Beyond Max-Cut: \lambda-Extendible Properties Parameterized Above the Poljak-Turzík Bound

Beyond Max-Cut: -Extendible Properties Parameterized Above the Poljak-Turzík Bound

Matthias Mnich Cluster of Excellence, Saarbrücken, Germany. m.mnich@mmci.uni-saarland.de Geevarghese Philip111Supported by the Indo-German Max Planck Center for Computer Science(IMPECS). MPII, Saarbrücken, Germany. gphilip@mpi-inf.mpg.de Saket Saurabh222Part of this work was done while visiting MPII supported by IMPECS. The Institute of Mathematical Sciences, Chennai, India. saket@imsc.res.in Ondřej Suchý333Part of this work was done while with the Saarland University, supported by the DFG Cluster of Excellence MMCI and the DFG project DARE (GU 1023/1-2), and while visiting IMSC Chennai supported by IMPECS. Technische Universität Berlin, Berlin, Germany. ondrej.suchy@tu-berlin.de
Abstract

Poljak and Turzík (Discrete Math. 1986) introduced the notion of -extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any and -extendible property , any connected graph on vertices and edges contains a spanning subgraph with at least edges. The property of being bipartite is -extendible for , and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdős bound for Max-Cut.

We define a variant, namely strong -extendibility, to which the Poljak-Turzík bound applies. For a strongly -extendible graph property , we define the parameterized Above Poljak-Turzík () problem as follows: Given a connected graph on vertices and edges and an integer parameter , does there exist a spanning subgraph of such that and has at least edges? The parameter is , the surplus over the number of edges guaranteed by the Poljak-Turzík bound.

We consider properties for which the Above Poljak-Turzík () problem is fixed-parameter tractable (FPT) on graphs which are vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík () is FPT for all . Our results hold for properties of oriented graphs and graphs with edge labels.

Our results generalize the recent result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards-Erdős bound, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max -Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem is FPT, thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; acyclic graphs.

1 Introduction

A number of interesting graph problems can be phrased as follows: Given a graph as input, find a subgraph of with the largest number of edges such that satisfies a specified property . Prominent among these is the Max-Cut problem, which asks for a bipartite subgraph with the maximum number of edges. A cut of a graph is a partition of the vertex set of into two parts, and the size of the cut is the number of edges which cross the cut; that is, those which have their end points in distinct parts of the partition.

Max-Cut Input: A graph and an integer . Question: Does have a cut of size at least ?

The Max-Cut problem is among Karp’s original list of 21 NP-complete problems [14], and it has been extensively investigated from the point of view of various algorithmic paradigms. Thus, for example, Goemans and Williamson showed [12] that the problem can be approximated in polynomial-time within a multiplicative factor of roughly , and Khot et al. showed that this is the best possible assuming the Unique Games Conjecture [15].

Our focus in this work is on the parameterized complexity of a generalization of the Max-Cut problem. The central idea in the parameterized complexity analysis [7, 11] of NP-hard problems is to associate a parameter with each input instance of size , and then to ask whether the resulting parameterized problem can be solved in time where is a constant and is some computable function. Parameterized problems which can be solved within such time bounds are said to be fixed-parameter tractable (FPT).

The standard parameterization of the Max-Cut problem sets the parameter to be the size of the cut being sought. This turns out to be not very interesting for the following reason: Let be the number of edges in the input graph . By an early result of Erdős [10], we know that every graph with edges contains a cut of size at least . Therefore, if then we can immediately answer YES. In the remaining case , and there are less than edges in the input graph. It follows from this bound on the size of the input that any algorithm—even a brute-force method—which solves the problem runs in FPT time on this instance.

The best lower bound known on the size of a largest cut for connected loop-less graphs on vertices and edges is , as proved by Edwards [8, 9]. This is called the Edwards-Erdős bound, and it is the best possible in the sense that it is tight for an infinite family of graphs, for example, the class of cliques of odd order . A more interesting parameterization of Max-Cut is, therefore, the following:

Max-Cut Above Tight Lower Bound (Max-Cut ATLB) Input: A connected graph , and an integer . Parameter: Question: Does have a cut of size at least ?

In the work which introduced the notion of “above-guarantee” parameterization, Mahajan and Raman [16] showed that the problem of asking for a cut of size at least is FPT parameterized by , and stated the fixed-parameter tractability of Max-Cut ATLB as an open problem. This question was resolved quite recently by Crowston et al. [5], who showed that the problem is in fact FPT.

We generalize the result of Crowston et al. by extending it to apply to a special case of the so-called -extendible properties. Roughly stated444See subsection 1.2 and section 2 for the definitions of various terms used in this section., for a fixed a graph property is said to be -extendible if: Given a graph graph , an “extra” edge not in , and any set of “extra” edges each of which has one end point in and the other in , there exists a graph which contains (i) all of , (ii) the edge , and (iii) at least a fraction of the edges in . The notion was introduced by Poljak and Turzík who showed [18] that for any -extendible property and edge-weighting function , any connected graph contains a spanning subgraph such that . Here denotes the total weight of all the edges in , and is the set of edges in a minimum-weight spanning tree of . It is not difficult to see that the property of being bipartite is -extendible for , and so—once we assign unit weights to all edges—the Poljak and Turzík result implies the Edwards-Erdős bound. Other examples of -extendible properties—with different values of —include -colorability and acyclicity in oriented graphs.

In this work we study the natural above-guarantee parameterized problem for -extendible properties , which is: given a connected graph and an integer as input, does contain a spanning subgraph such that ? To derive a generic FPT algorithm for this class of problems, we use the “reduction” rules of Crowston et al. To make these rules work, however, we need to make a couple of concessions. Firstly, we slightly modify the notion of lambda extendibility; we define a (potentially) stronger notion which we name strong -extendibility. Every strongly -extendible property is also -extendible by definition, and so the Poljak-Turzík bound applies to strongly -extendible properties as well. Observe that for each way of assigning edge-weights, the Poljak and Turzík result yields a (potentially) different lower bound on the weight of the subgraph. Following the spirit of the question posed by Mahajan and Raman and solved by Crowston et al., we choose from among these the lower bound implied by the unit-edge-weighted case. This is our second simplification, and for this “unweighted” case the Poljak and Turzík result becomes: for any strongly -extendible property , any connected graph contains a spanning subgraph such that .

The central problem which we discuss in this work is thus the following; here , and is an arbitrary—but fixed—strongly -extendible property:

Above Poljak-Turzík () (APT()) Input: A connected graph and an integer . Parameter: Question: Is there a spanning subgraph of such that ?

1.1 Our Results and their Implications

We show that that the Above Poljak-Turzík () problem is FPT for every strongly -extendible property for which APT() is FPT on a class of “almost-forests of cliques”. Informally, this is a class of graphs which are a small number () of vertices away from being a graph in which each block is a clique. This requirement is satisfied by the properties underlying a number of interesting problems, including Max-Cut, Max -Colorable Subgraph, and Oriented Max Acyclic Digraph.

The following is the main result of this paper.

{theorem}

The Above Poljak-Turzík () problem is fixed-parameter tractable for a -extendible property of graphs if

  • is strongly -extendible and

  • Above Poljak-Turzík () is FPT on almost-forests of cliques.

This also holds for such properties of oriented and/or labelled graphs.

We prove subsection 1.1 using the classical “Win/Win” approach of parameterized complexity. To wit: given an instance of a strongly -extendible property , in polynomial time we either (i) show that is a yes instance, or (ii) find a vertex subset of of size at most such deleting from leaves a “forest of cliques”. To prove this we use the “reduction” rules used by Crowston et al [5] in the context of Max-Cut.

Our main technical contribution is the proof that these rules are sufficient to show that every NO instance of APT() is at a vertex-deletion distance of from a forest of cliques. This proof requires several new ideas: a result which holds for all strongly -extendible properties is a significant step forward from Max-Cut. Our main result unifies and generalizes known results and implies new ones. Among these are Max-Cut, finding a -colorable subgraph of the maximum size, and finding a maximum-size acyclic subdigraph in an oriented graph. Using our theorem we also get a linear vertex kernel for maximum acyclic subdigraph, complementing the quadratic arc kernel by Gutin et al. [13].

Related Work

The notion of parameterizing above (or below) some kind of “guaranteed” values—lower and upper bounds, respectively—was introduced by Mahajan and Raman [16]. It has proven to be a fertile area of research, and Max-Cut is now just one of a host of interesting problems for which we now have FPT results for such questions [19, 17, 13, 1, 3, 5, 4, 2].

1.2 Preliminaries

We use “graph” to denote simple graphs without self-loops, directions, or labels, and use standard graph terminology used by Diestel [6] for the terms which we do not explicitly define. We use and to denote the vertex and edge sets of graph , respectively. For , we use (i) to denote the subgraph of induced by the set , (ii) to denote , (iii) to denote the set of edges in which have exactly one end-point in , and (iv) to denote ; we omit the subscript if it is clear from the context. A clique in a graph is a set of vertices such that between any pair of vertices in there is an edge in . A block of graph is a maximal 2-connected subgraph of and a graph is a forest of cliques, if the vertices of each of its blocks form a clique. Thus a graph is a forest of cliques if and only if the vertex set of any cycle in the graph forms a clique. A leaf clique of a forest of cliques is a block of the graph, which corresponds to a leaf in its block forest. In other words, it is a block which contains at most one cut vertex of the graph.

For , (i) we use to denote the graph , and (ii) for a weight function , we use to denote the sum of the weights of all the edges in . A graph property is a collection of graphs. For we use to denote the complete graph on vertices, and to denote the complete bipartite graph in which the two parts of vertices are of sizes .

Our results also apply to graphs with oriented edges, and those with edge labels. Subgraphs of an oriented or labelled graph inherit the orientation or labelling—as is the case—of in the natural manner: each surviving edge keeps the same orientation/labelling as it had in . For a graph of any kind, we use to denote the simple graph obtained by removing all orientations and labels from ; we say that is connected (or contains a clique, and so forth) if is connected (or contains a clique, and so forth).

2 Definitions

The following notion is a variation on the concept of -extendibility defined by Poljak and Turzík [18].

{definition}

[Strong -extendibility] Let be the class of (possibly oriented and/or labelled) graphs, and let . A property is strongly -extendible if it satisfies the following:

Inclusiveness

Block additivity

belongs to if and only if each block of belongs to .

Strong -subgraph extension

Let and be such that and . For any weight function there exists an with , such that .

The strong -subgraph extension requirement can be rephrased as follows: Let be a cut of graph such that , and let be the set of edges which cross the cut. For any weight function , there exists a subset such that (i) , and (ii) . Informally, one can pick a -fraction of the cut and delete the rest to obtain a graph which belongs to .

We recover Poljak and Turzík’s definition of -extendibility from the above definition by replacing strong -subgraph extension with the following property:

-edge extension

Let and be such that is isomorphic to and . For any weight function there exists an with , such that .

Observe from the definitions that any graph property which is strongly -extendible is also -extendible. It follows that Poljak and Turzík’s result for -extendible properties applies also to strongly -extendible properties.

{theorem}

[Poljak-Turzík bound] [18] Let be a class of (possibly oriented and/or labelled) graphs. Let , and let be a strongly -extendible property. For any connected graph and weight function , there exists a spanning subgraph of such that , where is the set of edges in a minimum-weight spanning tree of .

When all edges are assigned weight , we get:

Corollary \thetheorem.

Let be as in section 2. Any connected graph on vertices and edges has a spanning subgraph with at least edges.

Our results apply to properties which satisfy the additional requirement of being FPT on almost-forests of cliques.

Definition \thetheorem (Fpt on almost-forests of cliques).

Let , and let be a strongly -extendible property (of graphs with or without orientations/labels). The Structured Above Poljak-Turzík () problem is a variant of the Above Poljak-Turzík () problem in which, along with the graph and , the input contains a set such that and is a forest of cliques. We say that the property is FPT on almost-forests of cliques if the Structured Above Poljak-Turzík () problem is FPT.

In other words, a -extendible property is FPT on almost-forests of cliques, if for any constant there is an algorithm that, given a connected graph and a set of size at most such that is a forest of cliques, correctly decides whether is a yes-instance of APT() in time, for some computable function .

3 Fixed-Parameter Algorithms for Above Poljak-Turzík ()

We now prove Theorem 1.1 using the approach which Crowston et al. used for Max-Cut [5]. The crux of their approach is a polynomial-time procedure which takes the input of Max-Cut and finds a subset such that (i) is a forest of cliques, and (ii) if is a NO instance, then . Thus if , then one can immediately answer YES; otherwise one solves the problem in FPT time using the fact that Max-Cut is FPT on almost-forests of cliques (section 2).

The nontrivial part of our work consists of proving that the procedure for Max-Cut applies also to the much more general family of strongly -extendible problems, where the bound on the size of depends on . To do this, we show that each of the four rules used for Max-Cut is safe to apply for any strongly -extendible property. From this we get

Lemma \thetheorem.

Let , and let be a strongly -extendible graph property. Given a connected graph with vertices and edges and an integer , in polynomial time we can do one of the following:

  1. Decide that there is a spanning subgraph of with at least edges, or;

  2. Find a set of at most vertices in such that is a forest of cliques.

This also holds for strongly -extendible properties of oriented and/or labelled graphs.

We give an algorithmic proof of section 3. Let be an instance of Above Poljak-Turzík (). The algorithm initially sets , , , and then applies a series of rules to the tuple . Each application of a rule to produces a tuple such that (i) if is connected then so is , and (ii) if is a NO instance of APT() then so is ; the converse may not hold. The algorithm then sets , and repeats the process, till none of the rules applies. These rules—but for minor changes—and the general idea of “preserving a NO instance” are due to Crowston et al. [5].

We now state the four rules and show that they suffice to prove section 3. We assume throughout that and are as in section 3. For brevity we assume that the empty graph is in , and we let so that .

Rule 1.

Let be connected. If and are such that (i) is a connected component of , and (ii) is a clique in , then delete from to get ; set .

Rule 2.

Let be connected. Suppose Rule 1 does not apply, and let be the connected components of for some . If at least one of the s is a clique, and at most one of them is not a clique, then

  • Delete all the s which are cliques—let these be in number—to get , and

  • Set and .

Rule 3.

Let be connected. If are such that , , and is connected, then

  • Set and .

Rule 4.

Let be connected. Suppose Rule 3 does not apply, and let be such that

  1. ;

  2. Let be the connected components of . There is at least one such that both and are cliques in , and there is at most one for which this does not hold.

Then

  • Delete all the s which satisfy condition (2) to get , and,

  • Set , .

Let be the tuple which we get by applying these rules exhaustively to the input tuple . To prove section 3, it is sufficient to prove the following claims: (i) the rules can be exhaustively applied in polynomial time; (ii) is a forest of cliques; (iii) the rules transform NO-instances to NO-instances; and, (iv) if is a NO instance, then . We now proceed to prove these over several lemmata. Our rules are identical to those of Crowston et al. in how the rules modify the graph; the only difference is in how we change the parameter . The first two claims thus follow directly from their work.

Lemma \thetheorem.

[]555Proofs of results marked with a have been moved to Appendix A. Rules 1 to 4 can be exhaustively applied to an instance of Above Poljak-Turzík () in polynomial time. The resulting tuple has .

Lemma \thetheorem.

[5, Lemma 8] Let be the tuple obtained by applying Rules 1 to 4 exhaustively to an instance of Above Poljak-Turzík (). Then is a forest of cliques.

The correctness of the remaining two claims is a consequence of the -extendibility of property , and we make critical use of this fact in building the rest of our proof. This is the one place where this work is significantly different from the work of Crowston et al.; they could take advantage of the special characteristics of one specific property, namely bipartitedness, to prove the analogous claims for Max-Cut.

We say that a rule is safe if it preserves NO instances. {definition} Let be an arbitrary tuple to which one of the rules 1, 2, 3, or 4 applies, and let be the resulting tuple. We say that the rule is safe if, whenever is a YES instance of Above Poljak-Turzík (), then so is .

We now prove that each of the four rules is safe. For a graph we use to denote the maximum number of edges in a subgraph of , and to denote the Poljak-Turzík bound for . Thus if is connected and has vertices and edges then , and section 2 can be written as . For each rule we assume that has a spanning subgraph with at least edges, and show that has a spanning subgraph with at least edges.

We first derive a couple of lemmas which describe how contributions from subgraphs of a graph add up to yield lower bounds on .

Lemma \thetheorem.

[] Let be a cut vertex of a connected graph , and let be sets of vertices of such that for every we have , there is no edge between and and . For , let be a subgraph of with edges, and let . Then is a subgraph of , , and .

Lemma \thetheorem.

[] Let be a graph, and let be such that there exists a subgraph of with at least edges, and a subgraph of with at least edges. Then there is a subgraph of with at least edges.

This lemma has a useful special case which we state as a corollary:

Corollary \thetheorem.

Let be a graph, and let be such that there exists a subgraph of with at least edges, and the subgraph has a perfect matching. Then there is a subgraph of with at least edges.

Proof.

Recall that the graph is in by definition, and observe that . Thus has edges. The corollary now follows by repeated application of Lemma 3, each time considering a new edge of the matching as the graph . ∎

The safeness of Rule 1 is now a consequence of the block additivity property.

{lemma}

Rule 1 is safe.

Proof.

Let and . Observe that (i) is a cut vertex of , , there are no edges between and by assumptions of the rule, and . Also by assumption, there is a spanning subgraph of such that . By section 2 there is a subgraph of with . Hence section 3 applies and has a spanning subgraph with . ∎

We now prove some useful facts about certain simple graphs, in the context of strongly -extendible properties. Observe that every block of a forest is one of , which are both in . From this and the block additivity property of we get

Observation \thetheorem.

Every forest (with every orientation and labeling) is in .

The graph is a useful special case.

Observation \thetheorem.

The graph —also with any kind of orientation or labelling—is in , and it has edges.

The graph obtained by removing one edge from is another useful object, since it always has more edges than its Poljak-Turzík bound.

Lemma \thetheorem.

[] Let be a graph formed from the graph —also with any kind of orientation or labelling—by removing one edge. Then (i) , (ii) if , and (iii) if . As a consequence,

(1)

The above lemmata help us prove that Rules 2 and 3 are safe.

Lemma \thetheorem.

[] Rule 2 is safe.

Following the notation of Rule 3, observe that for the vertex subset we have—from section 3—that and . Since , if then applying section 3 we get that . Hence we get {lemma} Rule 3 is safe.

To show that Rule 4 is safe, we need a number of preliminary results. We first observe that—while the rule is stated in a general form—the rule only ever applies when it can delete exactly one component.

Observation \thetheorem.

[] Whenever Rule 4 applies, there is exactly one component to be deleted, and this component has at least 2 vertices.

Our next few lemmas help us further restrict the structure of the subgraph to which Rule 4 applies. We start with a result culled from Crowston et al.’s analysis of the four rules.

Lemma \thetheorem.

[5][] If none of Rules 1, 2, and 3 applies to , and Rule 4 does apply, then one can find

  • A vertex and a set such that is a connected component of , and the graph is 2-connected;

  • Vertices such that and

    • has exactly two components ,

    • ; are cliques, and each of is adjacent to some vertex in

From this we get the following.

Lemma \thetheorem.

[] Suppose Rules 1, 2, and 3 do not apply, and Rule 4 applies. Then we can apply Rule 4 in such a way that if are the vertices to be added to and the clique to be deleted, then contains at most one vertex such that is disconnected.

We now show that in such a case , and so the graph is not connected. First we need the following simple lemma.

Lemma \thetheorem.

[] Whenever Rule 4 applies, with the vertices to be added to and the clique to be deleted, every in is a cut vertex in and every in is a cut vertex in .

This allows us to enforce a very special way of applying Rule 4.

Lemma \thetheorem.

[] Suppose Rules 1, 2, and 3 do not apply, and Rule 4 applies. Then we can apply Rule 4 in such a way that if are the vertices to be added to and the clique to be deleted, then , and is disconnected.

These lemmas help us prove that Rule 4 is safe.

Lemma \thetheorem.

[] Rule 4 is safe.

The next lemma gives us a bound on the size of the set which we compute.

Lemma \thetheorem.

[] Let be a connected graph, , and , and let one application of Rule 1, 2, 3, or 4 to result in the tuple . Then .

Now we are ready to prove section 3, and thence our main theorem.

Proof (of section 3)..

Let be an input instance of Above Poljak-Turzík (), and let be the tuple which we get by applying the four rules exhaustively to the tuple . From section 3 we know that this can be done in polynomial time, and that the resulting graph satisfies .

Thus is either or the empty graph, and so and . Hence if then is a YES instance of Above Poljak-Turzík (). Since all the four rules are safe—Lemmas 333, and 3—we get that in this case is a YES instance, and we can return YES. On the other hand if then we know—using section 3—that , and—from section 3—that is a forest of cliques. This completes the proof. ∎

Proof (of Theorem 1.1)..

From section 3 we know that in polynomial time we can either answer YES, or find a set such that and is a forest of cliques. In the latter case we have reduced the original problem instance to an instance of Structured Above Poljak-Turzík () (See section 2). The theorem follows since—by assumption—this latter problem is FPT. ∎

4 Applications

In this section we use subsection 1.1 to show that Above Poljak-Turzík () is FPT for almost all natural examples of -extendible properties listed by Poljak and Turzík [18]. For want of space, we defer the definitions and all proofs to Appendix B.

4.1 Application to Partitioning Problems

First we focus on properties specified by a homomorphism to a vertex transitive graph. As a graph is -colorable if and only if it has a homomorphism to , searching for a maximal -colorable subgraph is one of the problems resolved in this section. In particular, a maximum cut equals a maximum bipartite subgraph and, hence, is also one of the properties studied in this section. We use to denote the class of graphs—oriented or edge-labelled—to which the property in question belongs.

It is not difficult to see that every vertex-transitive graph is a regular graph. In particular, if allows labels and/or orientations, then for every label and every orientation each vertex of a vertex transitive graph is incident to the same number of edges of the given label and the given orientation.

Lemma \thetheorem.

[] Let be a vertex-transitive graph with at least one edge of every label and orientation allowed in . Then the property “to have a homomorphism to ” is strongly -extendible in , where is the number of vertices of and is the minimum number of edges of the given label and the given orientation incident to any vertex of over all labels and orientations allowed in .

Note that while the above lemma poses no restrictions on the graphs considered, we can prove the following only for simple graphs.

Lemma \thetheorem.

[] If is an unoriented unlabeled graph, then the property “to have a homomorphism into ” is FPTon almost-forests of cliques.

subsection 4.1 and subsection 4.1, together with subsection 1.1 immediately imply the following corollary.

{corollary}

The problem APT(“to have a homomorphism into ”) is fixed-parameter tractable for every unoriented unlabeled vertex transitive graph .

In particular, by setting we get the following result.

{corollary}

Given a graph with edges and vertices and an integer , it is FPT to decide, whether has an -colorable subgraph with at least edges.

This shows that the Max -Colorable Subgraph problem is FPT when parameterized above the Poljak and Turzík bound [18].

4.2 Finding Acyclic Subgraphs of Oriented Graphs

In this section we show how to apply our result to the problem of finding a maximum-size directed acyclic subgraph of an oriented graph, where the size of the subgraph is defined as the number of arcs in the subgraph. Recall that an oriented graph is a directed graph where between any two vertices there is at most one arc. We show that subsection 1.1 applies to this problem. To this end we need the following two lemmata.

Lemma \thetheorem.

[] The property “acyclic oriented graphs” is strongly -extendible in the class of oriented graphs.

Lemma \thetheorem.

[] The property “acyclic oriented graphs” is FPT on almost-forests of cliques.

Combining Lemmata 4.2 and 4.2 with Theorem 1.1 we get the following corollary. {corollary} The problem APT(“acyclic oriented graphs”) is fixed-parameter tractable.

To put this result in some context, we recall a couple of open problems posed by Raman and Saurabh [19]: Are the following questions FPT parameterized by ?

  • Given an oriented directed graph on vertices and arcs, does it have a subset of at least arcs that induces an acyclic subgraph?

  • Given a directed graph on vertices and arcs, does it have a subset of at least arcs that induces an acyclic subgraph ?

In the first question, a “more correct” lower bound is the one of Poljak and Turzík, i.e., , and the lower bound is true only for connected graphs. Corollary 4.2 answers the corrected question. Without the connectivity requirement, one can show by adding sufficient number of disjoint oriented 3-cycles that the problem is -hard already for .

For the second question, observe that each maximal acyclic subgraph contains exactly one arc from every pair of opposite arcs. Hence we can remove these pairs from the digraph without changing the relative solution size, as exactly half of the removed arcs can be added to any solution to the modified instance. Thus, we can we can restrict ourselves to oriented graphs.

Now suppose that the oriented graph we are facing is disconnected. It is easy to check that picking two vertices from different connected components and identifying them does not change the solution size, as this way we never create a cycle from an acyclic graph. After applying this reduction rule exhaustively, the digraph becomes an oriented connected graph, and the parameter is unchanged. But then if then and we can answer YES due to section 2. Otherwise , we have a linear vertex kernel, and we can solve the problem by the well known dynamic programming on the kernel [20]. The total running time of this algorithm is . The smallest kernel previously known for this problem is by Gutin et al., and has a quadratic number of arcs [13].

5 Conclusion and Open Problems

In this paper we studied a generalization of the graph property of being bipartite, from the point of view of parameterized algorithms. We showed that for every strongly -extendible property which satisfies an additional “solvability” constraint, the Above Poljak-Turzík () problem is FPT. As an illustration of the usefulness of this result, we obtained FPT algorithms for the above-guarantee versions of three graph problems.

Note that for each of the three problems—Max-Cut,Max -Colorable Subgraph, and Oriented Max Acyclic Digraph—for which we used subsection 1.1 to derive FPT algorithms for the above-guarantee question, we needed to device a separate FPT algorithm which works for graphs that are at a vertex deletion distance of from forests of cliques. We leave open the important question of finding a right logic that captures these examples, and of showing that any problem expressible in this logic is FPT parameterized by deletion distance to forests of cliques. We also leave open the kernelization complexity question for -extendible properties.

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Appendix A Deferred Proofs

 section 3. Rules 1 to 4 can be exhaustively applied to an instance of Above Poljak-Turzík () in polynomial time. The resulting tuple has .

Proof.

Let be as in the description above. It is not difficult to verify that (i) each rule can be applied once in polynomial time, (ii) for each application of a rule, if is connected then so also is , and (iii) each rule strictly reduces the size of the graph —either by deleting vertices from , or by adding vertices to . Crowston et al. have shown [5, Lemma 7] that if is a connected graph with at least two vertices, then at least one of these four rules apply to the tuple . Since none of the conditions for applying a reduction rule depends on the value of , and since the only difference between our set of rules and theirs is the way in which is modified, their result implies this lemma. ∎

 section 3. Let be a cut vertex of a connected graph , and let be a family of sets of vertices of such that for every we have , there is no edge between and and . For , let be a subgraph of with edges, and let . Then is a subgraph of , , and .

Proof.

Since there are no edges between and for , and , every edge of is in exactly one . Therefore, is a subgraph of . Also as is a cut vertex in , it is a cut vertex in and the blocks of are exactly the blocks of ’s. Since each is in it follows from the block additivity property of that .

Since , we get

 section 3. Let be a graph, and let be such that there exists a subgraph of with at least edges, and a subgraph of with at least edges. Then there is a subgraph of with at least edges.

Proof.

Let , and consider the subgraph . Observe that , and . Thus the strong -subgraph extension property of applies to the pair , and for the weight function which assigns unit weights to all edges in , we get that the graph has a spanning subgraph which contains all the edges in and at least a -fraction of the edges in . Thus

 section 3. Let be a graph formed from the graph —also with any kind of orientation or labelling—by removing one edge. Then (i) , (ii) if , and (iii) if . As a consequence,

(2)
Proof.

A spanning tree of has three edges, and so claim (i) follows from section 3.

Let , and let . Consider the vertex subset , for which , , and . Applying the strong -subgraph extension property—for unit edge weights—on the set we get that there exists a subgraph of which has at least edges. Since , we get claim (ii).

Now consider the subgraph , and its vertex subset . We apply the strong -subgraph extension property—again for unit edge weights—to the pair . Since , and , there exists a subgraph of which has at least edges. For this is at least edges, and so in this case and . Hence we can use the strong -subgraph extension property for and to get a subgraph of with at least edges. For this means all the five edges of , proving claim (iii).

The second part of the lemma follows from these claims since . ∎

 section 3. Rule 2 is safe.

Proof.

We reuse the notation of the rule. Let be the cliques deleted by the rule, and let be the remaining component (if any) of . For let . Since , by assumption we have that . As we show below, for we have that . Applying section 3 to the graph and the family , we get