computed exactly as in the proof of Theorem 2 in O(n²) time in a preprocessing step. Moreover, note that after O(n) preprocessing time, least common ancestors of the source–target pairs can be found in constant time [12].

since O(∑ _w∈V deg _T (w)) = O(n) in trees.   □

Let m _e be the maximum number of paths that pass through an edge. We consider MTO instances where m _e is limited. We show that the problem is linear-time solvable for m _e ≤ 2, but NP-hard for m _e ≥ 3, thereby establishing a dichotomy on the complexity of MTO with respect to m _e .

For the polynomial-time algorithm, we employ the following lemma.

Lemma 1. If m _e ≤ 2, then the treewidth of G _c (T, P) is at most two.

We show that if m _e ≤ 2, then in the conflict graph G _c (T, P) we can find a vertex to which one of the above rules applies. Further, we show that the modified smaller conflict graph is still a conflict graph of some MTO instance. Thus, the claim follows by induction.

Clearly, if Rule (1) or (2) applies to a vertex v in the conflict graph, then we can just delete the corresponding pair in the MTO instance, and the conflict graph of the resulting MTO instance is identical to the graph that results by deleting v.

Next, we show that if neither Rule (1) nor Rule (2) applies, then we can find a vertex v in the conflict graph to which Rule (3) applies. To this end, let (T, P) with T = (V, E) denote an MTO instance and assume that T is rooted at an arbitrarily chosen inner vertex r. Moreover, among all vertices that are the least common ancestors of a pair in P, let x be one with maximum distance to the root r (that is, a deepest least common ancestor). We distinguish two cases based on whether x is an endpoint of a pair with both endpoints in T _x .

First, consider the case that x is the endpoint of a pair [s, t] ∈ P with s, t ∈ V (T _x ). Let y be the child of x that is contained in the path between s and t. Observe that by the choice of x there is no pair with both endpoints in T _y . Hence, for every pair that is in conflict with [s, t], the corresponding path contains the edge {x, y}. Thus, since m _e = 2, the pair [s, t] is in conflict with at most one other pair, and therefore the corresponding vertex has degree at most one in the conflict graph: a contradiction to the fact that neither Rule (1) nor (2) apply.

Second, consider the case that x is not an endpoint of any pair with both endpoints in V(T _x ). Moreover, let p = [s, t] ∈ P be an arbitrarily chosen pair with s, t ∈ V (T _x ). Let y₁ and y₂ denote the two children of x such that (without loss of generality) s ∈ V (T _y1 ) and t ∈ V (T _y2 ). Let v_[s,t]denote the vertex of G _c corresponding to [s, t]. First, note that by the assumption that Rule (1) does not apply, v_[s,t]has degree at least two. Moreover, by the choice of x there is no pair with both endpoints in V (T _y1 ) or in V (T _y2 ). Thus, every pair that is in conflict with [s, t] uses either the edge {x, y₁} or the edge {x, y₂}. Thus, since m _e ≤ 2 and ${deg}_{G_c}\left(v_{\left[s,t\right]}\right)\ge 2$, there are exactly two pairs p' = [s', t'], p'' = [s'', t''] ∈ P that are in conflict with [s, t]. Assume without loss of generality that t' ∈ V (T _y1 ) and s'' ∈ V (T _y2 ) (see Figure 1 for an illustration). Since Rule (2) does not apply to v_[s,t], we can assume that p' and p'' are not in conflict with each other. Hence, Rule (3) can be applied to v_[s,t]. Let $G_c^{\prime }$ denote the graph that results by first making the two neighbors of v adjacent and subsequently deleting v. It remains to show how to transform the MTO instance such that the conflict graph of the new instance is identical to $G_c^{\prime }$. To this end, consider the MTO instance that results by deleting the vertices in V (T _y1 ) ∪ V (T _y2 ), removing the pairs [s, t], [s', t'], and [s'', t''] and subsequently adding a vertex y', making y' adjacent to x, and adding the pairs [s', y'] and [y', t''] (see Figure 1 for an illustration). Clearly, [s', y'] and [y', t''] are in conflict. Moreover, since only the pairs p, p', and p'' have endpoints in V (T _y1 ) ∪ V (T _y2 ), this transformation does not change the conflicts with the other pairs. Further, we have that m _e ≤ 2 in the resulting MTO instance.

Since width-two tree decompositions can be constructed in linear time [21] and weighted VERTEX COVER can be solved in linear time on graphs with constant treewidth [15], this yields linear-time solvability for WEIGHTED MAXIMUM TREE ORIENTATION with m _e ≤ 2.

Theorem 4. If m _e ≤ 2, then WEIGHTED MAXIMUM TREE ORIENTATION can be solved in linear time.

Proof. To be able to determine the path between a pair [s, t] in O(n) time, we root the tree arbitrarily and calculate in linear time a data structure that allows least common ancestor queries in constant time [12]; the path can then be found by going upwards from s and t until hitting their least common ancestor, and then joining the two partial paths. We then construct the conflict graph by marking for each path the corresponding edges with the pair and the direction, and then registering a possible conflict for each tree edge. Since there can be only linearly many markings and conflicts, the construction takes O(n) time. A tree decomposition of width two can then be found in linear time [21], and, as mentioned above, solving weighted VERTEX COVER on a graph with treewidth at most two takes only linear time, too [15].   □

We can further prove that for m _e ≥ 3, MTO is NP-hard even on stars, that is, on trees where all leaves are attached to the same vertex. The proof is by reduction from MAX DI CUT.

Theorem 5. MAXIMUM TREE ORIENTATION on stars with m _e ≥ 3 is NP-complete.

Proof. As Medvedovsky et al. [3] pointed out, the NP-hard MAX DI CUT problem, defined as follows, can be reduced to MTO on stars.

MAX DI CUT

Given a directed graph G = (V, A) and a nonnegative integer k, is it possible to find a subset of vertices C ⊆ V such that there are at least |A| - k arcs (v, w) ∈ A with v ∈ C and w ∉ C?

From a MAX DI CUT instance (G = (V, A), k), one constructs an equivalent MTO instance (T = (V', E), P, k) by setting V':= V ∪ {r}, E := {{v, r} | v ∈ V}, and P := A, where r ∉ V is a new root vertex [3]. Clearly, if a MAX DI CUT instance has maximum degree three, then it reduces to an MTO instance with m _e ≤ 3. Thus, it remains to show that MAX DI CUT with maximum degree three is NP-hard. (Unfortunately, there seems to be no apt reduction from the undirected version MAX CUT, which is NP-hard for maximum degree three [22].)

MAX DI CUT can also be formulated as the problem to delete up to k arcs to obtain a graph where every vertex is only startpoint or only endpoint of arcs. We can characterize such graphs by a forbidden substructure consisting of three vertices u, v, w connected by the arcs (u, v) and (v, w) (the arcs (u, w) and (w, u) may or may not be present). Thus, if we ignore graphs with multiple arcs between two vertices, we have three forbidden induced subgraphs on three vertices. In this way, MAX DI CUT is similar to the TRANSITIVITY DELETION problem [23], which given a directed graph, asks for up to k arc deletions to make it transitive, that is, to fulfill for all u, v, w ∈ V that (u, v) ∈ A ∧ (v, w) ∈ A ⇒ (u, w) ∈ A. Transitive graphs are characterized by two of the three forbidden subgraphs for MAX DI CUT; the subgraph with {(u, v), (v, w), (u, w)} ⊆ A is not forbidden. However, if we examine the directed graphs that are produced in the reduction from 3-SAT that proves NP-hardness of TRANSITIVITY DELETION [23, Sect. 3.1], we notice that this substructure does not occur, and cannot be created by arc deletions. Thus, solving TRANSITIVITY DELETION and MAX DI CUT on these directed graphs is equivalent. Since the constructed instances also have degree at most three, we obtain the NP-hardness of MAX DI CUT with maximum degree three. It is easy to see that MTO is contained in NP, so we obtain the claimed theorem.   □

The goal in this section is to explore the space of practically meaningful parameterizations, here focusing on biological applications. We first performed experiments based on the same data as used by Medvedovsky et al. [3]. The network is a yeast protein-protein interaction network from the Database of Interacting Proteins (DIP) [24], containing 4,737 vertices and 15,147 edges. The cause–effect pairs were obtained from gene knockout experiments by Yeang et al. [2] and contain 14,502 pairs. After discarding small connected components and contracting cycles, we obtained a tree with 1,278 vertices and 5,569 pairs. (These numbers differ slightly from the ones stated by Medvedovsky et al. [3]. We do not use the additional kinase-substrate data, which is only meaningful to evaluate the orientations obtained, and which requires an arbitrary parameter choice not documented by Medvedovsky et al. [3].)

The resulting tree is, as already observed by Medvedovsky et al. [3], very star-like: there is one vertex of degree 1,151 and 1,048 degree-one vertices attached to it. The remaining 229 vertices have degree 1 to 4. All paths connecting cause-effect pairs pass through the central vertex.

We first note that this MTO instance is actually fairly easy to solve exactly. The Integer Linear Program (ILP) by Medvedovsky et al. [3, Sect. 3.1] and VERTEX COVER on the conflict graph solved by either an ILP or a simple branching strategy with data reduction all solve the instance in less than a second. More precisely, the running times are 0.09 s, 0.02 s, and 0.13 s, respectively, on a 2.67 GHz Intel Xeon W3520 machine, using GLPK 4.44 for the ILPs, and with the branching strategy implemented in Objective Caml. The branching strategy finds a vertex v of maximum degree and branches into the two cases of taking v into the vertex cover or taking all neighbors of v into the vertex cover. Before each branch, degree-1 vertices are eliminated by taking their neighbor into the vertex cover. The search in the second branch is cut short when the accumulated vertex cover is larger than that of the first branch.

Note that all three algorithms do not require the parameter k (number of unsatisfied pairs) as input, but will determine the minimum k such that there is a solution.

The reason that these strategies work so well is probably due to the low value of the parameter k: only 77 cause-effect pairs cannot be satisfied. This limits the size of the branch-and-bound tree that underlies all three methods.

We see that, here, the parameter k is not always a clear winner. When the network becomes sparser, the components that will be shrunk to a single vertex by the cycle contraction will be smaller, leaving fewer pairs with both endpoints on the same tree vertex, and thereby increasing the number of potential conflicts. Only for very high thresholds, the parameter becomes small again, since then the original instance is already much smaller. Still, all instances can be solved in less than one second by the three algorithms mentioned above, which exploit low values of k.

We also see that for denser graphs, the parameter values based on the number of cross pairs are quite low, e.g. $q_v^{\prime }$ = 3 for the whole graph. Thus, it seems likely that these instances can be quickly solved by the algorithm from Theorem 3, running in $O\left(2^{{q^{\prime }}_v}\cdot n^2\cdot q_v^{\prime }\right)$ time. One possible explanation for the low value for these parameters is that the networks exhibit a linear structure. For example, if each protein can be assigned a distance to the nucleus, and interactions mostly transport information to or from the nucleus, then we would expect to have only few cross pairs.

The parameter m _v could be expected to be not too high in biological networks, since otherwise this would make the network less robust, since elimination of one vertex would disrupt too many paths. However, one vertex in the tree under consideration can actually correspond to a very large component in the original graph, which weakens this effect. Therefore, this parameter is more useful in sparser graphs, where not too many graph vertices are joined into a tree vertex. However, for the given instances, it seems small enough to be exploited only for fairly small instances, where other parameters would give good results, too.

The parameter m _e could similarly be expected to be low in sparse networks; however, the NP-hardness result already for m _e ≥ 3 (Theorem 5) makes practical use of this parameter unlikely.

We started a parameterized complexity analysis of (WEIGHTED) MAXIMUM TREE ORIENTATION, obtaining a more fine-grained view on the computational complexity of this NP-hard problem. In this line, there are still several challenges for future investigations. For instance, it is open whether MTO is fixed-parameter tractable with respect to the parameter "number of satisfied pairs" (n - k). Further, in the spirit of "distance-from-triviality parameterization" [19, 20] it would be interesting to study the parameterized complexity of MTO with respect to the parameter "number of all possible pairs minus the number of input pairs"--recall that for parameter value zero MTO is polynomial-time solvable [11]. MTO restricted to stars is still NP-hard, but then at least one quarter of all input pairs can always be satisfied [3]. Hence, it would be interesting to study above guarantee parameterization [15, 20] with respect to the number of satisfied pairs. MTO can be translated into a vertex covering problem (see Proposition 1) on a graph class that is K₄-free--this motivates to study whether vertex covering on this graph class can be done faster than on general graphs. Clearly, MTO brings along numerous further parameters and parameter combinations which can make a more comprehensive multivariate complexity analysis [20] very attractive. Often, it is desirable to not only list a single solution, but to enumerate all optimal solutions. Our dynamic-programming-based algorithms seem suitable for this. Following Gamzu et al. [8] and extending the studies for MTO as pursued here to the more general case of mixed graphs with partially already oriented edges is of high interest. First steps in this direction have very recently been undertaken by Silverbush et al. [9] and Elberfeld et al. [26]. Finally, it seems promising to examine the parameters based on cross pairs in other networks such as communication networks, and to try to exploit these parameters for other hard network problems.