Towards constant-factor approximation for chordal / distance-hereditary vertex deletion

For a family of graphs $\mathcal{F}$, Weighted $\mathcal{F}$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G\setminus S\in\mathcal{F}$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $F$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.


Introduction
This problem captures many classical combinatorial optimization problems including Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal, and the problems corresponding to natural graph classes (e.g., planar graphs, chordal graphs, or graphs of bounded treewidth) also have been actively studied. Most of these problems, including the simplest Vertex Cover, are NP-hard, so polynomial-time exact algorithms are unlikely to exist for them.
Parameterized algorithms and approximation algorithms have been two of the most popular kinds of algorithms for NP-hard optimization problems, and F-Deletion has been actively studied from both viewpoints. There is a large body of work in the theory of parameterized complexity, where F-Deletion for many F's is shown to be in FPT or even admits a polynomial kernel. The list of such F's includes chordal graphs [24,17,3], interval graphs [8,7,4], distance-hereditary graphs [11,19], bipartite graphs [26,22], and graphs with bounded treewidth [14,21].
On the other hand, despite large interest, approximability for F-Deletion is not as well as understood as parameterized complexity. To the best of our knowledge, for all F's admitting parameterized algorithms in the above paragraph except Odd Cycle Transversal, the existence of a constant-factor approximation algorithm is not ruled out under any complexity hypothesis. When F can be characterized by a finite list of forbidden subgraphs or induced subgraphs (not minors), the problem becomes a special case of Hypergraph Vertex Cover with bounded hyperedge size, which admits a constant-factor approximation algorithm. Besides them, the only classes of graphs that currently admit constant-factor approximation algorithms are block graphs [1], 3-leaf power graphs [5], interval graphs [7], and graphs of bounded treewidth [14,15]. Weighted versions are sometimes harder than their unweighted counterparts, and within graphs of bounded treewidth, the only two nontrivial classes whose weighted version admits a constant-factor approximation algorithm are the set of forests (Weighted Feedback Vertex Set) and the set of graphs excluding a diamond as a minor [13]. See Figure 1.
When F is the set of perfect or weakly chordal graphs, it is known that a constant-factor approximation algorithm is unlikely to exist [16]. Therefore, there has been recent interest on identifying large subclasses of perfect graphs that admit constant-factor approximation algorithms. Among the subclasses of perfect graphs, chordal graphs and distance-hereditary graphs have drawn particular interest. Recall that chordal graphs are the graphs without any induced C ě4 1 , and distance-hereditary graphs are the graphs without any induced C ě5 , a gem, a house, or a domino. See Figure 1.
Chordal graphs are arguably the simplest graph class, apart from forests, which is characterized by infinite forbidden induced subgraphs. Structural and algorithmic aspects of chordal graphs have been extensively studied in the last decades, and it is considered one of the basic graph classes whose properties are well understood and on which otherwise NP-hard problems become tractable. As such, it is natural to ask how close a graph to a chordal graph in terms of graph edit distance and there is a large body of literature pursuing this topic [3,2,9,17,18,24,29].
Fixed-parameter tractability and the existence of polynomial kernel of F-Deletion for chordal graphs were one of important open questions in parameterized complexity [24,17]. An affirmative answer to the latter in [17] brought the approximability for chordal graphs to the fore as it uses an Opopt 2 log opt log nq-factor approximation algorithm as a crucial subroutine. It was soon improved to Opopt log nq-factor approximation [3,20]. An important step was taken by Agrawal et al. [2] who studied Weighted F-Deletion for chordal graphs, distance-hereditary graphs, and graphs of bounded treewidth. They presented polylogpnqapproximation algorithms for them, including Oplog 2 nq-approximation for chordal graphs, and left the existence of constant-factor approximation algorithms as an open question. For now, even the existence of Oplog nq-factor approximation is not known. This makes an interesting contrast with F-Deletion for forests, that is, Feedback Vertex Set. An algorithmic proof of Erdös-Pósa property 2 for cycles immediately leads to an Oplog nq-factor approximation for Feedback Vertex Set while the known gap function of Erdös-Pósa property for induced C ě4 is not low enough to achieve such an approximation factor [20].
Distance-hereditary graphs, in which any induced subgraph preserves the distances among all vertex pairs, form another important subclass of perfect graphs. It is supposedly the simplest dense graph class captured by a graph width parameter; distance-hereditary graphs are precisely the graphs of rankwidth 1 [25]. F-Deletion for distance-hereditary graphs has gained good attention for fixed-parameter tractability and approximability [2,19,11] particularly due to the recent surge of interest in rankwidth. An Oplog 3 nq-approximation is known [2].
Constant-factor approximation algorithms were designed for smaller subclasses of chordal and distance-hereditary graphs. They include block graphs (excluding C ě4 and a diamond) [1] and 3-leaf power graphs (excluding C ě4 , a bull, a dart, and a gem) [6]. See Figure 1. Recently, a p2` q-factor approximation for split graphs was announced [23].
In this paper, we take a step towards the (affirmative) answer of the question of [2] by presenting a constant-factor approximation algorithm for the intersection of chordal and distance-hereditary graphs, known as ptolemaic graphs. 3 They are precisely graphs without any induced C ě4 or a gem, so it is easy to see that they form a superclass of both 3-leaf power and block graphs.
Weighted Ptolemaic Deletion Input : A graph G " pV, Eq with vertex weights w : V Ñ R`Y t0u. Question : Find a set S Ď V of minimum weight such that GzS is ptolemaic. Theorem 1.1. Weighted Ptolemaic Deletion admits a polynomial-time constantfactor approximation algorithm.

Techniques
Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest. 2 Any graph has either a vertex-disjoint packing of k`1 cycles, or a feedback vertex set of size Opk log kq. 3 The name ptolemaic comes from the fact that the shortest path distance satisfies Ptolemy's inequality: For every four vertices u, v, w, x, the inequality dpu, vqdpw, xq`dpu, xqdpv, wq ě dpu, wqdpv, xq holds.

Inter-clique Digraphs
The starting point of our proof is to examine what we call an inter-clique digraph of G. Let CpGq be the collection of all non-empty intersections of maximal cliques in G, see Section 2 for the formal definition. An inter-clique digraph Ý Ñ T pGq of G, or simply Ý Ñ T , is a digraph isomorphic to the Hasse diagram of pCpGq, Ďq. A neat characterization of ptolemaic graphs was presented by Uehara and Uno [28]: a graph G is ptolemaic if and only if its inter-clique digraph is a forest. This immediately suggests the use of an Op1q-approximation algorithm for Feedback Vertex Set on the inter-clique digraph. Indeed, the black-box application of an Op1q-approximation algorithm for Feedback Vertex Set yields Op1q-approximation algorithms for subclasses of ptolemaic graphs including block graphs [1] and 3-leaf power graphs [5].
However, to leverage this characterization for Ptolemaic Deletion, two issues need to be addressed. First, a polynomial-time algorithm to construct an inter-clique digraph of the input graph G is needed, while the size of an inter-clique digraph can be exponentially large for general graphs. Second, even with the inter-clique digraph of polynomial size at hand, the application of Feedback Vertex Set remains nontrivial since (1) after deletion of vertices, the structure of the inter-clique digraph may drastically change, and (2) feedback vertex sets for the inter-clique digraph must satisfy additional constraints that a deletion of a node C P CpGq must imply the deletion of all nodes reachable from it (because they are subsets of C in G). Addressing each of these issues boils down to understanding the properties of an inter-clique digraph and elaborating the relationship between the input graph and its inter-clique digraph.
For general graphs, their inter-clique digraphs are acyclic digraphs in which each node can be precisely represented by all sources that have a directed path to the node. It turns out that eliminating from G all induced subgraphs isomorphic to C 4 and gem is key to tackling the aforementioned issues. We show that any hole of G indicates the existence of a cycle in Undp Ý Ñ T q, and vice versa when G is (C 4 , gem)-free (Lemmas 3.7-3.8). This in turn lets us to identify a variant of Weighted Feedback Vertex Set, termed Feedback Vertex Set with Precedence Constraints and defined in Section 1.1.2, which is essentially equivalent to Ptolemaic Deletion on G when it takes the inter-clique digraph of G as an input; see Proposition 3.11. Moreover, each subdigraph of Ý Ñ T induced by the ancestors of any node v of Ý Ñ T is a directed tree rooted at v, see Lemma 3.5. (Similar statement holds for the descendants of v.) This property is used importantly in analyzing our approximation for Feedback Vertex Set with Precedence Constraints. As Feedback Vertex Set with Precedence Constraints takes an inter-clique digraph as an input, we need to construct it in polynomial time. This is prohibitively time-consuming for general graphs. We show that the construction becomes efficient when G is both C 4 and gem-free, see Proposition 3.9.

Feedback Vertex Set with Precedence Constraints
Given acyclic directed graphs Ý Ñ G and a vertex v, let ancpvq and despvq be the set of ancestors and descendants respectively, and let Undp Ý Ñ G q denote the underlying undirected graph of Ý Ñ G . It remains to design a constant-factor approximation algorithm for the following problem: For each v P V , the subgraph induced by ancpvq is an in-tree rooted at v.
It is a variant of Undirected Feedback Vertex Set (FVS) on Undp Ý Ñ G q, with the additional precedence constraint on S captured by directions of arcs in A. This precedence constraint makes an algorithm for FVSP harder to analyze than FVS because a vertex v can be deleted "indirectly"; even when v does not participate in any cycle, deletion of any ancestor of v forces to v to be deleted, so the analysis for v needs to keep track of every vertex in ancpvq.
We adapt a recent constant-factor approximation algorithm for Subset Feedback Vertex Set by Chekuri and Madan [10] for FVSP. The linear programming (LP) relaxation variables are tz v u vPV , where z v is supposed to indicate whether v is deleted or not, as well as tx ue u ePA,uPe , where x ue is supposed to indicate that in the resulting forest Undp Ý Ñ G zSq rooted at arbitrary vertices, whether e is the edge connecting u and its parent.
Compared to the LP in [10], we added the z u ď z v for all pu, vq P A to encode the fact that u's deletion implies v's deletion. This LP is not technically a relaxation, but one can easily observe that in any integral solution, the graph induced by tv : z v " 0u has at most one cycle, which can be easily handled later. 4 The rounding algorithm proceeds as follows. Fix three parameters ε « 0.029, α « 0.514, β « 0.588. For notational convenience, letx ue :" 1´x ue . Also, for each e " pu, vq P A, let y e " z v´zu .
Slightly modifying the analysis of [10], one can show that after rounding, there is indeed at most one cycle remained in each connected component. In terms of the total weight of deleted vertices, it is easy to bound the total weight of deleted vertices in Step (i) and the final cleanup step for one cycle. The main technical lemma of the analysis bounds the weight of vertices deleted in Step (iii) by at most OpLPq.
Recall that ancpvq induces the directed tree Ý Ñ T rooted on v where all arcs are directed towards v, and deletion of any vertex in Ý Ñ T forces the deletion of v. The lemma is proved by showing that while ancpvq can be large, all vertices that can be possibly deleted during the rounding algorithm can be covered by at most two directed paths; it is proved by examining behaviors of the rounding algorithm on directed trees, followed by an application of Dilworth's theorem. The new LP constraint z u ď z v for all pu, vq P A ensures that the sum of the deletion probabilities along any path is at most Opz v q, so the total probability that v is deleted can be bounded by Opz v q.

62:6 A Constant-Factor Approximation of Weighted Ptolemaic Deletion 2 Preliminaries
For a mapping f : X Ñ Y between two finite sets and a set A Ď X, we denote Ť xPA tf pxqu by f pAq. For sets X and Y , we say that X and Y are overlapping if none of XzY , Y zX, and X X Y is empty. For a family F of sets, F is laminar if F has no overlapping two elements. In an undirected graph G, we say that two vertices u, v are true twins, or simply twins, if N G rus " N G rvs. Note that true twins must be adjacent. Since the true twin relation is an equivalence relation, the true twin classes of V is uniquely defined. For graphs G without an in-coming arc and a sink of Ý Ñ G is a vertex without an out-going arc. We denote by ancp

Clique and inter-clique digraph
We denote the set of maximal cliques in a graph G by MpGq. We define the set CpGq all non-empty intersections among maximal cliques, that is, We may write MpGq and CpGq as M and C respectively, if it is clear from the context. Cleary, CpGq defines a partially ordered set under the set containment relation Ď. A Hasse diagram Ý Ñ H of a poset pS, ďq represents each element of S as a vertex and adds an arc from y to x if and only if y ą x and there is no element z P S with y ą z ą x. We say that a digraph H q nodes instead of vertices in order to distinguish them from the vertices of G. For a vertex set X Ď V pGq, we define srcpXq as the set of all maximal cliques containing X. For v P V pGq, we may write srcpvq instead of srcptvuq. For a collection of sets X , srcpX q is defined as the collection of sets (without duplicates) srcpX q " tsrcpXq : X P X u. For the Hasse diagram Ý Ñ H of pCpGq, Ďq and a clique C P CpGq, we have srcpCq " srcp Ý Ñ H , Cq, and this justifies the reuse of the notation src for a vertex set, while srcp Ý Ñ G , vq is already defined to delineate the set of vertices with no in-coming arcs from which there is a directed path to v in Ý Ñ G . 62:7

Ptolemaic graphs
A graph is ptolemaic if for every four vertices a, b, c, and d in the same connected component, G satisfies the following inequality: Note that they can be equivalently defined as the set of (C ě4 , gem)-free graphs. The following theorem proves strong relationship between ptolemaic graphs and its inter-clique digraph.

3
Structures of Inter-clique digraphs

Basic properties of inter-clique digraphs
In this subsection, we investigate the properties of the Hasse diagram Ý Ñ H of the poset pCpGq, Ďq for a graph G " pV, Eq. All the results presented in this subsection assume no restriction on the input graph G.
We first observe that each vertex v of V can be uniquely associated to a clique C of CpGq with the property srcpvq " srcpCq. Ş M Psrcpvq M . Note that srcpvq " srcpCpvqq. In the following three lemmas, we investigate properties of descendants of nodes in Ý Ñ H . In this extended abstract, the proofs of some lemmas will be deferred to the full version.

Lemma 3.2.
If a node C has immediate descendants C 1 , . . . , C p with p ě 2 in Ý Ñ H , then we have srcpCq " srcpC i q X srcpC j q for every 1 ď i ă j ď p.

Inter-clique digraphs of (C 4 , gem)-free graphs
Here, we examine how the extra assumption that G is (C 4 , gem)-free brings about a new structure to emerge in the corresponding Hasse diagram Ý Ñ H . The following two lemmas will be crucially used in the proof of Proposition 3.11 to investigate the structure of minimal ptolemaic deletion set. Lemma 3.6. Let G " pV, Eq be a (C 4 , gem)-free graph. If G has a hole H and v P V pHq, then GrV pHq Y tv 1 uztvus contains a hole for every v 1 P Cpvq. Sketch of the proof. Let H " C 0 , P 1 , C 1 , P 2 , C 2 ,¨¨¨, C 2 ´1 , P 2 , C 2 p" C 0 q be a segment decomposition of H. We skip to prove the statement for ď 2. Suppose that " 3, and note that C 2i is a common ancestor of C 2i´1 and C 2i`1 for every i P r3s. For each i P r3s, choose an arbitrary clique C 1 i which is a sink in Ý Ñ H and a descendant of C 2i´1 . Then it is easy to see that C 1 i is a descendant of C 2i´1 only for each i. On the other hand, the cliques C 2i´1 and C 2i`1 are completely adjacent for every i P r3s, which implies that C 1 H with the shortest segment length with a segment decomposition H " C 0 , P 1 , C 1 , P 2 , C 2 ,¨¨¨, C 2 ´1 , P 2 , C 2 p" C 0 q. Then for any two nodes C, C 1 of H, C and C 1 are incomparable unless they belong to the same segment of H. In addition, for i, j P r s with |i´j| ě 2, there is no common ancestor of C 2i´1 and C 2j´1 in Ý Ñ H .

Constructing inter-clique digraphs for (C 4 , gem)-free graphs
As an arbitrary graph can have prohibitively many maximal cliques, we cannot expect a polynomial-time algorithm to construct inter-clique digraphs for general graphs. Instead, we present a polynomial-time algorithm for (C 4 , gem)-free graphs. Let C M :" tC P C : M P srcpCqu for each M P M.

Proposition 3.9.
There is a polynomial-time algorithm which, given a (C 4 , gem)-free graph G, constructs the Hasse diagram Ý Ñ H of pCpGq, Ďq.
Sketch of the proof. Let Z be the partition of V into true twin classes and n :" |V pGq|.
Since G has no C 4 as an induced subgraph, it has at most n 2 maximal cliques and these cliques can be enumerated with polynomial delay [12,27]. Thus, all of M, Z, and srcpZq can be computed in polynomial time. Observe that for certain cliques C P CpGq, srcpCq is already contained in srcpZq. By Lemma 3.3, we can easily show that if C P CpGq is a sink or has a unique immediate descendant in Ý Ñ H , then srcpCq P srcpZq. Let R X,0 :" srcpZq. For i ě 1, we define R X,i recursively as R X,i :" R X,i´1 Y tR X R 1 : R, R 1 P R X,i´1 u. Let the height of a node v of an acyclic digraph Ý Ñ G be the length of a longest directed path from v to a sink in Ý Ñ G . The height of Ý Ñ G is defined as the maximum over the heights of all nodes of Ý Ñ G .
Proof. It suffices to prove the following for each i ě 0: for any node C at height i in Ý Ñ H , we have srcpCq P R X,i . By the previous paragraph, we only need to consider a node C such that C has height i ą 0 and (at least) two immediate descendants C 1 , C 2 in Ý Ñ H . By induction hypothesis and because of that the height of the immediate descendants of C is at most i´1, we have srcpC 1 q, srcpC 2 q P R X,i´1 . Therefore, we have srcpC 1 q X srcpC 2 q P R X,i by definition. Then by Lemma 3.2, it holds that srcpCq P R X,i as claimed.
By Lemma 3.5, we can show that for each maximal clique M , the height of Ý Ñ H rC M s is at most |Z|, and therefore the height of Ý Ñ H is at most |Z|, that is, at most n. As we compute R X,i`1 from R X,i repeatedly, we need a guarantee that the sizes of the computed sets R X,i do not grow exponentially. For each maximal clique M , by the laminarity of C M , we can show that |C M | ď 2n, and therefore |CpGq| ď 2n 3 . Then we can compute each R X,i in polynomial time and srcpCpGqq can be computed in polynomial time by Claim 3.10. As we compute R X,i , the containment relations amongst the elements of R X,i can be determined as well. Then Ý Ñ H obviously comes from the Hasse diagram of psrcpCpGqq, Ďq.

Reduction from Ptolemaic Deletion to Feedback Vertex Set with Precedence Constraints
Let G " pV, Eq be a (C 4 , gem)-free graph with weight ω o : V Ñ R`Y t0u, Ý Ñ H be the Hasse diagram of pCpGq, Ďq, and Ý Ñ T " pN, Aq be an inter-clique digraph isomorphic to Ý Ñ H with an arc-preserving mapping γ : CpGq Ñ N . Notice that the canonical clique can be construed as a function which maps each vertex v of G to the clique C P CpGq such that srcpvq " srcpCq. We define a mapping C´1 : CpGq Ñ 2 V so that it maps each clique C of CpGq to its preimage under the canonical clique as a function from V to CpGq: if there is no vertex v P V with Cpvq " C, then the preimage of C under the canonical clique is H. We define φ : V Ñ N and φ´1 : N Ñ 2 V such that φpvq " γpCpvqq and φ´1pxq " C´1pγ´1pxqq. Now the node weight function ω : N Ñ R`Y t0u is defined as ωpxq :" ř vPφ´1pxq ω o pvq. For a set of nodes R of Ý Ñ T , the closure of R, denoted as R˚, is a minimal superset of R for which the following holds: (a) all descendants of R of weight zero are contained in R˚, (b) if all immediate descendants of a node v are in R˚and φ´1pvq " H, then v P R˚.
T " pN, Aq be an inter-clique digraph of G with an arc-preserving mapping γ : CpGq Ñ N and with node weight ω : C Ñ R`Yt0u, such that ωpxq :" ř vPφ´1pxq ω o pvq. Then the following two statements hold.

For any minimal ptolemaic deletion set S Ď V , (i) φpSq˚is downward-closed in
Sketch of the Proof. We first prove (1)-(i). We first observe that if S is a minimal deletion set, S contains the canonical clique Cpvq of v whenever S contains v P V .
Claim 3.12. If S Ď V is a minimal ptolematic deletion set, then Cpvq Ď S whenever v P S. Consequently, φ´1pxq Ď S for every x P φpSq.
Proof. Suppose Cpvq Ę S for some v P S. Since G is (C 4 , gem)-free, GzS is ptolemaic if and only if GzS is chordal. Since S is minimal, GzpSztvuq has a hole H intersecting v. By the assumption, there exists v 1 P CpvqzS. However, Lemma 3.6 implies that GrpV pHqztvuqYtv 1 us contains a hole and thus GzS contains a hole, a contradiction. The second statement is immediate from the first statement.
Consider a vertex v P S of G and an arbitrary descendant x of φpvq in Ý Ñ T . We claim that x P φpSq˚. If φ´1pxq " H, then by definition ωpxq " ř vPH ω o pvq " 0 and thus the claim trivially holds by definition of φpSq˚. Otherwise, let w P φ´1pxq and we have φ´1pxq Ď γ´1pxq Ď γ´1pφpvqq " Cpvq Ď S. Thus, w P φ´1pxq Ď S which implies x P φpSq, and φpSq˚is downward-closed in Ý Ñ T .

62:10 A Constant-Factor Approximation of Weighted Ptolemaic Deletion
To see that (1)-(ii), let H be a cycle of Ý Ñ T zφpSq˚with the least segment length and let x 0 , P 1 , x 1 , P 2 , x 2 ,¨¨¨, x 2 ´1 , P 2 , x 2 p" x 0 q be a segment decomposition of H. Consider the cliques γ´1px 2i´1 qzS of G for i P r s. We can show by (1)-(i) and the definition of the closure of a node set, that for every i P r s, there exists a vertex v i P γ´1px 2i´1 qzS of G.
We observe that all v i 's are distinct. Suppose that v i " v j for i ‰ j, and without loss of generality we may assume that 1 ď i ă j ď . Then the canonical clique Cpv i q is a common descendant of γ´1px 2i´1 q and γ´1px 2j´1 q, or equivalently, φpv i q is a common descendant of x 2i´1 and x 2j´1 . Let x˚be the greatest common descendant of x 2i´1 and x 2j´1 in Ý Ñ T , which is unique by Lemma 3.4. Let P and Q be the directed px 2i´1 , x˚q-path and the directed px 2j´1 , x˚q-path. Due to Lemma 3.8, both directed paths are disjoint from H except from the two starting vertex x 2i´1 and x 2j´1 . Then we can find a cycle from H with a shorter segment length by replacing a subpath of H between x 2i´1 and x 2j´1 with P, x˚, Q, a contradiction.
Furthermore, v i and v i`1 are adjacent because the cliques γ´1px 2i´1 q and γ´1px 2pi`1q´1 q are complete to each other in G due to the existence of common ancestor x 2i in Ý Ñ T . That is, J " v 1 , . . . , v , v 1 forms a cycle, and its length is at least four by Lemma 3.7. Furthermore, Lemma 3.8 implies that J is a hole, which altogether avoids S because of our choice of v i as a vertex of γ´1px 2i´1 qzS. This contradicts the assumption that S is a ptolemaic deletion set, which proves (1)-(ii). We skip to prove (1)-(iii).
To see (2), suppose that for a node set R of is not a ptolemaic deletion set of G. Let  H " v 1 , . . . , v s , v 1 be a hole of length s ě 5 in Gz Ť xPR φ´1pxq. Consider the canonical cliques Cpv 1 q, . . . , Cpv s q and their corresponding nodes x 1 , . . . , x s in Ý Ñ T . The adjacency of v i and v i`1 ensures that x i and x i`1 has a common ancestor for all i P rss, where s`1 " 1. Furthermore, none of the nodes from these common ancestors is contained in R since otherwise, some x i must belong to the downward-closed set R. This, however, means that x 1 , . . . , x s are contained in a closed walk of Ý Ñ T zR, contradicting (ii). We conclude that Ť xPR φ´1pxq is a ptolemaic deletion set of G and its weight is easily computed as suggested. Sketch of the proof. We skip the trivial runtime analysis. In order to turn the input graph into a (C 4 , gem)-free graph, we can design a simple linear programming (LP) to hit all C 4 and gem, and let X be the set of vertices whose LP value is at least 1{5. Since every copy of C 4 (resp. gem) must have a vertex with LP value at least 1{4 (resp. 1{5), G 1 :" GzX is (C 4 , gem)-free. Furthermore, the total weight of X is at most 5 times the LP value, which is at most 5OPT pto .
Each vertex of G 1 inherits its weight ω o v in G. We construct an inter-clique digraph Ý Ñ T " pN, Aq of G 1 with a node-weight ω as in Proposition 3.11. The node set ancpxq forms an in-tree rooted at x due to Lemma 3.5 It is a variant of Undirected Feedback Vertex Set on Undp Ý Ñ G q, with the additional precedence constraint on S is captured by the direction of arcs in A. The main result of this section is an Op1q-approximation algorithm for this problem. We consider the following linear programming (LP) relaxation. The relaxation variables are tz v u vPV , where z v is supposed to indicate whether v is deleted or not, as well as tx ue u ePA,uPe , where x ue is supposed to indicate that in the resulting forest Undp Ý Ñ G zSq rooted at arbitrary vertices, whether e is the edge connecting u and its parent.
Let OPT be the weight of the optimal solution, and LP ď OPT be the optimal value of the above LP. After solving the LP, we perform the following rounding algorithm. It is parameterized by three parameters ε, α, β P p0, 1q that satisfy 3p1´βq ě 1`8ε. If θ ąx ue , then add e to L u and say u points to e.
Though the above rounding algorithm is stated as a randomized algorithm, it is easy to make it deterministic, because there are at most Opmq subintervals of rα, βs such that two θ values from the same interval behave exactly the same in the rounding algorithm.
We first analyze the total weight of deleted vertices. In Step (i), we delete all vertices whose LP value z v ě ε, so the total weight of deleted vertices in Step (i) is at most LP{ε. The following lemma bounds the weight of vertices deleted in Step (iii) by at most 2LP{pβ´αq. β´α . Proof. Due to Step (i), we can assume that every vertex v satisfies z v ă ε and each arc e satisfies y e ă ε.

I S A
By the definition of Feedback Vertex Set with Precedence Constraints, Ý Ñ T is an in-tree rooted at v. We first prove the following claim that if we consider any directed path pu 0 , . . . , u k q of Ý Ñ T and the value of x ui,pui´1,uiq that u i gives to its incoming edge pu i´1 , u i q, the value at the end pi " kq is almost as large as the value at the beginning pi " 1q. T and e i " pu i´1 , u i q. Then for any i P rks, x uiei ě x u1e1´p z ui´zu1 q ě x u1e1´ε .
Proof. The proof proceeds by induction. The base case i " 1 is obviously true. When the claim holds for i´1, the constraint (2) of the LP (for u i´1 ) implies x ui´1ei´1`zui´1`xui´1ei ď 1, and the constraint (1) of the LP implies (for e i ) z ui`xui´1ei`xuiei " 1.
Subtracting the first inequality from the second equality yields which, by the induction hypothesis, is at least For e " pw, uq P Ap Ý Ñ T q, call e a target if Prru is directly deleted by es ą 0, which implies x ue´ye ă β ñ x ue ą 1´β´y e ą 1´β´ε. For two arcs e, f P Ap Ý Ñ T q, say they are incomparable if there is no directed path from the tail of one arc to tail of the other in Ý Ñ T (though they may share the head.) Proof. Assume towards contradiction that there exist three pairwise incomparable targets e 1 " pw 1 , u 1 q, e 2 " pw 2 , u 2 q, e 3 " pw 3 , u 3 q. It implies that x uiei ą 1´β´ε for each i. By Claim 4.3, for any i and any arc e 1 " pw 1 , u 1 q P Ap Ý Ñ T q that has a directed path from e i , we have For each i P r3s, consider the path P i from w i to v, and let g i be the last arc of P i that does not appear in any other P j 's. We consider the following two cases depending on how they intersect, and show both cannot happen. First, suppose all g 1 , g 2 , g 3 meet at the same vertex w; in other words, g i " pt i , wq for some t i 's. Then by (5), x wgi ą 1´β´2ε for each i. With (4), it implies ř i x wgi ą 3p1´βq´6ε ě 1, which violates the constraint (2) of the LP.
Finally, without loss of generality, suppose g 1 and g 2 meet at u, which is not incident on g 3 ; in other words, g 1 " pt 1 , uq, g 2 " pt 2 , uq, g 3 " pt 3 , wq for some t i 's, where w is an ancestor of u in Ý Ñ T and is the first vertex where all P 1 , P 2 , P 3 intersect. Let Ý Ñ T be the parent of u in the tree Ý Ñ T ( Ý Ñ T may be equal to w), and g " pu, tq. Then (5), implies x ugi ą 1´β´2ε for i P t1, 2u, which, combined with the LP constraint (2) for u, yields x ug ă 1´2p1´β´2εq " 2β´1`4ε. Together again with the LP constraint (1) for g, we have x tg ą 1´x ug´zt ą 2´2β´5ε.
Let h be the last arc of the path from u to w. Using Claim 4.3 again, we conclude that x wh ą 2´2β´6ε. Combined with x wg3 ą 1´β´2ε and h and g 3 are different, it implies x wh`xwg3 ą 3´3β´8ε ě 1 by (4), which contradicts the constraint (2) of the LP for w. Now we compute the probability that v is deleted by Step (iii) of the rounding algorithm. It happens whether v itself is directly deleted or some vertex u P ancpvq " V p Ý Ñ T q is directly deleted by a target e " pw, uq. By Claim 4.4, no three targets are pairwise comparable, and by Dilworth's Theorem, all targets are contained in two directed paths P 1 , P 2 in Ý Ñ T . By the choice of the rounding algorithm, for one path P 1 " pu 0 , . . . , u k " vq, for each i P rks, Prru i is directly deleted by pu i´1 , u i qs ď y pui´1,uiq β´α " z ui´zui´1 β´α .
Summing over all i's yields We can apply the same analysis to P 2 and use the union bound.
We now examine structure of the remaining graph after the rounding procedure. We first show that in the original graph, each arc, if not deleted, is pointed to by at least one of its endpoints.
Claim 4.5. For each e " pu, vq P A, if neither u nor v was deleted during the rounding, e is pointed to by at least one of them.
The following lemma shows that after the rounding, each connected component (in the undirected sense) has at most one cycle.