Fixed parameter tractability of multicut in directed acyclic graphs

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Kratsch, Stefan , Pilipczuk, Marcin, Pilipczuk, MichaŁ and Wahlström, Magnus . (2015)

Fixed-parameter tractability of multicut in directed acyclic graphs. SIAM Journal on

Discrete Mathematics, Volume 29 (Number 1). pp. 122-144. ISSN 0895-4801

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FIXED-PARAMETER TRACTABILITY OF MULTICUT IN

DIRECTED ACYCLIC GRAPHS∗

STEFAN KRATSCH†, MARCIN PILIPCZUK‡, MICHAL PILIPCZUK§, AND

MAGNUS WAHLSTR ¨OM¶

Abstract. TheMulticut_{problem, given a graph}G_{, a set of terminal pairs}T ₌{₍s_i, t_i)|₁≤

i ≤r}, and an integerp, asks whether one can find acutset consisting of at mostp nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t_i is not reachable froms_i for each 1≤i ≤r. The fixed-parameter tractability of Multicutin undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355–388] and, independently, by Bousquet, Daligault, and Thomass´e [Proceedings of STOC, ACM, 2011, pp. 459–468], after resisting attacks as a long-standing open problem. In this paper we prove thatMulticutis fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remainsW[1]-hard.

Key words. fixed-parameter tractability, multicut, directed acyclic graph, important separa-tors, shadow removal

AMS subject classifications.05C85, 68R10, 68W05

DOI.10.1137/120904202

1. Introduction. Parameterized complexity is an approach for tackling hard problems by designing algorithms that perform robustly, when the input instance is in some sense simple; its difficulty is measured by an integer that is additionally appended to the input, called the parameter. Formally, we say that a problem is

fixed-parameter tractable (FPT) if it can be solved by an algorithm that runs in time

f(_k)_nc_{fornbeing the length of the input andkbeing the parameter, wheref is some} computable function and_c is a constant independent of the parameter.

The search for fixed-parameter algorithms resulted in the introduction of a num-ber of new algorithmic techniques and gave fresh insight into the structure of many classes of problems. One family that has received a lot of attention recently is the so-called graph cut problems, where the goal is to make the graph satisfy a global separation requirement by deleting as few edges or vertices as possible (depending on

∗_{Received by the editors January 2, 2013; accepted for publication (in revised form) October 21,} 2014; published electronically January 15, 2015. A preliminary version of this paper was presented at the 39th International Colloquium on Automata, Languages and Programming (ICALP 2012).

http://www.siam.org/journals/sidma/29-1/90420.html

†_{Technische Universit¨at Berlin, 10623 Berlin, Germany ([email protected]). This} re-search was done while the author was at the University of Utrecht, Netherlands, and was supported by the Netherlands Organization for Scientific Research (NWO), project “KERNELS: Combinatorial Analysis of Data Reduction.”

‡_{University of Warwick, Coventry CV4 7AL, UK ([email protected]). This research was} done while the author was at the University of Warsaw, Poland, and was supported by NCN grant N206567140 and the Foundation for Polish Science.

§_{University of Warsaw, Krakowskie Przedmiescie 26/28, 00-927 Warszawa, Poland (michal.}

[email protected]). This research was done while the author was at the University of Bergen, Norway, and was supported by European Research Council (ERC) grant “Rigorous Theory of Pre-processing,” reference 267959.

¶_{Royal Holloway University of London, Egham Hill, Egham, Surrey TW20 0EX, UK} ([email protected]). This research done while the author was at the Max-Planck Insti-tute for Informatics, Saarbr¨ucken, Germany.

122

the variant). Graph cut problems in the context of fixed-parameter tractability were to our knowledge first introduced explicitly in the seminal work of Marx [12], where it was proved that (i) Multiway Cut (separate all terminals from each other by

a cutset of size at most _p) in undirected graphs is FPT when parameterized by the size of the cutset; (ii)Multicut in undirected graphs is FPT when parameterized

by both the size of the cutset and the number of terminal pairs. Fixed-parameter tractability of Multicut_{parameterized by the size of the cutset only was left open}

by Marx [12] and resolved only much later (see below).

Probably the most fruitful contribution of the work of Marx [12] is the concept of

important separators, which proved to be a tool almost perfectly suited to capturing the bounded-in-parameter character of sensible cutsets. The technique proved to be extremely robust and serves as the key ingredient in a number of FPT algorithms [13,12,4,8,14,3,10,11,6]. In particular, the fixed-parameter tractability of Skew Multicut in directed acyclic graphs (DAGs), obtained via a simple application of

important separators, enabled the first FPT algorithm for Directed Feedback Vertex Set[4], resolving another long-standing open problem.

However, important separators have a drawback in that not all graph cut problems admit solutions with “sensible” cutsets in the required sense. This is particularly true in directed graphs, where, with the exception of the aforementionedSkew Multicut

problem inDAGs, for a long time few FPT graph cut problems were known; in fact,

up until very recently it was open whetherMultiway Cutin directed graphs admits

an FPT algorithm even in the restricted case of two terminals. The same complication arises in the undirectedMulticut_{problem parameterized by the size of the cutset.}

After a long struggle,Multicut_{was shown to be FPT by Marx and Razgon [13]}

and, independently, by Bousquet, Daligault, and Thomass´e [2]. The key component in the algorithm of Marx and Razgon [13] is the technique ofshadow removal, which, in some sense, serves to make the solutions to cut problems more well-behaved. This was adapted to the directed case by Chitnis, Hajiaghayi, and Marx [6], who proved that

Multiway Cut, parameterized by the size of the cutset, is FPT for an arbitrary

number of terminals, by a simple and elegant application of the shadow removal technique. A subsequent work by Chitnis et al. [5] improves the complexity bounds of the shadow removal technique and applies it to show fixed-parameter tractability of Directed Subset Feedback Vertex Set. This gives hope that, in general,

shadow removal may be helpful for the application of important separators to the directed world.

As for the directed Multicut_{problem, it was shown by Marx and Razgon [13]}

to be_W[1]-hard when parameterized only by the size of the cutset but otherwise had unknown status, even for a constant number of terminals in a DAG_{. We note that}

this case is known to be NP-hard and APX-hard [1].

Our results. The main result of this paper is the proof of fixed-parameter tractabil-ity of theMulticut in DAGsproblem, formally defined as follows:

Multicut in DAGs Parameter: p+r

Input: DAG _G, set of terminal pairsT ={(_{si, ti})|1≤_i≤_r},_{si, ti} ∈_V(_G) for 1≤_i≤_r, and an integer_p.

Question: Does there exist a set_Z of at most_pnonterminal vertices of_G, such that for any 1≤_i≤_rthe terminal_ti is not reachable from_si in _G\_Z?

More precisely, we prove the following theorem.

Theorem 1.1. Multicut in DAGscan be solved in O∗(2O(rp(r+p2)))time.

Note that throughout the paper we use_O∗-notation to suppress polynomial fac-tors. Note also that we focus on vertex cuts; it is well known that in the directed acyclic setting the arc- and vertex-deletion variants are equivalent (cf. [6]).

Our algorithm makes use of the shadow removal technique introduced by Marx and Razgon [13], adjusted to the directed setting by Chitnis, Hajiaghayi, and Marx [6] with the improved bounds of [5], as well as the basic important separators toolbox that can be found in [6]. We remark that the shadow removal is but one of a number of ingredients of our approach: in essence, the algorithm combines the shadow removal technique with a degree reduction for the sources in order to carefully prepare the structure of the instance for a simplifying branching step. For a more detailed overview of a single step of the algorithm we also refer to Figure1later in this work.

We complement the main result with two lower bounds. First, we show that the dependency on_rin the exponent is probably unavoidable.

Theorem 1.2. Multicut in DAGs, parameterized by the size of the cutset

only, is_W[1]-hard.

Thus, we complete the picture of parameterized complexity of Multicut in DAGs. We hope that it is a step toward fully understanding the parameterized

complexity of Multicutin general directed graphs.

As a related result, we establish NP-completeness of Skew Multicut, which is

the special case of Multicut in DAGswhere we are givendsources (s_i)d

i=1 and d sinks (_ti)d_i=1, and the set of terminal pairs is defined asT ={(_{si, tj}) : 1 ≤_i≤_j ≤

d}. The significance of this problem is mainly that it is used as a core subroutine in the FPT algorithm for Directed Feedback Vertex Set _{of Chen et al. [4].}

A polynomial-time algorithm for Skew Multicut _{might have given new insights}

into Directed Feedback Vertex Set_{, with potential consequences both for the}

running time of the FPT algorithm and, arguably, for the existence of a polynomial kernel (which is now one of the major open problems in kernelization; see, e.g., [9]).

Theorem 1.3. Skew Multicut is NP-complete even in the restricted case of

two sinks and two sources.

Organization of the paper. In section 2 we introduce some notation and recall

the framework of important separators and shadow removal. Section 3 contains our algorithm forMulticut in DAGsand the proof of Theorem 1.1, whereas section4

contains the proofs of Theorems1.2and1.3. We conclude with a discussion on open problems in section5.

2. Preliminaries. For a directed graph _G, by _V(_G) and _E(_G) we denote its vertex and arc set, respectively. For a vertex_v∈_V(_G), we define itsin-neighborhood

NG−(v) = {u: (u, v) ∈ E(G)} and out-neighborhood N

+

G(v) ={u : (v, u)∈ E(G)}; these definitions are extended to sets _X ⊆ _V(_G) by _NG−(_X) = (_v∈XNG−(_v))\_X and _NG+(_X) = (_v∈XNG+(_v))\_X. The in-degree and out-degreeof _v are defined as

|NG−(v)| and|N

+

G(v)|, respectively. In this paper we consider simple directed graphs only; if at any point a modification of the graph results in a multiple arc, we delete all copies of the arc except for one. By_Grev we denote the graph_Gwith all the arcs reversed, i.e.,_Grev = (_V(_G)_,{(_{v, u}) : (_{u, v})∈_E(_G)}).

A path in _G is a sequence of pairwise distinct vertices_P = (_v1, v2, . . . , vd) such that (_{vi, vi}+1)∈_E(_G) for any 1≤_{i < d}. If _v1 is the first vertex of the path _P and

vd is the last vertex, we say thatP is a v1vd-path. We extend this notion to sets of vertices: if _v1 ∈ X and vd ∈ Y for some X, Y ⊆ V(G), then P is an XY-path as well. For a path _P = (_v1, v2, v3, . . . , vd) the verticesv2, v3, . . . , vd−1 are the internal vertices of_P. The set of internal vertices of a path _P is theinterior of_P. We say

that a vertex _v is reachablefrom a vertex _uin _Gif there exists a_uv-path in_G. As the considered digraphs are simple, each path_P = (_v1, v2, . . . , vd) has a uniquefirst

arc (_v1, v2) and a uniquelast arc (vd−1, vd).

Let (_G,T_{, p}) be a Multicut in DAGsinstance with a set of r terminal pairs

T ={(_{si, ti}) : 1≤_i≤_r}. We call the terminals_si source terminalsand the terminals

tisink terminals. We letTs={si: 1≤i≤r},Tt={ti: 1≤i≤r}andT =Ts∪Tt. Fix a topological order of _G. For any sets _{X, Y} ⊆ _V(_G), we may order the vertices of_X and _Y with respect toand compare_X and _Y lexicographically; we refer to this order on subsets of_V(_G) as thelexicographical order.

A set_Z ⊆_V(_G) is called a multicutin (_G,T_{, p}) if_Z contains no terminals, but for each 1≤_i≤_r,_tiis not reachable from_siin_G\_Z. Given aMulticut in DAGs

instance I = (_G,T_{, p}) a multicut _Z is called a solution if |_Z| ≤ _p. A solution _Z is called a lex-min solution if _Z is lexicographically minimum solution in I among solutions of minimum possible size.

Consider the following easy reduction.

Lemma 2.1. There exists a polynomial-time algorithm that, given a Multicut

in DAGsinstance(G,T, p), computes an equivalent instance(G,T, p) such that

1. |T |=|T| and_p=_p;

2. T={(_{si, ti}) : 1≤_i≤_r} and all terminals _si and_ti are pairwise distinct;

3. for each1≤_i≤_rwe have _NG−(s_i) = 0and N_G+(t_i) = 0.

Proof. To construct the graph_G, start with the graph_Gand for each terminal_vof

T replace_vwith_p+1 copies of_v(i.e., vertices with the same in- and out-neighborhoods as _v; these copies are not terminals in T). Moreover, for each 1≤_i≤_r, add to_G a new terminal_si with arcs {(_{si, u}) : _u∈ _NG+(_si)} and a new terminal _ti with arcs

{(_{u, ti}) :_u∈_NG−(_ti)}. Finally, takeT={(_{si, ti}) : 1≤_i≤_r} and_p =_p.

To see the equivalence, first take a multicut_Z in (_G,T_{, p}) and an arbitrary_siti -path _P = (_{si, v1, v2, . . . , vd, ti}) in _G. Assume without loss of generality (w.l.o.g.) that_P visits at most one copy of each terminal_vin T, or else_P contains a shorter

siti-path as a strict subpath. The path P induces a path P = (si, v1, v2, . . . , vd, ti) in _G: _vk =_vk if_vk is present in_G, and_vk =_v if _vk is one of the _p+ 1 copies of a terminal_v in T. The path_P is intersected by_Z on some nonterminal vertex _vk; as

vk =vk,Z intersectsP. We infer thatZ is a multicut in (G,T, p) as well. In the other direction, let _Z be a multicut in (_G,T_{, p}) of size at most _pand let_P = (_{si, v}1, v2, . . . , vd, ti) be an arbitrarysiti-path in G. Construct a path P = (_{si, v1, v2, . . . , vd, ti}) by taking _vk =_vk if_vk is a nonterminal in (_G,T_{, p}) and taking

vk to be one of the copies of vk that is not contained in Z otherwise; note that at least one copy is not contained in_Z, as|_Z| ≤ _p. As _Z is a multicut in (_G,T_{, p}),

Z intersects_P. By the choice of the internal vertices of_P,_Z intersects_P on some vertex _vk = _vk for _vk being a nonterminal in (_G,T_{, p}). Therefore _Z ∩_V(_G) is a multicut in (_G,T_{, p}).

In our algorithm, the set of terminal pairsT is never modified, and we never add an arc into a source or out of a sink. Thus, we may assume that during the course of our algorithm all terminals are pairwise distinct and that _NG−(_si) =_NG+(_ti) =∅ for all 1≤_i≤_r.

For _v ∈_V(_G), by _S(_{G, v}) we denote set of source terminals _si for which there exists an_siv-path in_G. For a set_S⊆_Ts_{byV(G, S) we denote the set of nonterminal} vertices_v for which_S(_{G, v}) =_S.

2.1. Important separators and shadows. In the rest of this section we recall the notion of important separators by Marx [12], adjusted to the directed case by

Chitnis, Hajiaghayi, and Marx [6], as well as the shadow removal technique of Marx and Razgon [13] and Chitnis, Hajiaghayi, and Marx [6,5].

Definition 2.2 (separator [6, Definition 2.2]). Let G be a directed graph with

terminals _T ⊆_V(_G). Given two disjoint nonempty sets _{X, Y} ⊆_V(_G), we call a set

Z ⊆V(_G) an _X−_Y separatorif (i) _Z∩_T =∅,(ii) _Z∩(_X∪_Y) =∅,(iii) there is no path from _X to_Y in_G\_Z. An_X−_Y separator_Z is calledminimalif no proper subset of _Z is an _X−_Y separator.

By cut_G(_{X, Y}) we denote the size of a minimum_X−_Y separator in_G; cut_G(_{X, Y}) =∞if_Gcontains an arc going directly from_X to_Y (recall that in our instances all terminals have in- or out-degree zero). By Menger’s theorem, cut_G(_{X, Y}) equals the maximum possible size of a family of_XY-paths with pairwise disjoint interiors.

Definition 2.3 (important separator [6, Definition 4.1]). Let G be a directed

graph with terminals _T ⊆ _V(_G) and let _{X, Y} ⊆ _V(_G) be two disjoint nonempty

sets. Let _Z and _Z be two _X −_Y separators. We say that _Z is behind _Z if any

vertex reachable from _X in _G\_Z is also reachable from _X in _G\_Z. A minimal

X−Y separator _Z is animportant separatorif no other_X−_Y separator_Z satisfies |Z| ≤ |Z| while being also behind_Z.

We also need some known properties of minimum size cuts (cf. [3,6]).

Lemma 2.4 ([6, Lemma B.4]). Let G be a directed graph with terminals T ⊆

V(_G). For two disjoint nonempty sets _{X, Y} ⊆ _V(_G), there exists exactly one

mini-mum size important _X−_Y separator.

Definition 2.5 (_{closest mincut}). _{Let G be a directed graph with terminals}

T ⊆ V(_G). For two disjoint nonempty sets _{X, Y} ⊆ _V(_G), the unique minimum

size important _X−_Y separator is called the_X−_Y mincut closest to_Y. The_X−_Y

mincut closest to_X is the _Y −_X mincut closest to_X in _Grev.

Lemma 2.6. _{Let Gbe a directed graph with terminals T} ⊆_{V(G)and letX, Y} ⊆

V(_G) be two disjoint nonempty sets. Let _B be the unique minimum size important

X−Y separator, that is, the_X−_Y mincut closest to_Y, and let_v∈_Bbe an arbitrary vertex. Construct a graph_G from_Gas follows: delete_vfrom_Gand add an arc(_{x, w})

for each _x∈_X and_w∈ _NG+(_v)\_X, where _X is the set of vertices reachable from

X in_G\_B. Then the size of any _X−_Y separator in _G is strictly larger than |_B|. Proof. The claim is obvious if_Y ∩_NG+(_v)=∅, as then _G contains a direct arc from_X to_Y. Therefore, let us assume that_Y ∩_NG+(_v) =∅.

We prove the lemma by exhibiting more than |_B| _XY-paths in _G that have pairwise disjoint interiors. Recall that_X ⊇_X is the set of vertices reachable from

X in _G\_B; note that_NG+(_X) =_B. Since _B is a minimum size _X−_Y separator, there exist a set of _XY-paths (_Pu)_u∈B such that_Pu intersects _B only in _uand the interiors of paths _Pu are pairwise disjoint. Note that each path _Pu can be split into two parts: _PuX, between_X and_u, with all vertices except for_ucontained in_X, and

PY

u, betweenuandY, with all vertices except forucontained inY=V(G)\(X∪B). Consider a graph _G defined as follows: we take the graph _G[_V(_G)\_X] and add a terminal_sand arcs (_{s, w}) for all_w∈_NG+(X) = (B\ {v})∪(N_G+(v)\X). Let

B be a minimum size_s−_Y separator in_G.

We claim that_B is an _X−_Y separator in_G. Let _P be an arbitrary_XY-path in _Gand let_ube the last (closest to _Y) vertex of_B on_P. Then in_G there exists a shortened version_P of_P: if _u=_v and_wis the vertex directly after_uon_P, then

P starts with the arc (_{s, w}) and then follows_P to _Y (observe that_{w /}∈_X, as_v was the last vertex of_Bon_P), while if_u=_v, then_P starts with the arc (_{s, u}) and then follows_P to_Y. As_B has to intersect_P, then_B also intersects_P.

Moreover, we claim that_B is behind_B. This follows directly from the fact that

Gdoes not contain_X, so _X is still reachable from_X in _G\_B.

As_B is an important separator,_B is behind_B, and_B=_B (as_v∈_B\_B), we have that|_B|_>|_B|. Therefore, there exists a familyP of at least|_B|+ 1_sY-paths in _G that have pairwise disjoint interiors. Observe that all these paths are disjoint with_X by the construction of_G. Each path from P that starts with an arc (_{s, w}) for_w∈_NG+(_v)\_X is present (with the first vertex replaced by an arbitrary vertex of _X) in_G as well. Moreover, each other path starts with an arc (_{s, u}) for _u∈_B; in _G such a path can be concatenated with the _Xu-path _PuX. All paths _PuX are entirely contained in_X except for the endpoint_u, so we obtain the desired family of

XY-paths in_G.

We use the technique of shadows [13,6,5] to identify vertices separated from all sources in a given Multicut in DAGsinstance. We note that we do not use the

full power of the shadow removal technique in directed graphs: the delicate part of the results of Chitnis, Hajiaghayi, and Marx [6, 5] is to remove forward and reverse shadows at once; in our work we need to remove only one type of shadows, namely, forward ones.

We now recall the necessary definitions of the shadow removal technique from [6] and the improved bounds of [5]. Although we give full definitions for completeness, we will chiefly need Definition2.10and Lemma2.11in the rest of the paper.

Definition 2.7 (shadow [6, Definition 2.3]). Let Gbe a directed graph andT ⊆

V(_G)be a set of terminals. Let_Z⊆_V(_G)be a subset of vertices. Then for_v∈_V(_G)

we say that

1. _v is in theforward shadowof_Z (with respect to _T) if _Z is a _T−_v separator in_G, and

2. _v is in thereverse shadow of_Z (with respect to_T) if _Z is a_v−_T separator in_G.

Definition 2.8 (thin [6, Definition 4.4]). Let G be a directed graph and T ⊆

V(_G) a set of terminals. We say that a set _Z ⊆_V(_G) is thin in _G if there is no

v∈Z such that_v belongs to the reverse shadow of _Z\ {_v} with respect to_T.

Theorem 2.9 (derandomized random sampling [5]1). There is an algorithm that,

given a directed graph _G, a set of terminals_T ⊆_V(_G), and an integer_p, produces in time_O∗(2O(p2)_{) a familyAof size 2}O(p2)_{log|V(G)|of subsets of V(G)\T such that}

the following holds. Let _Z ⊆_V(_G)\_T be a thin set with |_Z| ≤_p and let _Y be a set such that for every _v∈_Y there is an important _v−_T separator _Zv ⊆_Z. For every such pair (_{Z, Y})there exists a set_A∈ Asuch that_A∩_Z =∅ but _Y ⊆_A.

We use Theorem 2.9 to identify vertices separated from all sources in a given

Multicut in DAGs_instance.

Definition 2.10 (_{source shadow}). _{Let (G,}T_{, p) be a} Multicut in DAGs

instance and_Z ⊆_V(_G) be a set of nonterminals in _G. We say that _v ∈_V(_G) is in

source shadowof_Z if _Z is a_Ts_{−v separator.}

Lemma 2.11 (derandomized random sampling for source shadows). There is

an algorithm that, given a Multicut in DAGsinstance (G,T, p), produces in time

O∗(2O(p2)_{)a family A of size2}O(p2)_{log|V(G)| of subsets of nonterminals ofG such}

that if (_G,T_{, p}) is a YES instance and _Z is the lex-min solution to (_G,T_{, p}), then there exists_A∈ Asuch that _A∩_Z =∅ and all vertices of source shadows of _Z in _G are contained in _A.

1_{We cite here improved yet unpublished bounds of [5]. See [6, Theorem 4.1 and section 4.3] for} the same result but with double-exponential bounds.

Proof. Let_Y be the set of vertices in source shadow of_Z. To prove the lemma it is sufficient to apply Theorem2.9for the graph_Grev with terminals_Ts_{and budgetp.} Thus, we need to prove that the pair (_{Z, Y}) satisfies properties given in Theorem2.9. First, assume that_Zis not thin in_Grevw.r.t. _Ts_{. Letv∈Zbe a witness: Z\{v}} is a_v−_Tsseparator in _Grev. We infer that_S(_G\(_Z\ {_v})_{, v}) =∅and_Z\ {_v} is a multicut in (_G,T), a contradiction to the choice of_Z.

Second, take an arbitrary vertex _u ∈ _Y, that is, _uis in source shadow of _Z in

G. Let _B ⊆_Z be the set of those vertices _v ∈ _Z for which there exists a _vu-path in _G\(_Z \ {_v}). Clearly, _B is a _u−_Ts separator in _Grev. We claim that it is an important one.

If_B is not a minimal_u−_Ts_{separator inG}rev_{, letv∈B be such thatB\ {v}is} a _u−_Ts_{separator in G}rev_{; thenB\ {v} is av−T}s_{separator in G}rev _{(as there is a}

vu-path in _G\(_Z\ {_v})) and_Z\ {_v} is a multicut in (_G,T), a contradiction to the choice of_Z.

Assume then that there exists a_u−_Ts_{separatorBinG}rev _{that isbehindBand}

|B| ≤ |B|, _B =_B. We claim that _Z = (_Z\_B)∪_B is a multicut in (_G,T). This would lead to a contradiction with the choice of_Z, as|_Z| ≤ |_Z|and_Z is smaller in the lexicographical order than _Z. Assume then that _Z is not a multicut in (_G,T), that is, there is an _siti-path _P in _G\_Z for some 1≤_i ≤_r. As _Z is a multicut in (_G,T),_P contains at least one vertex_v ∈_B\_B =_Z\_Z. By the choice of_B and the fact that _B is behind_B in _Grev, we infer that there exists a_vu-path in_G\_Z. As_P contains_v, there exists an_siu-path in_G\_Z, a contradiction to fact that_B is an_u−_Tsseparator in_Grev. This concludes the proof of the lemma.

3. The algorithm. We now present the FPT algorithm forMulticut in DAGs.

The algorithm is an intricate, multistep branching process using several tools, hence we split the presentation into several parts. In section3.1, we give the basic notions and results and introduce the potential function we will use to analyze the branching process. In section3.2, we present an importantdegree reductionprocess, by which we can arrange so that the source terminals have bounded degree. Section3.3 contains an overview of the branching process, and finally section3.4gives the full details and correctness proofs. A diagram of the algorithm is given in Figure1.

3.1. Potential function and simple operations. Our algorithm consists of a number of branching steps. To measure the progress of the algorithm, we introduce the following potential function.

Definition 3.1 (potential). Given aMulticut in DAGsinstanceI= (G,T, p),

we define its potential _φ(I)as_φ(I) = (_r+ 1)_p−r_i=1cut_G(_{si, ti}).

Observe that if I = (_G,T_{, p}) is a Multicut in DAGs instance, in which

cut(_{si, ti}) _{> p} for some (_{si, ti}) ∈ T, then we can immediately conclude that I is a NO instance. Therefore, w.l.o.g. we can henceforth assume that cut(_{si, ti})≤_pfor all (_{si, ti})∈ T in all the appearing instances of Multicut in DAGs_.

In many places we perform the following simple operations on Multicut in DAGs_{instances (G,}T_{, p). We formalize their properties in subsequent lemmata.}

Definition 3.2 (_{killing a vertex}). _{For a}Multicut in DAGs_instance(G,T_{, p)}

and a nonterminal vertex_v of _G, bykilling the vertex_v we mean the following oper-ation: we delete the vertex_v and decrease _pby one.

Definition 3.3 (bypassing a vertex). For a Multicut in DAGs instance

(_G,T_{, p}) and a nonterminal vertex _v of _G, by bypassing the vertex _v we mean the following operation: we delete the vertex_v and for each in-neighbor_v− of _vand each out-neighbor _v+ of_v we add an arc (_v−_{, v}+).

Lemma 3.4. Let I = (G,T, p−1) be obtained from Multicut in DAGs

instanceI = (_G,T_{, p})by killing a vertex_v. ThenI is a YES instance if and only if I is a YES instance that admits a solution that contains_v. Moreover, _φ(I)_{< φ}(I).

Proof. Let_Z be a multicut inIthat contains_v. As_G\_Z=_G\(_Z\{_v}),_Z\{_v} is a multicut inI of size|_Z| −1. In the other direction, if_Z is a multicut inI, then

G\Z =_G\(_Z∪ {_v}) and_Z∪ {_v}is a multicut inI of size|_Z|+ 1. To see that the potential strictly decreases, note that cut_G(_{si, ti})≥ cut_G(_{si, ti})−1 for all 1 ≤_i ≤

r.

Lemma 3.5. _Let I_{= (G,}T_{, p)be obtained from} Multicut in DAGs_instance

I= (_G,T_{, p})by bypassing a vertex_v. Then

1. any multicut in I is a multicut inI as well;

2. any multicut in I that does not contain _v is a multicut in I as well;

3. _S(_{G, u}) =_S(_{G, u})for any_u∈_V(_G) =_V(_G)\ {_v};

4. _φ(I)≤_φ(I).

Proof. The lemma follows from the following observations on relations between paths in _Gand_G. For a path_P in _Gwhose first and last points are different from

v, we define _PG asP with a possible occurrence ofv removed. By the definition of

G, _PG is a path inG. In the other direction, for a path P in G, we define P_G as

a path obtained from _P by inserting the vertex_v between any consecutive vertices

vi, vi+1 for which (vi, vi+1)∈E(G)\E(G). SinceGandG are acyclic, the vertexv is inserted at most once. By the construction of_G,_PG is a path in_G.

Now, for a multicut_Z in I and an arbitrary _siti-path_P in _G, the path _PG is intersected by _Z; thus _P is intersected by _Z as well and _Z is a multicut in I. For a multicut _Z in I with _{v /}∈ _Z, and an arbitrary _siti-path _P in _G, the path _PG is intersected by _Z. As _{v /}∈_Z, we infer that _P is intersected by_Z as well and _Z is a multicut inI.

To prove the third claim, note that for any _u ∈ _V(_G), any _siu-path _P in _G yields an _siu-path _PG in G and vice versa. Finally, the last claim follows from

the fact that any family P of _siti-paths in _G with pairwise disjoint sets of internal vertices can be transformed into a similar family P = {_PG : P ∈ P} in G. Thus

cut_G(s_i, t_i)≥cut_G(s_i, t_i) for 1≤i≤r.

We note that bypassing a vertex corresponds to the torso operation of Chitnis, Hajiaghayi, and Marx [6] and, if we perform a series of bypass operations, the result does not depend on their order.

Lemma 3.6. _Let I_{= (G,}T_{, p) andX}⊆_{V(G)be a subset of nonterminals ofG.}

Let I = (_G,T_{, p})be obtained from I by bypassing all vertices of _X in an arbitrary order. Then (_{u, v})∈_E(_G)for _{u, v}∈_V(_G) =_V(_G)\_X if and only if there exists a

uv-path in _Gwith internal vertices from_X (possibly consisting only of an arc(_{u, v})). In particular,Idoes not depend on the order in which the vertices of_X are bypassed.

Proof. We perform induction with respect to the size of the set _X. For _X =∅ the lemma is trivial.

LetI= (_G,T_{, p}) be an instance obtained by bypassing all vertices of_X\{_w}in

I in an arbitrary order for some_w∈_X. Take_{u, v}∈_V(_G) =_V(_G)\_X. Assume first that (_{u, v})∈_E(_G). By the definition of bypassing, (_{u, v})∈_E(_G) or (_{u, w})_,(_{w, v})∈

E(_G). In the first case there exists a_uv-path in_Gwith internal vertices from_X\{_w} by the induction hypothesis. In the second case, by the induction hypothesis, there exist a_uw-path and a_wv-path in_G, both with internal vertices from_X\ {_w}; their concatenation is the desired_uv-path (recall that_Gis acyclic).

In the other direction, let _P be a_uv-path in _Gwith internal vertices in_X. If_w does not lie on _P, by the induction hypothesis (_{u, v})∈_E(_G). Otherwise, _P splits

into a_uw-path and a_wv-path, both with internal vertices in_X\{_w}. By the induction hypothesis (_{u, w})_,(_{w, v})∈_E(_G). By the definition of bypassing, (_{u, v})∈_E(_G) and the lemma is proved.

3.2. Degree reduction. We now introduce the second main tool used in the algorithm (the first one being the source shadow reduction of Lemma 2.11). In an instance (_G,T_{, p}), let_Bi be the _si−_ti mincut closest to _si and let _Z be a solution. If we know that a _vti-path survives in _G\_Z for some _v ∈_Bi, we may add an arc (_{v, ti}) and then bypass the vertex_v, strictly increasing the value cut_G(_{si, ti}) (and thus decreasing the potential) by Lemma 2.6. Therefore, we can branch: we either guess the pair (_{i, v}) or guess that none such exist; in the latter branch we do not decrease potential but instead we may modify the set of arcs incident to the sources to get some structure, as formalized in the following definition.

Definition 3.7 (_{degree-reduced graph}). _{For a} Multicut in DAGs_instance

(_G,T_{, p})thedegree-reducedgraph_G∗is a graph constructed as follows. For1≤_i≤_r, let _Bi be the _si−_ti mincut closest to _si. We start with _V(_G∗) = _V(_G), _E(_G∗) =

E(_G\_Ts_{) and then, for each1≤i≤r, we add an arc (s}

i, v)for all v∈Bi and for

all_v∈_1≤i≤rBi for which si∈S(G, v)butv is not reachable fromBi inG. The following two lemmata formalize the properties of the degree-reduced graph and the aforementioned branching step. Recall that we assume that each vertex_si (_ti) has in- (out-) degree zero.

Lemma 3.8 (properties of the degree-reduced graph). For any Multicut in DAGs instance I = (G,T, p) and the degree-reduced graph G∗ of I, the following

holds:

1. |_NG+∗(Ts)| ≤rp.

2. For each 1≤_i≤_r,_Bi is the_si−_ti mincut closest to_si in_G∗.

3. _φ(I) =_φ(I), where I= (_G∗_,T_{, p}).

4. _Z ⊆_V(_G) is a multicut in (_G∗_,T) if and only if _Z is a multicut in (_G,T)

satisfying the following property: for each 1 ≤ _i ≤ _r, for each_v ∈ _Bi, the vertex_v is either in_Z or_Z is an_v−_ti separator; in particular, I is a YES instance if and only ifI is a YES instance that admits a solution satisfying the above property.

5. For each_v∈_V(_G)we have_S(_G∗_{, v})⊆_S(_{G, v}); moreover, if(_{si, v})is an arc in_G∗ for some1≤_i≤_r, then_S(_G∗_{, v}) =_S(_{G, v}).

Proof. For claim 1, note that for each 1 ≤ _i ≤ _r, _NG+∗(si) ⊆

1≤i≤rBi and

|Bi|= cut_G(s_i, t_i)≤p.

For Claim2, we first prove that_Biis an_si−_tiseparator in_G∗. Assume otherwise; let_P be an_siti-path in_G∗ that avoids_Bi. Let (_{si, v}) be the first arc on _P. By the construction of_G∗, in_Gthe vertex_vis reachable from_sibut not from_Bi. Therefore, there exists an_siv-path in_G\_Bi; together with_P truncated by_siit gives an_siti-path in_G\_Bi, a contradiction.

Now note that for each 1 ≤_i ≤_r, there exist cut_G(_{si, ti}) _siti-paths in _G with pairwise disjoint interiors; each path visits a different vertex of _Bi. These paths can be shortened in_G∗ using arcs (_{si, v}) for_v∈_Bi; thus,_Bi is an_si−_ti separator in_G∗ of minimum size. The fact that it is an important _ti−_si separator in _Grev follows from the fact that_Bi⊆_NG+∗(si).

Claim 3 follows directly from claim 2, as cut_G(_{si, ti}) = cut_G∗(s_i, t_i) for all 1 ≤

i≤r.

For claim4, let_Z be a multicut in (_G∗_,T). Take arbitrary 1≤_i≤_r and let_P be an arbitrary_siti-path in_G. This path intersects_Bi; let_v be the last (closest to

ti) vertex ofBi onP. LetP∗ be ansiti-path in G∗ defined as follows: we start with the arc (_{si, v}) and then we follow _P from _v to _ti. As _Z is a multicut in (_G∗_,T),_Z intersects_P∗. We infer that_Z intersects_P and_Z is a multicut in_G. Moreover, for each 1≤_i≤_Bi and_v∈_Bi, if_{v /}∈_Z, then_Z is a _v−_ti separator in_G∗ (and in_Gas well, as _Gand_G∗ differ only on arcs incident to the sources), as otherwise_Z would not intersect an_siti-path in_G∗ that starts with the arc (_{si, v}).

In the second direction, let_Z be a multicut in (_G,T) that satisfies the conditions given in claim4. Let_P∗be an arbitrary_siti-path in_G∗. As_Biis an_si−_ti separator in_G∗,_P∗ intersects_Bi; let_v be the last (closest to_ti) vertex of_Bi on_P∗. Note that the part of the path_P∗from _vto_ti(denote it by _Pv) is present also in the graph _G. By the properties of_Z, _v ∈_Z or _Z intersects _Pv. Thus _Z intersects _P∗ and _Z is a multicut in (_G∗_,T).

To see the first part of claim5note that if (_{si, v}) is an arc in_G∗, then_si∈_S(_{G, v}): clearly this is true for _v ∈ _Bi, and otherwise _si ∈ _S(_{G, v}) is one of the conditions required to add arc (_{si, v}). The second part follows directly from the construction: if (_{si, v}) is an arc in _G∗, then _v∈_Bi for some 1≤i ≤r. Ifs_i ∈S(G, v), then either

vis reachable from some vertex of_Bi inG∗ or the arc (s_i, v) is present inG∗.

We note that in the definition of the degree-reduced graph, the arcs between a source _si and vertices in _Bi for _i = _i are added only to ensure claim 5. For the remaining claims, as well as the branching described at the beginning of the section (formalized in the subsequent lemma) it would suffice to add only arcs (_{si, v}) for 1≤_i≤_rand_v∈_Bi.

Lemma 3.9. _{There exists an algorithm that, given a} Multicut in DAGs

in-stance I = (_G,T_{, p}), in polynomial time generates a sequence of instances (I_j = (_Gj,T_{j, pj}))d_j=1 satisfying the following properties. LetI0= (_G∗_,T_{, p});

1. if _Z is a multicut in I_j for some 0 ≤ _j ≤ _d, then _Z ⊆ _V(_G) and _Z is a multicut inI too;

2. for any multicut_Z in I, there exists 0≤_j ≤_d such that_Z is a multicut in Ij too;

3. for each1≤_j≤_d,_pj=_p,T_j=T and_φ(I_j)_{< φ}(I);

4. _d≤_rp.

Proof. Let_Bibe as in Definition3.7. Informally speaking, we guess an index 1≤

i≤rand a vertex_v∈_Bisuch that_tiis reachable from_vin_G\_Z, where_Zis a solution toI(in particular,_{v /}∈_Z). If we have such_v, we can add an arc (_{v, ti}) and then bypass

ti; by the choice ofBiand Lemma2.6, the value cut(si, ti) strictly increases during this operation. The last branch—where such a choice (_{i, v}) does not exists—corresponds to the degree-reduced graph_G∗. We now proceed to the formal arguments.

For 1≤_i≤_rand_v∈_Biwe define the graph_Gi,vas follows: we first add an arc (_{v, ti}) to_Gand then bypass the vertex_v. We now apply Lemma2.6for_ti−_sicuts in the graph_Grev: _Bi is the unique minimum size important _ti−_si separator, _v∈_Bi, and _Grev_i,v contains the same set of vertices and a superset of arcs of the graph _G from the statement of the lemma. Therefore cut_Gi,v(_{si, ti})_>cut_G(_{si, ti}). Moreover, cut_Gi,v(_si, t_i) ≥ cut_G(s_i, t_i) for i = i, as adding an arc and bypassing a vertex

cannot decrease the size of the minimum separator. Therefore_φ((_Gi,v,T_{, p}))_{< φ}(I); we output (_Gi,v,T_{, p}) as one of the output instancesI_j. Clearly,_d=r_i=1|_Bi| ≤_rp. To finish the proof of the lemma we need to show the equivalence stated in the first two points of the statement.

In one direction, note that as the graphs_Gi,v are constructed from _Gby adding an arc and bypassing a vertex, any multicut in (_Gi,v,T) is a multicut in (_G,T) as well. Moreover, by Lemma3.8, claim4, any multicut ofI0is a multicut ofI as well.

In the other direction, let_Z be a solution toI. Consider two cases. First assume that there exists 1≤_i≤_rand_v∈_Bisuch that_{v /}∈_Zand_Zisnota_v−_ti separator. As _Z is a multicut in (_G,T), _Z is a _si−_v separator in _G. Therefore _Z is also a multicut in a graph _Gwith the arc (_{v, ti}) added. As _{v /}∈ _Z, by Lemma 3.5, _Z is a multicut in (_Gi,v,T). In the second case, if for each 1≤_i≤_rand _v ∈_Bi, we have

v∈Z or_Z is a_v−_ti separator in_G, we conclude that_Z is a multicut in (_G∗_,T) by Lemma3.8, claim4.

3.3. Overview on the branching step. In order to prove Theorem 1.1, we show the following lemma that encapsulates a single branching step of the algorithm.

Lemma 3.10. _{There exists an algorithm that, given a} Multicut in DAGs

instance I = (_G,T_{, p}) with |T |=_r, in time _O∗(2r+O(p2))either correctly concludes that I is a NO instance or computes a sequence of instances (I_j = (_Gj,T_{j, pj}))d_j=1

such that

1. I is a YES instance if and only if at least one instanceI_j is a YES instance;

2. for each1≤_j≤_d,_V(_Gj)⊆_V(_G), _pj ≤_p,T_j=T, and_φ(I_j)_{< φ}(I);

3. _d≤2r+O(p2)_rplog|V(G)|.

The algorithm of Theorem1.1applies Lemma3.10and solves the output instances recursively.

Proof of Theorem 1.1. Let I = (_G,T_{, p}) be a Multicut in DAGs _instance.

Clearly, if_φ(I)_<0, then cut(_{si, ti})_{> p}for some 1≤_i≤_r and the instance is a NO instance. Otherwise, we apply Lemma3.10and solve the output instances recursively. Note that the potential of I is an integer bounded by (_r+ 1)_p; thus the search tree of the algorithm has depth at most (_r+ 1)_p. Using a simple fact that for_{k, n >}1 we have

logk_n= 2klog logn≤232k3/2+13(log logn)3 = 223k3/2_no(1)_,

we obtain that the number of leaves of the search tree is bounded by

2r+O(p2)_rplog|_V(_G)|

(r+1)p

= 2O(r2p)·2O(rp3)·2O(r3/2p3/2)|_V(_G)|o(1)

=_O∗(2O(rp(r+p2)))_.

The last equality follows from the fact that_r3/2_p3/2_≤r2_p+rp3_.

In rough overview of the proof of Lemma3.10, we describe a sequence of steps, as illustrated in Figure 1, where in each step, either the potential of the instance is decreased or more structure is forced onto the instance. For example, consider Lemma 3.9. The result is a branching into polynomially many branches, where in every branch but one the potential strictly decreases, and in the remaining branch, the degrees of the source terminals are bounded. Thus we may treat this step as “creating” a degree-reduced instance. The ultimate goal is to pin down a set of not too many vertices, at least one of which is guaranteed to be deleted in an optimal solution. In total, the entire process can be seen as producing some_f(_{p, r}) branches with new instances, with a strictly decreased potential in each branch.

In somewhat more detail, let _Z be the lex-min solution to I. To illustrate the principles, consider first a single terminal_s∈_Ts_{, and assume that we know (or have} guessed) that there is a vertex _v ∈ _Z such that _S(_{G, v}) = {_s}. Let us suppose that _V(_G,{_s}) is too large to branch on directly; we reduce its size by a sequence of modifications. We first observe that for every_v∈_V(_G,{_s}) with_{v /}∈_Z, either_s

Fig. 1_{.A summary of one branching step.}

reaches _v in _G\_Z or _v is in source shadow of_Z. Lemma2.11lets us guess a set_A which contains all vertices of the latter type; we apply this lemma, bypass the vertices

A, and assume in what follows that every_v ∈_V(_G,{_s}) is either contained in _Z or reached by_sin _G\_Z.

The next observation is that under this assumption, we may (1) add an arc _sv for every_v∈_V(_G,{_s}), and (2) remove every arc_uv where_{u, v}∈_V(_G,{_s}). It can be verified that this preserves _Z as a (lex-min) solution, without creating any new