How to Secure Matchings Against Edge Failures

05/03/2018 ∙ by Viktor Bindewald, et al. ∙ 0

The matching preclusion number of a graph is the minimal number of edges whose removal destroys all perfect matchings. We provide algorithms and hardness results for the task of increasing the matching preclusion number from one to two in bipartite graphs at minimal cost. Our motivation is to make matchings of a graph robust against the failure of a single edge. Our methods rely on a close relationship to the classical strong connectivity augmentation problem. For the unit weight problem we provide a deterministic _2 n-factor approximation algorithm, as well as polynomial-time algorithms for graphs of bounded treewidth and chordal-bipartite graphs. For general weights we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm.

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1 Introduction

An augmentation problem asks for a minimum-cost set of edges to be added to a graph in order to establish a certain property. We say that a bipartite graph is robust if it admits a perfect matching after the removal of any edge. Our goal is to make a bipartite graph robust at minimal cost and we study the complexity of the corresponding augmentation problem. We refer to this problem informally as robust matching augmentation. As a motivation, note that in many situations some kind of infrastructure is already available, so we may prefer upgrading it instead of designing robust infrastructure from scratch. Assume we have some assignment-type application, such as staff or task scheduling, so our infrastructure is given in terms of a bipartite graph. The application requires that we choose a perfect matching that assigns, say, tasks to machines. By buying additional edges, we would like to ensure that no matter which edge fails, the resulting graph has a perfect matching, i.e., the infrastructure remains useable. In such an application, buying edges may correspond for example to training staff or upgrading machines.

A complementary approach to creating robust infrastructure is captured by design problems. A design problem asks for a minimum-cost subgraph with a certain property, for instance a minimum-cost -edge-connected subgraph [11, 21]. Robust matching augmentation can be stated also as a design problem, where the given infrastructure is available at zero cost and the host graph is a complete bipartite graph. In fact, our problem is a special case of the bulk-robust assignment problem, a design problem introduced in [2]. Bulk-robustness is a redundancy-based robustness concept proposed by Adjiashvili, Stiller and Zenklusen [3], which allows to specify a list of failure scenarios. The bulk-robust assignment problem is known to be -hard even if only one of two fixed edges may fail [2]. Here we consider the setting that any single edge may fail.

A central theme in our algorithmic results is the occurrence of the classical strong connectivity augmentation problem, which asks for the minimal number of arcs that are needed to make a given digraph strongly connected. It was shown by Eswaran and Tarjan that this problem admits a polynomial-time algorithm, but its edge-weighted variant is -hard [15]. We show that also for robust matching augmentation the weighted problem is much harder than its cardinality version. To this end, we give a -factor approximation algorithm for the cardinality version which is essentially tight and prove that the weighted problem admits no -factor approximation under standard complexity assumptions.

Our Contribution

Recall that we call a graph robust if it admits a perfect matching after the removal of any single edge. For a bipartite graph , we denote by the edge-set of its bipartite complement. We provide algorithms and hardness results for several restrictions of the following problem.

Robust Matching Augmentation
instance: Undirected bipartite graph that admits a perfect matching.
task: Find a set of minimum cardinality, such that the graph is robust.

By a close relation of robust matching augmentation and connectivity augmentation, we provide a deterministic -factor approximation for Robust Matching Augmentation, as well as a fixed parameter tractable (FPT) algorithm for the same problem parameterized by the treewidth of the input graph. We also give a polynomial-time algorithm for instances on chordal-bipartite graphs, which are bipartite graphs without induced cycles of length at least six. Furthermore, we show that Robust Matching Augmentation admits no polynomial-time sublogarithmic-factor approximation algorithm unless , so our approximation guarantee is essentially tight.

Let us give an overview of the high-level ideas behind our algorithmic results and make some connections to other problems. We first show that we may restrict our attention to an arbitrary fixed perfect matching of the input graph. That is, it suffices to prevent the adversary from destroying a given fixed matching. From the input graph and the perfect matching we construct an auxiliary digraph. In this digraph we select certain sources and sinks which we connect using the Eswaran-Tarjan algorithm to obtain a strongly connected subgraph. It turns out that strong connectivity in the auxiliary digraph implies robustness in the original graph. We obtain an optimal solution to our Robust Matching Augmentation instance if the selection of sources and sinks was optimal.

We model the task of properly selecting sources and sinks as a variant of the Set Cover problem with some additional structure. Given an acyclic digraph, the task is to select a minimum-cardinality subset of the sources, such that each sink is reachable from at least one of the selected sources. We refer to this problem as Source Cover and remark that its complexity may be of independent interest, since it generalizes Set Cover but is a special case of Directed Steiner Tree. We give an FPT algorithm for the Source Cover problem parameterized by the treewidth of the input graph (ignoring orientations). This FPT algorithm is single exponential in the treewidth. As a by-product, we obtain FPT algorithms for the node-weighted and arc-weighted versions of the Directed Steiner Tree problem on acyclic digraphs, which are exponential in the treewidth and linear in the number of nodes of the input graph.

Finally, we relax the requirement of having a perfect matching to having a matching of cardinality at least . In fact, all of our algorithmic results for Robust Matching Augmentation generalize to the setting where we desire to have a matching of cardinality after deleting any single edge from a graph.

We refer by Weighted Robust Matching Augmentation to the generalization of Robust Matching Augmentation, where each edge has a non-negative cost . The task is to find a minimum-cost set , such that is robust. First, we show that the approximability of Weighted Robust Matching Augmentation is closely linked to that of Directed Steiner Forest. In particular we show that an -factor approximation algorithm for Weighted Robust Matching Augmentation implies an -factor approximation algorithm for Directed Steiner Forest, where is the number of terminal pairs. By a result of Halperin and Krauthgamer [22] it follows that there is no -factor approximation for Weighted Robust Matching Augmentation, unless . On the positive side, we show that an -factor approximation for the Directed Steiner Forest problem yields an -factor approximation Weighted Robust Matching Augmentation. Hence, the algorithms from [9, 17] give an approximation guarantee of for Weighted Robust Matching Augmentation, for every .

Second, we prove a complexity dichotomy based on graph minors. Let be a class of connected graphs closed under connected minors. We show that Weighted Robust Matching Augmentation restricted to input graphs from is -complete if contains at least one of two simple graph classes, which will be defined in Section 5, and admits a polynomial-time algorithm otherwise. The polynomial-time algorithm for the remaining instance classes uses a reduction to the Directed Steiner Forest problem with a constant number of terminal pairs, which in turn admits a (slice-wise) polynomial-time algorithm due to a result by Feldman and Ruhl [16]. The terminal pairs of the instance are computed by the Eswaran-Tarjan algorithm.

Related work

Adjiashvili, Bindewald and Michaels in [2] proposed an LP-based randomized algorithm for the bulk-robust assignment problem. They claim an -factor approximation guarantee for their algorithm. Since the robust assignment problem generalizes Weighted Robust Matching Augmentation, an -factor approximation for our problem is implied. However, due to our inapproximability result for Weighted Robust Matching Augmentation, this can not be true, unless . The authors of [2] agree that their analysis is incorrect.

A connectivity augmentation problem related to strong connectivity, but of a different flavor, is the tree augmentation problem (TAP). The TAP asks for a minimum-cost edge-set that increases the edge-connectivity of a given tree from one to two. In contrast to robust matching augmentation, the TAP admits a constant-factor approximation [20]. The constant has recently been lowered to for bounded-weight instances [1, 18]. Similar to robust matching augmentation, the input graph is available at zero cost. Let us briefly remark that there is more conceptual similarity. The matching preclusion number of a graph is the minimal number of edges to be removed, such that the remaining graph has no perfect matching. Robust matching augmentation can be stated as the task of finding a minimum-cost edge-set that increases the matching preclusion number of a bipartite graph from one to two, while the TAP aims to increase connectivity from one to two. The matching preclusion number is considered to be a measure of robustness of interconnect networks [8, 10]. Determining the matching preclusion number of a graph is -hard [14, 24].

Robust perfect matchings with a given recovery budget were studied by Dourado et al. in [14]. Our notion of robustness corresponds to 1-robust -recoverable in their terminology. They provide hardness results and structural insights mainly for fixed recovery budgets, which bound the number of edges that can be changed in order to repair a matching, after a certain number of edges has been removed from the graph.

Notation

Undirected and directed graphs considered here are simple. For sets , , we denote by their disjoint union. For an undirected bipartite graph with bipartition , we denote by the edge-set of its bipartite complement. Let be a directed graph. We refer by to the arcs not present in . That is, we let . By we refer to the underlying undirected graph of . For , we write for the graph . Simple paths in graphs are given by a sequence of vertices. For graphs we write if is a subgraph of . Recall that a graph is an induced minor of a graph if it arises from by a sequence of vertex deletions and edge contractions. Similarly, the graph is a minor of if we additionally allow edge deletion. Furthermore, the graph is a connected minor of if is connected and a minor of . In general, contractions may result in parallel edges or loops, which we simply discard in order to keep our graphs simple. Let be a class of graphs. We will refer to the restriction of (Weighted) Robust Matching Augmentation to instances where the graph is bipartite, admits a perfect matching, and belongs to the class as (Weighted) Robust Matching Augmentation on . Given a set of items and sets , the Set Cover problems asks for a minimum-cardinality subset , such that each is contained in some . The incidence graph of a Set Cover instance is an undirected bipartite graph on the vertex set that has an edge if and only if the item is contained in the set .

Organization of the Paper

The remainder of the paper is organized as follows. We illustrate the relation between robust matching augmentation and strong connectivity augmentation in Section 2. Algorithms for the Source Cover problem are given in Section 3. Based on the results from Sections 2 and 3, we present our results on robust matching augmentation with unit costs in Section 4. In Section 5 we give the complexity classification for the weighted version of the problem and Section 6 concludes the paper.

2 Robust Matchings and Strong Connectivity Augmentation

In this section we give some preliminary observations on the close relationship between robust matching augmentation with unit costs and strong connectivity augmentation. For this purpose, we fix an arbitrary perfect matching and construct an auxiliary digraph that is somewhat similar to the alternating tree used in Edmond’s blossom algorithm. We show that the original graph is robust if the auxiliary graph is strongly connected (but not vice versa). Furthermore, we show that there is an optimal edge-set making the given graph robust, that corresponds to a set of arcs connecting sources and sinks in the auxiliary digraph. Finally, if no source or sink of the auxiliary digraph corresponds to a non-trivial robust part of the original graph, then we may use the algorithm for strong connectivity augmentation by Eswaran and Tarjan [15] to make the original graph robust. As a consequence, we have that Robust Matching Augmentation on trees can be solved efficiently by using the Eswaran-Tarjan algorithm. In Section 4, we will generalize this result.

Let be a bipartite graph that admits a perfect matching and let be an arbitrary but fixed perfect matching of . We call an edge critical if admits no perfect matching. Observe that an edge is critical if and only if it is not contained in an -alternating cycle. Furthermore, no edge in is critical. Since is perfect, each edge is incident to a unique vertex of . We consider the following auxiliary digraph , whose arc-set is given by

We first note that the choice of the bipartion of is irrelevant.

Fact 1.

Let , where is a bipartition of . Then is isomorphic to .

Note that we may perform the reverse construction as well. That is, from any digraph we may obtain a corresponding undirected graph and a perfect matching of such that . In fact, augmenting edges to is equivalent to augmenting arcs to .

Fact 2.

Let be the set of arcs that are not present in . Then there is a 1-to-1 correspondence between and .

(a) Graph and matching (wiggly edges).

(b) Digraph .
Figure 1: Illustration of the correspondence between the dotted edges in and .

An example of the correspondence mentioned in Fact 2 is shown in Figure 1. In order to keep our notation tidy, we will make implicit use of Fact 2 and refer to and interchangeably. Observe that for edges there is an -alternating path containing and in if and only if is connected to in . This implies the following characterization of robustness.

Fact 3.

is robust if and only if each strongly connected component of is non-trivial, that is, it contains at least two vertices.

Let be a digraph. A vertex of is called a source (sink) if it has no incoming (outgoing) arc. We refer to the set of sources (sinks) of by (). Furthermore, we denote by the condensation of , that is, the directed acyclic graph of strongly connected components of . We call a source or sink of strong if the corresponding strongly connected component of is non-trivial. From Fact 3 it follows that a subgraph of that corresponds to a strong source or a strong sink is robust against the failure of a single edge. Furthermore, observe that the choice of the perfect matching of is irrelevant in the following sense.

Fact 4.

Let and be perfect matchings of . Then is isomorphic to .

Fact 4 is of key importance for our algorithmic results, for which we generally assume that some fixed perfect matching is given. Next, we observe that for unit costs we may restrict our attention to connecting sources and sinks of in order to make robust. It is easy to check that this does not hold for general non-negative costs.

Fact 5.

Let such that is robust. Then there is some of cardinality at most , such that is robust and connects only sinks to sources of .

We remark that the construction of given in the proof of Fact 5 can be performed in polynomial time.

We denote by the minimal number of arcs to be added to a digraph in order to make it strongly connected. Eswaran an Tarjan have proved the following min-max relation [15].

Fact 6.

Let be a digraph. Then .

From the proof of Fact 6 it is easy to obtain a polynomial-time algorithm that, given a digraph , computes an arc-set of cardinality such that is strongly connected [19]. We will refer to this algorithm by Eswaran-Tarjan. The following proposition illustrates the usefulness of the algorithm Eswaran-Tarjan for Robust Matching Augmentation, and at the same time its limitations.

Fact 7.

Suppose that contains no strong sources or sinks. Then Eswaran-Tarjan computes a set of minimum cardinality such that is robust.

Fact 7 implies that Eswaran-Tarjan solves Robust Matching Augmentation on trees. If strong sources or sinks are present in , then we may or may not need to consider them in order to make robust. This is precisely what makes the problem Robust Matching Augmentation hard. We will formalize the task of selecting strong sources and sinks in terms of the Source Cover problem, which is discussed in the next section.

3 The Source Cover Problem

To present our algorithmic results in Section 4 in a concise fashion it will be convenient to introduce the Source Cover problem. Given an acyclic digraph, the Source Cover problem asks for a minimum-cardinality subset of its sources, such that each sink is reachable from at least one selected source. It is easy to see that Source Cover is a special case of the Directed Steiner Tree problem and that it generalizes Set Cover. We give a simple polynomial-time algorithm for Source Cover if the input graph is chordal-bipartite (ignoring orientations). Furthermore, we show that Source Cover parameterized by treewidth (again ignoring orientations) is FPT. As a by-product, we obtain a simple FPT algorithm for the arc-weighted and node-weighted versions of the Directed Steiner Tree problem on acyclic digraphs, whose running time is linear in the size of the input graph and exponential in the treewidth of the underlying undirected graph. To the best of our knowledge, the parameterized complexity of the general Directed Steiner Tree problem with respect to treewidth is open. For the corresponding undirected Steiner Tree problem, an FPT algorithm was given by Bodlaender et al. in [7].

The Source Cover problem is formally defined as follows.

Source Cover
instance: Weakly connected acyclic digraph with at least one arc.
task: Find a minimum-cardinality subset of the sources of , such that for each sink there is an --path in .

The assumptions that is connected and contains at least one arc are present only for technical reasons. By “flattening” the input digraph, we can turn an instance of Source Cover into a Set Cover instance as follows. Let be an acyclic digraph, where is given by

Then is the incidence graph of a Set Cover instance on , such that the feasible solutions of I and are in 1-to-1 correspondence.

As illustrated in Figure 2, useful properties of the input digraph may not be present in the corresponding flattened digraph. In particular, if has treewidth at most , then the treewidth of cannot be bounded by a constant in general. Furthermore, the graph is not necessarily balanced111A graph is called balanced if the length of each induced cycle is divisible by four. (or planar) if is. Therefore, we cannot take advantage of polynomial-time algorithms for Set Cover on balanced incidence graphs or incidence graphs of bounded treewidth. Motivated by the example in Figure 1(b) we leave as an open question, whether Source Cover on balanced graphs admit polynomial-time algorithms. By Theorem 11, the existence of such an algorithm implies a polynomial-time algorithm for Robust Matching Augmentation on balanced graphs.

(a) A digraph such that is balanced, but is not.

(b) Digraphs such that has treewidth one, but the treewidth of is unbounded.
Figure 2: Examples showing that flattening does not preserve balancedness or bounded treewidth.

3.1 Source Cover on Chordal Bipartite Graphs

We show that in contrast to the treewidth and balancedness, chordal-bipartiteness is indeed preserved by the flattening operation introduced above. From this we obtain the following result.

Theorem 8.

Source Cover on chordal-bipartite graphs admits a polynomial-time algorithm.

To prove the theorem, we show that if is chordal-bipartite, so is . The graph is the incidence graph of a Set Cover instance, whose optimal solutions correspond canonically to the optimal solutions of the Source Cover instance . It is known that Set Cover on chordal-bipartite incidence graphs (and more generally, balanced graphs) admits a polynomial-time algorithm: It is possible to use LP-methods and the fact that covering polyhedra of balanced matrices are integral, see [25, pp. 562-573]. Alternatively we can use a combinatorial algorithm by Hoffman et al. [23].

3.2 Source Cover on Graphs of Bounded Treewidth

We provide a fixed-parameter algorithm for Node Weighted Directed Steiner Tree on acyclic digraphs that is single-exponential in the treewidth of the underlying undirected graph and linear in the instance size. Since Source Cover is a restriction of Node Weighted Directed Steiner Tree on acyclic graphs, we have a polynomial-time algorithm for Source Cover parameterized by the treewidth of the underlying undirected graph. Let us first recall some definitions related to Steiner problems and tree decompositions.

Node Weighted Directed Steiner Tree
instance: Acyclic digraph , costs , terminals , root .
task: Find a minimum-cost subset , such that is connected to each terminal in .

Arc Weighted Directed Steiner Tree is the corresponding problem, where the costs are on the arcs of the graph. A tree decomposition of a graph is a tree as follows. Each node of has a bag of vertices of such that the following properties hold.

  • .

  • If and both contain a vertex , then the bags of all nodes of in the path between and contain as well. Equivalently, the tree nodes containing vertex form a connected subtree of .

  • For each edge in there is some bag that contains both and . That is, for vertices adjacent in , the corresponding subtrees have a node in common.

The width of a tree decomposition is the size of its largest bag minus one. The treewidth of is the minimum width among all possible tree decompositions of .

To solve the Node Weighted Directed Steiner Tree on acyclic digraphs, we use a simple dynamic-programming algorithm over the tree decomposition of the underlying undirected graph of the input digraph with vertices.

Theorem 9.

Node Weighted Directed Steiner Tree on acyclic digraphs can be solved in time if a tree decomposition of of width is provided.

Note that an optimal tree-decomposition of a graph can be computed in time by Bodlaender’s famous theorem [6]. Our algorithm intuitively works in the following way and is similar to the dynamic programming algorithm for Dominating Set (see, e.g., [12, Section 7.3.2]). We interpret a solution to Node Weighted Directed Steiner Tree as follows: each vertex of may be active or not. Each active vertex needs a predecessor that is also active, unless it is the root. The cost to activate a vertex is given by the cost function of the Node Weighted Directed Steiner Tree instance. Starting with all terminals active, it is easy to see that Node Weighted Directed Steiner Tree on acyclic graphs is equivalent to the problem of finding a minimum cost active vertex set satisfying the above conditions. We compute an optimal solution in a bottom-up fashion using a so-called nice tree decomposition of the input graph.

By a simple reduction, we also obtain an -time algorithm for Arc Weighted Directed Steiner Tree on acyclic digraphs. We just subdivide each arc and assign the cost of the arc to the corresponding new vertex. Each old vertex receives cost zero. This transformation does not increase the treewidth.

Furthermore, we can reduce Source Cover to Node Weighted Directed Steiner Tree by adding a new vertex and connecting to each source by an arc. The sources have cost one, while all other vertices have cost zero. The root vertex is and the set of terminals is the set of sinks. By adding only one new vertex, the treewidth is increased by at most one. As a consequence of this reduction and Theorem 9, we obtain the following result.

Corollary 10.

Source Cover can be solved in time if a tree-decomposition of of width is provided.

4 Robust Matching Augmentation

In this section we present our main results on the problem Robust Matching Augmentation. Let us first redefine the problem in a slightly different way.

Robust Matching Augmentation
instance: Bipartite graph and perfect matching of .
task: Find a minimum-cardinality set such that is robust.

Fixing the perfect matching in the instance is just for notational convenience, since we can compute a perfect matching in polynomial time and our results do not depend on the exact choice of , according to the discussion in Section 2. The next theorem is our main technical result of this section. By combining the theorem with the results in Section 3 we obtain our algorithmic results.

Theorem 11.

There is a polynomial-time algorithm that, given an instance of Robust Matching Augmentation, computes two instances and of Source Cover such that the following holds.

  1. and are induced minors of .

  2. From a solution of and a solution of we can construct in polynomial time a solution of I of cardinality .

Proof.

Let be an instance of Robust Matching Augmentation, where . Our goal is to obtain from solutions of the Source Cover instances a suitable selection of sources and sinks of , such that we can make robust by connecting the selected sources and sinks, using the algorithm Eswaran-Tarjan. Let us denote by the vertex in that is incident to an edge . Furthermore, let . We construct the Source Cover instance as follows. For each critical edge , we remove from each vertex , such that is reachable from in . Let be the resulting graph and let the Source Cover instance be given by . The construction of is as for , but with the arcs of reversed. This turns the sources of into sinks. Clearly, the acyclic digraphs of and are induced minors of , since they were constructed by deleting vertices of and contracting strong components. By Fact 3, the set of critical edges can be obtained efficiently by Tarjan’s classical algorithm for computing strongly connected components. In order to generate and , observe that and can both be obtained by applying a breadth-first search starting at each vertex of or , respectively. So it remains to prove Statement 2 and 3.

Let () be a solution to (). We show how to construct in polynomial time a solution of I of cardinality . Let be the set of vertices incident to critical edges. Moreover, let be the graph induced by the vertices of that are on -paths or on -paths in . Note that can be computed by a depth-first search applied on each source and sink. By running Eswaran-Tarjan on we obtain an arc-set such that is strongly connected. Hence, each is on some directed cycle in . From we can obtain in a straight-forward way an arc-set of the same cardinality, such that each is on some directed cycle of . For each , we add to an arc , where () is some vertex in the strong component () of . By the construction of , each is on some directed cycle of . By Fact 2 and 6 we have constructed a solution of I of cardinality . This completes the proof of Statement 3.

It remains to prove that . Suppose for a contradiction that . Without loss of generality, let attain the maximum. Due to Fact 5, we may assume that an optimal solution of I connects sources and sinks of . Let be the corresponding sources of . Then for each critical edge , the vertex must be reachable from some source . But then is a solution of of cardinality , a contradiction. ∎

As a first consequence of Theorem 11 we obtain a simple -factor approximation algorithm for Robust Matching Augmentation. We “flatten” the graph of the Source Cover instances as described in Section 3 to obtain Set Cover instances and then use the classic Greedy-Algorithm to achieve a -factor approximation.

Corollary 12.

Robust Matching Augmentation admits a polynomial-time -factor approximation algorithm, where is the number of vertices of the input graph.

In a similar fashion we obtain a polynomial-time algorithm on chordal-bipartite graphs by combining Theorems 11 and 8 and the observation that is chordal-bipartite if is. Furthermore, we give an FPT algorithm parameterized by the treewidth by combining Theorems 11 and Corollary 10 and the observation that treewidth is monotone under taking minors.

Corollary 13.

Robust Matching Augmentation admits a polynomial-time algorithm on chordal-bipartite graphs and an FPT algorithm parameterized by the treewidth of the input graph.

We now show that our algorithms are also applicable in the following more general setting. Suppose we would like to have a matching of a given cardinality in the graph, no matter which edge is deleted by the adversary.

Robust -Matching Augmentation
instance: Bipartite graph that admits a matching of size .
task: Find a minimum-cardinality set such that for , the graph admits a matching of size .

Note that if is not the size of a maximum matching, then is feasible due to the existence of a larger matching. We give a polynomial-time reduction from Robust -Matching Augmentation to Robust Matching Augmentation that increases the treewidth by at most two. On the other hand, chordal-bipartiteness of the input graph is not preserved However, the corresponding digraph contains no induced cycle of length at least six, so Theorem 8 is still applicable. By Proposition 14 and the previous corollaries, we obtain for Robust -Matching Augmentation a -factor approximation algorithm, a polynomial-time algorithm on chordal-bipartite graphs, and an FPT algorithm parameterized by the treewidth.

Proposition 14.

There is a polynomial-time reduction from Robust -Matching Augmentation to Robust Matching Augmentation, such that the following holds. Let be an instance of Robust -Matching Augmentation and let . Then

  1. and from a solution of we can construct in polynomial time a solution of I such that .

  2. If is chordal-bipartite then has no induced cycle of length at least six.

5 Weighted Robust Matching Augmentation

As shown above, Robust Matching Augmentation is tightly linked to Set Cover in terms of approximation. Our first result in this section shows that Weighted Robust Matching Augmentation is substantially more complicated, as its approximability is closely linked to Directed Steiner Forest. This problem is formally defined as follows:

Directed Steiner Forest
instance: Directed graph , terminal pairs , costs .
task: Find a minimum-cost subgraph such that for each , the vertex is connected to in .

Proposition 15.

Let be the number of vertices of the Weighted Robust Matching Augmentation instance and and be the number of vertices and terminals of the Directed Steiner Forest instance, respectively.

An -factor approximation algorithm for Weighted Robust Matching Augmentation implies an -factor approximation algorithm for Directed Steiner Forest. An - or an -factor approximation algorithm for Directed Steiner Forest imply an - or -factor approximation algorithm for Weighted Robust Matching Augmentation, respectively.

On the one hand this result implies an -factor approximation algorithm for Weighted Robust Matching Augmentation for every , due to [9, 17], who achieve a guarantee of , for every . On the other hand, an algorithm achieving a guarantee of or better for Weighted Robust Matching Augmentation implies a better approximation algorithm for Directed Steiner Forest, as the number of distinct terminal pairs is at most and the current best approximation factor in terms of is due to Berman et al. [4]. Additionally, by a result of Halperin and Krauthgamer [22], the above proposition implies the following lower bound.

Corollary 16.

For every Weighted Robust Matching Augmentation does not admit a -factor approximation algorithm unless .

Given this negative result we proceed to the analysis of structural restrictions that make Weighted Robust Matching Augmentation more accessible. The main result of this section is a classification of the complexity of the problem Weighted Robust Matching Augmentation on minor-closed graph classes. In particular we show that the problem is -hard on a minor-closed class of graphs if and only if contains at least one of the two graph classes and , which we will define next. Let be the star graph with leaves and let be the path on vertices. For any graph let be the graph obtained by attaching a leaf to each vertex of . Then and . Note that each graph in and has a unique perfect matching. See Figure 3 for an illustration of the graphs and .

Figure 3: The graphs and , each with its unique perfect matching.
Lemma 17.

Weighted Robust Matching Augmentation is -hard on each of the classes and .

We complement Lemma 17 by showing that Weighted Robust Matching Augmentation on a class of graphs admits a polynomial-time algorithm if contains neither nor .

Theorem 18.

Let be a class of connected graphs that is closed under connected minors. Then Weighted Robust Matching Augmentation on admits a polynomial-time algorithm if and only if there is some such that contains neither the graph nor . The only if part holds under the assumption that .

In order to prove Lemma 17, we first show that Weighted Robust Matching Augmentation is -hard for graphs consisting only of a perfect matching by a reduction from Robust Matching Augmentation. The hardness of Weighted Robust Matching Augmentation on and follows from this result.

Before we give the proof of Theorem 18, we need the following key lemma. The polynomial-time algorithm described in the proof of the lemma uses the fact that Directed Steiner Forest can be solved in polynomial time if the number of terminal pairs is constant [16].

Lemma 19.

Let be constant and let be a class of perfectly matchable trees, each with at most leaves. Then Weighted Robust Matching Augmentation on admits a polynomial-time algorithm.

We remark that the running time of the algorithm given in Lemma 19 slicewise polynomial in the number of leaves of the input graph. We can now state the proof of our main result.

Proof of Theorem 18.

According to Lemma 17, Weighted Robust Matching Augmentation is NP-hard if completely contains the class or the class . Assuming , this proves the only if statement of the theorem.

To see the if statement, let us consider such that does not contain or . First we will reduce the problem to the case when contains only trees. For this, let be the class of all trees in that admit a perfect matching.

Claim 1.

There is a polynomial time reduction of Weighted Robust Matching Augmentation on to Weighted Robust Matching Augmentation on .

The key idea for the proof is to define an equivalent instance on an arbitrary tree of on an adapted cost function. We may hence restrict our attention to Weighted Robust Matching Augmentation on the class . As the next claim shows, the relevant trees contained in have a bounded number of leaves.

Claim 2.

There is some number depending only on such that every tree in has at most many leaves.

According to the above claims, there is a polynomial reduction of Weighted Robust Matching Augmentation on to Weighted Robust Matching Augmentation on a class of trees with a bounded number of leaves. Hence, Lemma 19 implies that Weighted Robust Matching Augmentation on can be solved in polynomial time. ∎

6 Conclusion

We presented algorithms for the task of securing matchings of a graph against the failure of a single edge. For this, we established a connection to the classical strong connectivity augmentation problem. Not surprisingly, the unit weight case is more accessible, and we were able to give a -factor approximation algorithm, as well as polynomial-time algorithms for graphs of bounded treewidth and chordal-bipartite graphs. For general non-negative weights, we showed a close relation to Directed Steiner Forest in terms of approximability and gave a dichotomy theorem characterizing minor-closed graph classes which allow a polynomial-time algorithm.

In our opinion, the case of a single edge failure is well understood now and so one might go for the case of a constant number of edge failures next. Let us remark that if the number of edge failures is a part of the input, even checking feasibility is -hard [14, 24].

References

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Appendix A Omitted Proofs

a.1 Proofs Omitted from Section 2

Proof of Fact 4.

Let and be two distinct perfect matchings of . Then their symmetric difference is a sum of -alternating cycles. But each cycle is in some strong component of and , so both condensations must be isomorphic. ∎

Proof of Fact 5.

Let be an arc in . Let be a copy of , where the arc is replaced by an arc from a sink of reachable from to a source of from which is reachable. We show that is robust. Suppose for a contradiction that this is not the case. Then there is some edge , such that , , and is not on an -alternating cycle in . Equivalently, is not contained in a directed cycle of . However, since is robust, we have that and the arc are contained in some directed cycle of . That is, there are , such that , , and . Let () be a path connecting and ( and ). Then is a closed walk that contains a simple directed cycle visiting . This contradicts our assumption that is not on a directed cycle in . By iterating this argument we obtain an arc-set such that and is robust. By construction, contains only arcs that connect sources and sinks of . ∎

Proof of Fact 7.

By assumption, we have that contains no strong sources or sinks. Therefore, each source and each sink of corresponds to a critical edge of the matching . Let of minimum cardinality, such that is robust. By Fact 5, we may assume that connects only sinks to sources of . If , then at least one sink or at least one sources is not incident to an arc of . Therefore, the graph is not robust. ∎

a.2 Proofs Omitted from Section 3

Proof of Theorem 8.

Let be a Source Cover instance such that is connected, has at least one arc, and contains no induced cycle of length at least six. If is chordal-bipartite, then we can apply the polynomial-time algorithm for Set Cover on chordal-bipartite incidence graphs, see [25, pp. 562-573] and [23]. It remains to show that is chordal-bipartite. Suppose for a contradiction, that contains an induced cycle , where and , and . In order to keep the notation concise, let .

Since is a cycle in connecting sources and sinks, we have that for , there are directed paths and in such that connects to and connects to . We now construct a cycle in and then show that is chordless and has length at least . Let be any shortest path from to in . Let us assume we already constructed the paths and for . We now define the paths and in the following way: is a shortest path from to in . If there exist more than one shortest path, then we pick the one whose endpoint is closest to on . We refer to this endpoint by . Similarly, is a shortest path from to in . If there is more than one shortest path, then we pick the one whose starting point is closest to on . We refer to this starting point by . Finally (= ) is a shortest path from to . Again, if there is more than one such shortest path, then we first pick the one whose starting point is closest to on and then whose endpoint is closest to on . We refer to these two vertices by and , respectively. Now let .

We have that is by construction a cycle in . Note that and , since otherwise were adjacent to or were adjacent to in . Therefore, is simple and has length at least . Now assume for a contradiction that has some chord . Observe that connects two distinct paths and (without loss of generality, and ) only if and or and , respectively, since otherwise is not chordless. On the other hand and contradicts the choice of the starting vertex of on . Similarly, and contradicts the choice of the endvertex of on . Therefore, is an induced cycle in of length at least , which contradicts our assumption that has no induced cycles of length . ∎

a.3 Source Cover on graphs with bounded treewidth

We now present the -time algorithm for Node Weighted Directed Steiner Tree on acyclic digraphs that is single-exponential in the treewidth of the underlying undirected graph and linear in the instance size. Let us first again recall some definitions. The problem node-weighted Directed Steiner Tree problem is defined as follows.

Node Weighted Directed Steiner Tree
instance: Acyclic digraph , costs , terminals , root .
task: Find a minimum-cost subset , such that is connected to each terminal in .

Our algorithm is presented best using a so-called nice tree decomposition. This kind of decomposition limits the structure of the difference of two adjacent nodes in the decomposition. Formally, consider a tree decomposition of a graph , rooted in a leaf of . We say that is a nice tree decomposition if every node is of one of the following types.

  • Leaf: has no children and .

  • Introduce: has exactly one child and there is a vertex of with .

  • Forget: has exactly one child and there is a vertex of with .

  • Join: has two children and such that .

Such a nice decomposition is easily computed given any tree decomposition of . We define to be the subtree of rooted in : the tree of all vertices not connected to the root in the forest , together with . By we denote the set of vertices contained in all bags of nodes in .

A coloring of a bag is a mapping , where the individual colors have the following meaning.

  • Active and already covered, represented by a 1, means that the vertex is active and that there is at least one predecessor of it that is either labeled 1 or .

  • Active and not yet covered, represented by a , means that the vertex is active but every predecessor is labeled 0.

  • Not active, represented by a , means that the vertex is not contained in the solution.

Note that there are colorings of the bag . For a coloring of we denote by the minimum cost222Here, a vertex has a cost if it is colored or and 0 otherwise. of a coloring satisfying the following conditions.

  1. each vertex in is colored 1, or 0 according to .

  2. every vertex of is colored 0 or 1.

  3. each sink is colored either 1 or .

  4. each with is either a source or at least one predecessor of in is colored either or .

To present the individual steps of the algorithm, assume that we are given a nice tree decomposition of our input graph. Let us say we are currently considering the node in and distinguish between the type of node .

  • Leaf: put if it is not the root.

  • Introduce: let be the unique child of and let such that . The value depends on the type of vertex is and on the coloring of . By definition, sinks have to be active and therefore the optimal value is if . The same is true for sources labeled in (those do not have predecessors and need to be labeled either 1 or 0). Finally, we set the cost to be if is labeled 1 in and not a source, but non of its predecessors is active in . Thus we set

    (1)

    where the pair is compatible to if the following conditions hold.

    • If , then . As the introduced vertex is not considered to be part of the solution, we can simply keep the coloring of the child node.

    • If , then , , and . This condition makes sure that the introduced vertex can only be labeled if none of its predecessors is labeled 1 or .

    • If , then , , and, moreover, or . This conditions says that the introduced vertex can only be labeled if at least one of its predecessors is labeled 1 or , unless it is a source.

  • Forget: let be the unique child of and let such that . Then we put

    We do not allow a vertex labeled to be forgotten, as we can not assure to cover it in later bags. For the remaining cases we simply keep the optimal value.

  • Join: let and be the two children of the join node with . We put

    (2)

    where the minimum runs over all colorings of and of with and .

  • Root: as the graph is connected and the root node is a leaf, the root node is a forget node, where its child node contains exactly one vertex in its bag. The algorithm terminates with the output

where is the unique coloring of the empty bag .

Having presented the algorithm, we need to prove Theorem 10 by showing the correctness and bounding the running time of the algorithm.

Proof of Theorem 10.

We need to show that the algorithm works correctly and is fixed parameter tractable when parameterized by the treewidth of the underlying graph. Let be a nice tree decomposition of of width with nodes.

Claim 1.

The algorithm correctly computes an optimal solution to Node Weighted Directed Steiner Tree in Acyclic Graphs.

Proof.

We show the statement by a straight-forward inductive proof on the decomposition tree. The induction hypothesis states that is the minimum cost of a solution induced by the vertices of , satisfying the conditions (a)-(d) (see p. A.3). The base case are the leaf nodes where the hypothesis clearly holds. Now let the induction hypothesis be true for all descendants of . We distinguish between the remaining three node types and argue that the induction hypothesis holds in .

  • Introduce: let be the unique child of the introduce node and let such that . Clearly (a) holds and (b) holds by the induction hypothesis. By putting to if for a sink , (c) also holds.

    For (d) observe that the notion of compatibility is defined correctly. If this is trivial. For observe that has to satisfy the condition that . Thus the condition (d) holds for . Now for a given coloring we have to check if is calculated correctly. This is true for the cases in which is set to . So it remains to show that we identify all compatible colorings for to calculate the minimum. The case is trivial. For the cases observe that has to satisfy and . Calculating the minimum over all pairs compatible to is hence correct. Finally it is clear that that we have to add to the minimum of all compatible colorings for if .

  • Forget: let be the unique child of and let such that . For a forget node we put . Clearly (a), (c) and (d) hold by the induction hypothesis. (b) also holds as we only allow colorings that satisfy . Finally it is easily verified that the calculation of is correct.

  • Join: let and be the two children of the join node with . By (2), a vertex may only be colored if it is colored 1 either in or . As the induction hypothesis holds for and , (a)-(d) also hold for . It remains to show that is calculated correctly. The considered colorings and of and have to satisfy