How did the shape of your network change? (On detecting anomalies in static and dynamic networks via change of non-local curvatures)

08/16/2018 ∙ by Bhaskar DasGupta, et al. ∙ University of Illinois at Chicago 0

Notions of local and non-local curvatures of higher-dimensional geometric shapes and topological spaces play a fundamental role in physics and mathematics in characterizing anomalous behaviours of these higher dimensional entities. However, using curvature measures to detect anomalies in networks is not yet very common due to several reasons such as lack of preferred geometric interpretation of networks and lack of experimental evidences that may lead to specific desired curvature properties. To this end, our goal in this paper to formulate and analyze curvature analysis methods to provide the foundations of systematic approaches to find critical components and anomaly detection in networks. For this purpose, we use two measures of network curvatures: (i) the Gromov-hyperbolic combinatorial curvature based on the properties of exact and approximate geodesics and higher-order connectivities, and (ii) geometric curvatures based on defining k-complex-based Forman's combinatorial Ricci curvature for elementary components, and using Euler characteristic of the complex that is topologically associated with the given graph. Based on these measures, we precisely formulate several computational problems related to anomaly detection in static or dynamic networks, and provide non-trivial computational complexity results (e.g., exact or approximation algorithms, approximation hardness) for these problems.

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1 Introduction

Useful insights for many complex systems are often obtained by representing them as networks and analyzing them using graph-theoretic and combinatorial algorithmic tools [25, 57, 1]

. In principle, we can classify these networks into

two major classes:

  1. [label=]

  2. Static networks that model the corresponding system by one fixed network. Examples of such networks include biological signal transduction networks without node dynamics, and most social networks.

  3. Dynamic networks where elementary components of the network (such as nodes or edges) are added and/or removed as the network evolves over time. Examples of such networks include biological signal transduction networks with node dynamics, causal networks reconstructed from DNA microarray time-series data, biochemical reaction networks and dynamic social networks.

Typically, such networks may have so-called critical (elementary) components whose presence or absence alters some significant non-trivial non-local property of these networks. For example:

  1. [label=]

  2. For a static network, there is a rich history in finding various types of critical components dating back to quantifications of fault-tolerance or redundancy in electronic circuits or routing networks. Recent examples of practical application of determining critical and non-critical components in the context of systems biology include quantifying redundancies in biological networks [51, 67, 5] and confirming the existence of central influential neighborhoods in biological networks [2].

  3. For a dynamic network, critical components may correspond to a set of nodes or edges whose addition and/or removal between two time steps alters a significant topological property of the network. Popularly also known as the anomaly detection or change-point detection [7, 48] problem, these types of problems have been studied over the last several decades in data mining, statistics and computer science mostly in the context of time series data with applications to areas such as medical condition monitoring [75, 14], weather change detection [28, 60] and speech recognition [20, 63].

In this paper we seek to address research questions of the following generic nature:

“Given a static or dynamic network, identify the critical components of the network that “encode” significant non-trivial global properties of the network”.

To identify critical components, one first needs to provide details for following four specific items: 0.6cm

[leftmargin=0.3cm]

(i)

network model selection,

(ii)

network evolution rule for dynamic networks,

(iii)

definition of elementary critical components, and

(iv)

network property selection (i.e., the global properties of the network to be investigated).

The specific details for these items for this paper are as follows: 0.6cm

[leftmargin=0.3cm]

(i) Network model selection:

Our network model will be undirected graphs.

(ii) Network evolution rule for dynamic networks:

Our dynamic networks follow the time series model and are given as a sequence of networks over discrete time steps, where each network is obtained from the previous one in the sequence by adding and/or deleting some nodes and/or edges.

(iii) Critical component definition:

Individual edges are elementary members of critical components.

(iv) Network property selection:

The network measure for this paper will be appropriate notions of “network curvature”. More specifically, we will use (a) Gromov-hyperbolic combinatorial curvature based on the properties of exact and approximate geodesics distributions and higher-order connectivities and (b) geometric curvatures based on identifying network motifs with geometric complexes (“geometric motifs” in systems biology jargon) and then using Forman’s combinatorializations.

1.1 Some basic definitions and notations

For an undirected unweighted graph of nodes , the following notations related to are used throughout:

  1. [label=]

  2. denotes a path of length consisting of the edges , , , .

  3. and denote a shortest path and the distance (i.e., number of edges in ) between nodes and , respectively.

  4. denotes the diameter of .

  5. denotes the graph obtained from by removing the edges in from .

A -approximate solution (or simply an -approximation) of a minimization (resp., maximization) problem is a solution with an objective value no larger than (resp., no smaller than) times (resp., times) the value of the optimum; an algorithm of performance or approximation ratio produces an -approximate solution. A problem is -inapproximable under a certain complexity-theoretic assumption means that the problem does not admit a polynomial-time -approximation algorithm assuming that the complexity-theoretic assumption is true. We will also use other standard definitions from structural complexity theory as readily available in any graduate level textbook on algorithms such as [69].

1.2 Why use network curvature measures?

Prior researchers have proposed and evaluated a number of established network measures such as degree-based measures (e.g., degree distribution), connectivity-based measures (e.g., clustering coefficient), geodesic-based measures (e.g., betweenness centrality) and other more novel network measures [21, 52, 5, 9] for analyzing networks. The network measures considered in this paper are “appropriate notions” of network curvatures. As provably demonstrated in published research works such as [2, 73, 72, 65], these network curvature measures saliently encode non-trivial higher-order correlation among nodes and edges that cannot be obtained by other popular network measures. Some important characteristics of these curvature measures that we consider are [2, Section (III)][47]:

  1. [label=]

  2. These curvature measures depend on non-trivial global network properties, as opposed to measures such as degree distributions or clustering coefficients that are local in nature or dense subgraphs that use only pairwise correlations.

  3. These curvature measures can mostly be computed efficiently in polynomial time, as opposed to measures such as community decompositions, cliques or densest--subgraphs.

  4. When applied to real-world biological and social networks, these curvature measures can explain many phenomena one frequently encounters in real network applications that are not easily explained by other measures such as:

    1. [label=]

    2. paths mediating up- or down-regulation of a target node starting from the same regulator node in biological regulatory networks often have many small crosstalk paths, and

    3. existence of congestions in a node that is not a hub in traffic networks.

    Further details about the suitability of our curvature measures for real biological or social networks are provided in Section 2.1.1 for Gromov-hyperbolic curvature and Section 2.2.3 for geometric curvatures.

Curvatures are very natural measures of anomaly of higher dimensional objects in mainstream physics and mathematics [15, 11]. However, networks are discrete objects that do not necessarily have an associated natural geometric embedding. Our paper seeks to adapt the definition of curvature from the non-network domains in a suitable way for detecting network anomalies. For example, in networks with sufficiently small Gromov-hyperbolicity and sufficiently large diameter a suitably small subset of nodes or edges can be removed to stretch the geodesics between two distinct parts of the network by an exponential amount leading to extreme implications on the expansion properties of such networks [10, 24], which is akin to the characterization of singularities (an extreme anomaly) by geodesic incompleteness (i.e., stretching all geodesics passing through the region infinitely) [39]. It is our hope that research works in this paper will stimulate further research concerning the exciting interplay between curvatures from network and non-network domains, a much desired goal in our opinion.

1.2.1 Scalar vs. vector curvature

In this paper, we consider a scalar curvature measure

. The standard Gromov-hyperbolic curvature measure is always a scalar value. Geometric curvatures however could also be defined by a vector by looking at local curvatures at all

elementary components (e.g., nodes or edges) of a network, and defining the overall curvature as a vector of these values. We leave algorithmic analysis of such geometric vector curvatures, which seems to require considerably different combinatorial and optimization tools, for future research. Both scalar and vector versions of curvatures are used in physics and mathematics to study higher-dimensional objects with their own pros and cons. For example, for a two-dimensional curve, the standard curvature as defined by Cauchy is a scalar curvature whereas the normal vector used in the study of differential geometry of curves is a vector curvature. Even though a casual glance may seem to suggest that the scalar curvature is a weak concept with inadequate influence on the global geometry of the higher-dimensional object that is being studied, there exists non-trivial results (e.g., the positive mass theorem of Schoen, Yau and Witten) that suggest that this may not be the case.

1.3 Why only the edge-deletion model?

In this paper we add or delete edges from a network while keeping the node set the same. This scenario captures a wide variety of applications such as inducing desired outcomes in disease-related biological networks via gene knockout [64, 77], inference of minimal biological networks from indirect experimental evidences or gene perturbation data [4, 3, 70], and finding influential nodes in social and biological networks [5], to name a few. However, the node addition/deletion model or a mixture of node/edge addition/deletion model is also significant in many other applications. Although some of our complexity results can be easily extended for bounded-degree graphs to the node deletion model, we do not outline these generalizations here but leave it as a separate future research topic.

1.4 Two examples in which curvature measures detect anomaly where other simpler measures do not

It is obviously practically impossible to compare our curvatures measures for anomaly detection with respect to every possible other network measure that has been used in prior research works. However, we do still provide two illustrative examples of comparing our curvature measures to the well-known densest subgraph measure which is defined as follows. Given a graph , the densest subgraph measure find a subgraph induced by a subset of nodes that maximizes the ratio (density) . Let denote the density of a densest subgraph of . An efficient polynomial time algorithm to compute using a max-flow technique was first provided by Goldberg [37]. We urge the readers to review the definitions of the relevant curvature measures (in Section 2) and the anomaly detection problems (in Section 3) in case of any confusion regarding the examples we provide.

1.4.1 Extremal anomaly detection for a static network

Consider the extremal anomaly detection problem (Problem Eadp in Section 3.1) for a network of nodes and edges as shown in Fig. 1 using the geometric curvature as defined by Equation (1). It can be easily verified that and . Let and suppose that we set our targeted decrease of the curvature or density value to be of the original value, i.e., we set for the geometric curvature measure and for the densest subgraph measure. It is easily verified that , thus showing . However, many more than just one edge will need to be deleted from to bring down the value of to .

Figure 1: Toy example of extremal anomaly detection discussed in Section 1.4.1.

1.4.2 Targeted anomaly detection for a dynamic biological network

Figure 2: Toy example of targeted anomaly detection discussed in Section 1.4.2.

Consider the targeted anomaly detection problem (Problem Tadp in Section 3.2) for a dynamic biological network of variables as shown in Fig. 2, where affects with a delay, using the Gromov-hyperbolic curvature (Definition 1). Suppose that the network inference from microarray data is done by incorporating a time delay of two in the hitting-set approach of Krupa [43]. It can be easily verified that , , and . Since it follows that ; however, edges will need to be deleted from to bring down the value of to .

1.5 Algebraic approaches for anomaly detection

In contrast to the combinatorial/geometric graph-property based approach investigated in this paper and elsewhere, an alternate approach for anomaly detection is the

algebraic tensor-decomposition based approach

studied in the contexts of dynamic social networks [66] and pathway reconstructions in cellular systems and microarray data integration from several sources [6, 58]. This approach is quite different from the ones studied in this paper with its own pros and cons.

1.6 Remarks on the organization of our proofs

Many of our proofs in Sections 45 are long, complicated and/or involve tedious calculations. For easier understanding and to make the paper more readable, when appropriate we have included a subsection generically titled “Proof techniques and relevant comments regarding Theorem ” before providing the actual detailed proofs. The reader is cautioned however that these brief subsections are meant to provide some general idea and subtle points behind the proofs and should not be considered as a substitution for more formal proofs.

2 Two notions of graph curvature

For this paper, a curvature for a graph is a scalar-valued function . There are several ways in which network curvature can be defined depending on the type of global properties the measure is desired to affect; in this paper we consider two such definitions as described subsequently.

2.1 Gromov-hyperbolic curvature

This measure for a metric space was first suggested by Gromov in a group theoretic context [38]. The measure was first defined for infinite continuous metric space [15], but was later also adopted for finite graphs. Usually the measure is defined via geodesic triangles as stated in Definition 1. For this definition, it would be useful to consider the given graph as a metric graph, i.e., we identify (by an isometry) any edge with the real interval and thus any point in the interior of the edge can also be thought as a (virtual) node of . Define a geodesic triangle to be an ordered triple of three shortest paths , and for the three nodes in .

Definition 1 (Gromov-hyperbolic curvature measure via geodesic triangles).

For a geodesic triangle , let be the minimum number such that lies in a -neighborhood of , i.e., for every node on , there exists a node on or such that . Then the graph has a Gromov-hyperbolic curvature (or Gromov hyperbolicity) of where .

An infinite collection of graphs belongs to the class of -Gromov-hyperbolic graphs if and only if any graph has a Gromov-hyperbolic curvature of . Informally, any infinite metric space has a finite value of if it behaves metrically in the large scale as a negatively curved Riemannian manifold, and thus the value of can be related to the standard scalar curvature of a hyperbolic manifold. For example, a simply connected complete Riemannian manifold whose sectional curvature is below has a value of (see [62]). This is a major justification of using as a notion of curvature of any metric space.

For an -node graph , and a -approximation of can be computed in and in time, respectively [32]. It is easy to see that if is a tree then . Other examples of graph classes for which is a small constant include chordal graphs, cactus of cliques, AT-free graphs, link graphs of simple polygons, and any class of graphs with a fixed

diameter. A small value of Gromov-hyperbolicity is often crucial for algorithmic designs; for example, several routing-related problems or the diameter estimation problem become easier for networks with small

values [18, 16, 17, 36]. There are many well-known measures of curvature of a continuous surface or other similar spaces (e.g., curvature of a manifold) that are widely used in many branches of physics and mathematics. It is possible to relate Gromov-hyperbolic curvature to such other curvature notions indirectly via its scaled version, e.g., see [46, 56, 45].

2.1.1 Is Gromov-hyperbolic curvature a suitable statistically significant measure for real-world networks ?

Recently, there has been a surge of empirical works measuring and analyzing the Gromov curvature of networks, and many real-world networks (e.g., preferential attachment networks, networks of high power transceivers in a wireless sensor network, communication networks at the IP layer and at other levels) were observed to have a small constant value of  [56, 59, 44, 46, 8]. The authors in [2] analyzed well-known biological networks and well-known social networks for their values and found all but one network had a statistically significant small value of . These references also describe implications of range of on the actual real-world applications of these networks. As mentioned in the following subsection, the Gromov-hyperbolicity measure is fundamentally different from the commonly used topological properties for a graph.

2.1.2 Some clarifying remarks regarding Gromov-hyperbolicity measure

As pointed out in details by the authors in [24, Section 1.2.1], the Gromov-hyperbolicity measure enjoys many non-trivial topological characteristics. In particular, the authors in [24, Section 1.2.1] point out the following:

  1. [label=]

  2. is not a hereditary or monotone property.

  3. is not necessarily the same as tree-width measure (see also [26, 2]), or other standard combinatorial properties (e.g., betweenness centrality, clustering coefficient, dense sub-graphs) that are commonly used in the computer science literature.

  4. “Close to hyperbolic topology” is not necessarily the same as “close to tree topology”.

2.2 Geometric curvatures

In this section, we describe generic geometric curvatures of graphs by using correspondence with topological objects in higher dimension.

2.2.1 Some basic topological concepts

We first review some basic concepts from topology; see introductory textbooks such as [40, 33] for further information. Although not necessary, the reader may find it useful to think of the underlying metric space as the -dimensional real space be for some integer .

  1. [label=]

  2. A subset is convex if and only if for any , the convex combination of and is also in .

  3. A set of points are called affinely independent if and only if for all and implies .

  4. The -simplex generated by a set of affinely independent points is the subset of generated by all convex combinations of .

    1. [label=]

    2. Each -subset defines the -simplex that is called a face of dimension (or a -face) of . A -face, -face and -face is called a facet, an edge and a node, respectively.

  5. A (closed) halfspace is a set of points satisfying for some . The convex set obtained by a bounded non-empty intersection of a finite number of halfspaces is called a convex polytope (convex polygon in two dimensions).

    1. [label=]

    2. If the intersection of a halfspace and a convex polytope is a subset of the halfspace then it is called a face of the polytope. Of particular interests are faces of dimensions , and , which are called facets, edges and nodes of the polytope, respectively.

  6. A simplicial complex (or just a complex) is a topological space constructed by the union of simplexes via topological associations.

2.2.2 Geometric curvature definitions

Informally, a complex is “glued” from nodes, edges and polygons via topological identification. We first define -complex-based Forman’s combinatorial Ricci curvature for elementary components (such as nodes, edges, triangles and higher-order cliques) as described in [31, 73, 72], and then obtain a scalar curvature that takes an appropriate linear combination of these values (via Gau-Bonnet type theorems [12]) that correspond to the so-called Euler characteristic of the complex that is topologically associated with the given graph. In this paper, we consider such Euler characteristics of a graph to define geometric curvature.

To begin the topological association, we (topologically) associate a -simplex with a -clique ; for example, -simplexes, -simplexes, -simplexes and -simplexes are associated with nodes, edges, -cycles (triangles) and -cliques, respectively. Next, we would also need the concept of an “order” of a simplex for more non-trivial topological association. Consider a -face of a -simplex. An order association of such a face, which we will denote by the notation with the additional subscript , is associated with a sub-graph of at most nodes that is obtained by starting with and then optionally replacing each edge by a path between the two nodes. For example,

  • is a node of for all .

  • is an edge, and for is a path having at most nodes between two nodes adjacent in .

  • is a triangle (cycle of nodes or a -cycle), and for is obtained from nodes by connecting every pair of nodes by a path such that the total number of nodes in the sub-graph is at most .

Naturally, the higher the values of and are, the more complex are the topological associations. Let be the set of all ’s in that are topologically associated. With such associations via -faces of order , the Euler characteristics of the graph and consequently the curvature can defined as

(1)

It is easy to see that both and are too simplistic to be of use in practice. Thus, we consider the next higher value of in this paper, namely when . Letting denote the number of cycles of at most nodes in , we get the measure

2.2.3 Are geometric curvatures a suitable measure for real-world networks ?

The usefulness of geometric curvatures for real-world networks was demonstrated in publications such as [73, 72, 65].

3 Formalizations of two anomaly detection problems on networks

In this section, we formalize two versions of the anomaly detection problem on networks. An underlying assumption on the behind these formulations is that the graph adds/deletes edges only while keeping the same set of nodes.

3.1 Extremal anomaly detection for static networks

The problems in this subsection are motivated by a desire to quantify the extremal sensitivity of static networks. The basic decision question is: “is there a subset among a set of prescribed edges whose deletion may change the network curvature significantly?”. This directly leads us to the following decision problem:

Problem name: Extremal Anomaly Detection Problem (Eadp)
Input: A curvature measure
A connected graph , an edge subset such that
      is connected and a real number (resp., )
Decision question: is there an edge subset such that
                                             (resp., ) ?
Optimization question: if the answer to the decision question is “yes” then minimize
Notation: if the answer to the decision question is “yes” then
      the minimum possible value of is denoted by

The following comments regarding the above formulation should be noted:

  1. [label=,leftmargin=*]

  2. For the case (resp., ) we allow (resp., ), thus need not be a feasible solution at all.

  3. The curvature function is only defined for connected graphs, thus we require to be connected.

  4. The edges in can be thought of as “critical” edges needed for the functionality of the network. For example, in the context of inference of minimal biological networks from indirect experimental evidences [4, 3], the set of critical edges represent direct biochemical interactions with concrete evidence.

3.2 Targeted anomaly detection for dynamic networks

These problems are primarily motivated by change-point detections between two successive discrete time steps in dynamic networks [7, 48], but they can also be applied to static networks when a subset of the final desired network is known. Fig. 2 illustrates targeted anomaly detection for a dynamic biological network.


Problem name: Targeted Anomaly Detection Problem (Tadp)
Input: Two connected graphs and with
A curvature measure
Valid solution: a subset of edges such that .
Objective: minimize .
Notation: the minimum value of is denoted by

4 Computational complexity of extremal anomaly detection problems

4.1 Geometric curvatures: exact and approximation algorithms for Eadp

Theorem 2.

(a) The following statements hold for Eadp when : 0.1in

(a1)

We can decide in polynomial time the answer to the decision question (i.e., if there exists any feasible solution or not).

(a2)

If a feasible solution exists then the following results hold:

(a2-1)

Computing is -hard for all that are multiple of .

(a2-2)

If is sufficient larger than then we can design an approximation algorithm that approximates both the cardinality of the minimal set of edges for deletion and the absolute difference between the two curvature values. More precisely, if for some , then we can find in polynomial time a subset of edges such that

(b) The following statements hold for Eadp when : 0.1in

(b1)

We can decide in polynomial time the answer to the decision question (i.e., if there exists any feasible solution or not).

(b2)

If a feasible solution exists and is not too far below then we can design an approximation algorithm that approximates both the cardinality of the minimal set of edges for deletion and the absolute difference between the two curvature values. More precisely, letting denote the number of cycles of of at most nodes that contain at least one edge from , if for some then we can find in polynomial time a subset of edges such that

(b3)

If then, even if (i.e., a trivial feasible solution exists), computing is at least as hard as computing Tadp and therefore all the hardness results for Tadp in Theorem 10 also apply to .

4.1.1 Proof techniques and relevant comments regarding Theorem 2

  1. [label=,leftmargin=*]

  2. (on proofs of (a1) and (b1)) After eliminating a few “easy-to-solve” sub-cases, we prove the remaining cases of (a1) and (b1) by reducing the feasibility questions to suitable minimum-cut problems; the reductions and proofs are somewhat different due to the nature of the objective function. It would of course be of interest if a single algorithm and proof can be found that covers both instances and, more importantly, if a direct and more efficient greedy algorithm can be found that avoids the maximum flow computation.

  3. (on proofs of (a2-2) and (b2)) Our general approach to prove (a2-2) and (b2) is to formulate these problems as a series of (provably -hard and polynomially many) “constrained” minimum-cut problems. We start out with two different (but well-known) polytopes for the minimum cut problem (polytopes (4) and (4)). Even though the polytope (4) is of exponential size for general graphs, it is of polynomial size for our particular minimum cut version and so we do not need to appeal to separation oracles for its efficient solution. We subsequently add extra constraints corresponding to a parameterized version of the minimization objective and solve the resulting augmented polytopes (polytopes (5) and (5)) in polynomial time to get a fractional solution and use a simple deterministic rounding scheme to obtain the desired bounds.

    1. [label=]

    2. Our algorithmic approach uses a sequence of linear-programming () computations by using an obvious binary search over the relevant parameter range. It would be interesting to see if we can do the same using computations.

    3. Is the factor in “” an artifact of our specific rounding scheme around the threshold of and perhaps can be improved using a cleverer rounding scheme? This seems unlikely for the case when since the inapproximability results in (b3) include a -inapproximability assuming the unique games conjecture is true. However, this possibility cannot be ruled out for the case when since we can only prove -hardness for this case.

    4. There are subtle but crucial differences between the rounding schemes for (a2-2) and (b2) that is essential to proving the desired bounds. To illustrate this, consider an edge with a fractional value of for its corresponding variable. In the rounding scheme (6) of (a2-2) will only sometimes be designated as a cut edge, whereas in the rounding scheme (6) of (b2) will always be designated as a cut edge.

  4. (on the bounds over in (a2-2) If then the condition on is redundant (i.e., always holds). Thus indeed the -approximation is likely to hold unconditionally for practical applications of this problem since anomaly is supposed to be caused by a large change in curvature by a relatively small number of elementary components (edges in our cases).

    Furthermore, if then the condition on always holds irrespective of the value of , and the smaller is with respect to the better is our approximation of the curvature difference. As a general illustration, when the assumptions are , and the corresponding bounds are .

  5. (on the hardness proof in (a2-1)) Our reduction is from the densest--subgraph (DS) problem. We use the reduction from the CLIQUE problem to DS detailed by Feige and Seltser in [30] which shows that DS is -hard even if the degree of every node is at most . For convenience in doing calculations, we use the reduction of Feige and Seltser starting from the still -hard version of the CLIQUE problem where the input instances are -regular -node graphs. Pictorially, the reduction is illustrated in Fig. 3. Note that DS is not known to be -inapproximable assuming P (though it is likely to be), and thus our particular reduction cannot be generalized to -inapproximability assuming P.

Figure 3: Illustration of the reduction in the proof of Theorem 2(a2-1). (a) The original instance of the -CLIQUE problem on an -node graph. (b) Illustration of the -hardness reduction from -CLIQUE to DS by Feige and Seltser [30]. (c) Illustration of associations of nodes with unique -cycles such that an edge between two nodes are adjacent correspond to sharing an unique edge of their associated -cycles. (d) Splitting of every edge of into a path of edges for the case when for some integer .

4.1.2 Proof of Theorem 2

(a1) and (a2-2) Let the notation denote the set of cycles having at most nodes in a graph . Assume and let ; thus where and . Since is fixed, and all the cycles in can be explicitly enumerated in polynomial () time. Let be the set of cycles in that involve one of more edges from . We first observe that the following sub-cases are easy to solve:

  • If then we can assert that there is no feasible solution. This is true because for any it is true that is at most .

  • If and then there exists a trivial optimal feasible solution of the following form:

    select any set of edges from where is the least positive integer satisfying .

Thus, we assume that and . Consider a subset of edges for deletion and suppose that removal of the edges in removes cycles from (i.e., ). Then,

(2)

and consequently one can observe that

(3)

Note that and is the number of edges in that are not in and therefore not selected for deletion. Also, note that is a quantity that depends on the problem instance only and does not change if one or more edges are deleted. Based on this interpretation, we construct the following instance (digraph) of a (standard directed) minimum - cut problem (where is the capacity of a directed edge ):

  • The nodes in are as follows: a source node , a sink node , a node (an “edge-node”) for every edge and a node (a “cycle-node”) for every cycle . The total number of nodes is therefore , i.e., polynomial in .

  • The directed edges in and their corresponding capacities are as follows:

    • For every edge , we have a directed edge (an “edge-arc”) of capacity .

    • For every cycle , we have a directed edge (a “cycle-arc”) of capacity .

    • For every cycle and every edge such that is an edge of , we have a directed edge (an “edge-cycle-arc”) of capacity .

For an - cut of (where and ), let and denote the edges in the cut and the capacity of the cut, respectively. It is well-known how to compute a minimum - cut of value in polynomial time [22]. The following lemma proves part (a1) of the theorem.

Lemma 3.

There exists any feasible solution of Eadp if and only if . Moreover, if is a minimum - cut of of value then is a feasible solution for Eadp.

Proof.  Suppose that there exists a feasible solution with edges for Eadp, and suppose that removal of the edges in removes cycles from . Consider the cut where

Note that no edge-cycle-arc belongs to and therefore

and thus by Inequality (3) we can conclude that

For the other direction, consider a minimum - cut of of value . Consider the solution for Eadp, and suppose that removal of the edges in removes cycles from . Since admits a trivial - cut of capacity , no edge-cycle-arc can be an edge of any minimum - cut of , i.e., contains only edge-arcs or cycle-arcs. Let . Consider an edge and let be a cycle in containing . Since contains no edge-cycle-arc, it does not contain the arc . It thus follows that the cycle-node must also belong to and thus . Now note that

This completes a proof for (a1). We now prove (a2-2). Let be an optimal solution of the optimization version of Eadp having nodes. Note that and thus in polynomial time we can “guess” every possible value of , solve the corresponding optimization problem with this additional constraint, and take the best of these solutions. In other words, it suffices if we can find, under the assumption that for some , find a solution satisfying the claims in (a2-2).

We showed in part (a1) that the feasibility problem can be reduced to finding a minimum - cut of the directed graph . Notice that is acyclic, and every path between and has exactly three directed edges, namely an edge-arc followed by a edge-cycle-arc followed by a cycle-arc. The minimum - cut problem for a graph has a well-known associated convex polytope of polynomial size (e.g., see [69, pp. 98-99]). Letting to be the variable corresponding to each node , and to be the variable associated with the edge , this minimum - cut polytope for the graph is as follows:

  minimize subject to      for every edge         for every node         for every edge   (4)

It is well-known that all extreme-point solutions of (4) are integral. An integral solution of (4) generates a - cut by letting and . For our case, we have an additional constraint in that the number of edges to be deleted from is , which motivates us to formulate the following polytope for our problem:

  minimize subject to      for every edge         for every node         for every edge   (5)

Let denote the optimal objective value of (5).

Lemma 4.

.

Proof.  Suppose that removal of the edges in the optimal solution removes cycles from . We construct the following solution of (5) with respect to the optimal solution of Eadp having nodes:

It can be verified as follows that this is indeed a feasible solution of (5):

  • Since and , it follows that is satisfied.

  • No edge-cycle-arc belongs to . Thus, if is an edge-cycle-arc then and it is not the case that and . Thus for every edge-cycle-arc the constraint is satisfied.

  • Consider an edge-arc ; note that . If that and , otherwise and . In both cases, the constraint is satisfied. The case of a cycle-arc is similar.

  • The constraint is trivially satisfied since by our assumption.

Note that does not contain any edge-cycle-arcs. Thus, the objective value of this solution is