An undirected graph is defined as , where is a finite set of vertices and is a set of unordered pairs of vertices called edges. Two vertices are said to be adjacent to each other if they have an edge between them, i.e., . The neighborhood of a vertex is the set of all vertices adjacent to it. A set of vertices such that none of them is adjacent to each other is called an independent set. An independent set of maximum cardinality is called a maximum independent set and its cardinality is denoted as , called the independence number of . An independent set is said to be maximal if it not a strict subset of another independent set. If weights are assigned to each vertex, then a weighted maximum independent set is an independent set which maximizes over all independent sets and the value of this maximum is denoted by , called the -weighted independence number of .
This paper is about a relation between graphs and a class of continuous optimization problems called Linear Complementarity Problems (LCPs). Given a matrix
and a vector, LCP() is the following problem:
Any such is called a solution of LCP(). Our work is motivated by previous results in  which established an LCP based characterization of . For a graph on vertices, let denote its adjacency matrix: i.e., if , else it is Now consider the LCP given by LCP() where is the identity matrix, is the adjacency matrix of and is a vector of ’s. The authors of  show that the maximum weighted norm amongst the solutions of LCP equals the weighted independence number of . Formally,
For any simple graph on vertices and a vector of weights , we have
It easy to argue that the characteristic vector of any maximal independent set in solves . Consequently, one trivially has that for all that solve . The above result is nontrivial because it shows that no fractional solution of can attain a strictly greater value for than that attained by the characteristic vector of a -weighted maximum independent set.
Indeed there is a third connection which concerns the interpretation of solutions of LCP in terms of Nash equilibria of a public goods game defined on a network . In the model considered in , each vertex of the graph is an agent who exerts a scalar effort and its utility for the effort profile is
where denotes the neighborhood of vertex in graph , is a differentiable strictly concave increasing benefit function and is the marginal cost of exerting unit effort. Thus each agent benefits from its own effort and the effort exerted by its neighbors (more details on this model can be found in ). Assume for simplicity that , i.e., marginal benefit equals marginal cost at unit effort. It is shown in  that the Nash equilibria of the above public goods game are given by solutions of LCP. Moreover, each maximal independent set corresponds to an equilibrium of the above game wherein each vertex in the maximal independent set exerts unit effort and all other vertices exert no effort. It follows that for a given vector of non-negative weights , the maximum weighted effort amongst all equilibria, which is the maximum weighted norm amongst the solutions of LCP, is achieved by the equilibrium corresponding to the -weighted maximum independent set. These results illustrate how Nash equilibria of this game are intimately related to combinatorial structures on the underlying graph.
In this paper, we are concerned with perturbations of the above model. Specifically, we ask, what happens if the argument of the benefit function is instead of ? Here is a substitutability factor which captures the case when the benefit agent derives from its neighbors is proportional to times the sum of efforts exerted by its neighbors. We view as a small perturbation from unity, i.e., is close to unity, but may not be exactly unity. Thus we think of as the idealized case where efforts of neighbors substitute exactly for the agent’s own effort, and can be thought of as arising due to small losses or misspecifications from this idealized situation. The existence of equilibria in this case has been studied in . It is easy to argue that an equilibrium exists and all the equilibria of this new game are given by solutions of LCP. With this motivation, in this paper, we characterize norm maximizing solutions of for , but close to .
Our effort is to relate these solutions to combinatorial structures on the underlying graph. Before we mention our contributions, we make a few observations about First, we observe that small perturbations around unity have a nontrivial effect on the combinatorial structure of solutions. In particular, for , it is not true that the characteristic vector of every maximal independent set solves . In fact, a binary vector solves if and only if it is the characteristic vector of a -dominating independent set, an object which may not exist in . In the case when a -dominating independent set does not exist, identifying combinatorial structures that support solutions and maximize the norm becomes challenging and forces one to expand the search space of combinatorial structures that can be identified with solutions of .
In this paper we do precisely this. We introduce a new concept called independent cliques solutions (ICS) which are defined as solutions of whose support is a union of independent cliques. Two cliques in a graph are said to be independent if no vertex of one clique has any vertex of the other clique as its neighbor. Independent cliques can be thought of as a generalization of independent sets since when each clique is degenerate (a single vertex), a union of independent cliques is an independent set. We prove that the maximum norm amongst all solutions of is achieved by an ICS with cliques for , where
and is the size of the largest clique in . Thus while norm maximizing solutions of include those supported by a maximum independent set, i.e., degenerate cliques, for the corresponding solutions of comprise of not-necessarily-degenerate cliques. Moreover, in the case when when a unique maximum independent set exists in a graph the characteristic vector of the unique maximum independent set is a solution of and is, in fact, its norm maximizing solution for . Lastly, we show that for , the results of  continue to hold, i.e., the maximum weighted norm amongst the solutions of is the weighted independence number achieved by the characteristic vector of a -weighted maximum independent set.
These results are proved as follows. We show the existence of ICSs via an algorithm (Algorithm 1) that constructs an ICS for any graph. We prove that Algorithm 1 outputs a vector, as a function of with support as a union of independent cliques. Moreover, the support does not depend on . From this we show that these independent cliques support an ICS for all , where
We also prove that the lower bound on , namely , is tight by showing via examples that when this , an ICS need not exist. Next, we prove that the ICS of which achieves the maximum norm amongst all the ICSs also achieves the maximum norm amongst all solutions of for .
The problem of characterizing norm maximizing solutions of is highly complex since the solution set of the LCP is not convex (it is a union of polyhedra ) and no known graph structures directly provide solutions to . Our results show that for , an ICS always exists and is also norm maximizing amongst all solutions of . It is also easy to show that for , admits a unique solution that is related to centrality notions on graphs (see ). It would be fascinating to ascertain the combinatorial structures that characterize norm maximizing solutions of for the entire range of from to unity. Though our results do not span this range, we believe they nonetheless provide interesting relations between the structural properties of graphs and solutions of .
1.1 Related Work
Our results, in effect, give a characterization of the solutions to a special class of Linear Programs with Complementary Constraints (LPCC). In its most general form, an LPCC is defined as
When we take to be in the LPCC, and take , and , the LPCC reduces to the problem we consider. LPCCs provide a generalization to problem classes such as linear programming and finding sparse (minimum norm) solutions of linear equations [5, 6]. The results in  show that it is hard to find approximate solutions of an LPCC. While the LPCC is a newly explored topic, LCPs are deeply studied subjects, book-length treatments of which can be found in  and .
The idea that Nash equilibria of games can be related to solutions of LCPs is not new. Consider a simultaneous move game with two players (I,II), where player I has possible actions and the player II has possible actions. The cost matrices are such that when player I chooses action and player II chooses action , they incur costs and
respectively. Players can also choose to play mixed strategies which are vectors defined over the probability simplex in adimensional space for player I and dimensional space for player II. A Nash equilibrium in mixed strategies in this game is defined as a pair of vectors , such that
where is a probabilty simplex in , . Assuming and are entrywise positive matrices, we define () as
It can be shown () that () satisfy LCP() with
More generally, certain equilibria of games involving coupled constraints  also reduce to LCPs.
There has been prior effort at relating LCPs with independent sets (, ). Particularly, the authors in  show that is a characterestic vector of a maximal independent set if and only if solves LCP() with
On the contrary, our work is to characterize norm maximizing solutions of LCP, solutions to which lie in a different space () compared to those of LCP() (). It is shown in  that the characterestic vectors of maximal independent sets are solutions of LCP, but the same is not the case with solutions of LCP for general . Our work is distinct from both  and  since we consider a different class of LCPs and relate their equilibria to combinatorial structures in the graph.
Generalizations of independent sets have been studied in other contexts. The authors of  generalize independent sets to -independent sets where a set of vertices of is said to be independent if is independent and every independent subset of with is a subset of . It is easy to note that -independent sets are independent sets and -independent sets are unique maximum independent sets. This generalization is more restrictive, and -independent sets need not even exist for all graphs for all . In contrast, we provide a more inclusive generalization to a union of independent cliques which includes all independent sets as a special case. To the best of our knowledge, such a generalization is the first of its kind. We also study the special case when unique maximum independent sets exist. These sets need not always exist for a graph. The authors of  provide sufficient conditions on the graph under which such independent sets exist.
1.2 Organization of the paper
The rest of the paper is organized as follows. In Section 2, we provide the preliminaries and the notation used throughout the paper along with an introduction to LCPs. In Section 3, we discuss properties of LCP and its solutions. In Section 4, we introduce the notion of Independent Clique Solutions (ICS), provide an algorithm to find them and prove that they achieve the maximum norm amongst all LCP solutions for . In Section 5, we extend the results of  to the case when . The paper concludes in Section 6.
2 Preliminaries and Notation
For a graph , we denote by and its vertex and edge sets respectively. For a vertex , let denote the neighborhood of in , i.e. . For a set of vertices , . Also, let denote the closed neighborhood of in . For , let denote the graph restricted to the vertex set and for a vector indexed by , let denote the subvector of with components indexed by the vertex set . For a vector indexed by , let be its support, i.e.
Let denote the characteristic vector of set , i.e.
We use the standard notation for the norm. Let denote the adjacency matrix of a graph given by
An independent set of a graph is defined as a set of vertices such that none of them are neighbors. A maximal independent set is defined as an independent set such that all the vertices which are not in have at least one neighbor in , i.e. . A maximum independent set is an independent set which has the highest cardinality amongst all independent sets. The cardinality of the maximum indepdendent set is denoted by and is also called the independence number of the graph . Let denote the weighted independence number for which is the maximum sum of weights amongst all indepdendent sets, i.e. . A dominating set () is a set such that . Thus, an independent set which is also dominating is a maximal independent set. A -dominating set is defined as a dominating set such that . A set which is both -dominating and independent is called a -dominating indepdendent set. It can be observed that a -dominating independent set is also a -dominating independent set for all . For , these are equivalent to maximal independent sets and they always exist. For , they may or may not exist depending on the graph. Examples of both the cases is shown in Fig. 1.
For a graph with adjacency matrix recall the problem where is a positive parameter. In this section we prove some properties of LCP) that will be used later in the paper.
Define and denote by the component of . is called as the discounted sum of the closed neighborhood of with respect to ,
We denote the set of solutions of LCP) by SOL). Clearly if and only if,
. We first show a lemma that establishes some basic properties of SOL).
Consider the . Then,
[label = ()]
If a graph G is a disjoint union of graphs and , then ,
For a graph , if , is a -dominating set of ,
For a graph , if , and .
See Section A.1.
The following lemma characterizes integer (binary) solutions of .
For a graph and , is an integer solution of LCP if and only if it is a characteristic vector of a -dominating independent set of
Let be an integer solution of LCP. Then, by Lemma 1 (c), is a binary vector, and hence for a subset of vertices in . If is not an independent set, then there exist such that , i.e. , whereby . Thus, and imply Eq. 4 is violated. This gives a contradiction. Hence, must be an independent set. From Lemma 1 (e) we have that must also be a -dominating set, as required.
For the converse, let be a -dominating indepdendent set of . Then, for , since . Also, for , . Thus, we have and for all . Hence, if is a characteristic vector of a -dominating indepdendent set of , then it is an integer solution of LCP.
Consider the following optimization problem which gives the norm maximizing solution amongst SOL:
We next show that if a -dominating independent set is contained in the support of a solution then has -norm no lesser than that of .
For a graph and , if is a solution of LCP and is a -dominating independent set of , then .
Let . From Eq. 4, we know that
Summing over , we get
Consequently, if is such that every solution of contains a -dominating independent in its support, then the solution of maxSOL would attained a characteristic vector of a -dominating independent set. Clearly, such a property does not hold for all graphs since -dominating independent sets do not exist in every graph. This requires us to analyze the solutions of maxSOL more carefully.
We now introduce a function called the potential function, the stationary points of which are the solutions of LCP. This was proved by  in the context of games on networks and is known more generally in the LCP literature . Define,
equals the set of stationary points of in .
We note that, when , where
represents the minimum eigenvalue of the adjacency matrix of the graph, the potential function is a strictly concave function. Thus, in this case, SOL is a singleton.
4 Solutions of LCP for
In this section, we assume throughout that . Our goal is to prove that the solution of maxSOL is achieved by a member of the class of solutions that we call independent clique solution (ICS). To prove this, we first formally define independent clique solutions and prove their existence constructively (Section 4.1). We also prove some results relating this special class of solutions to independent sets. We then show that the maximum norm amongst the solutions in this class is monotonically increasing under graph inclusion (Section 4.2). Using this, we inductively argue that the solution of maxSOL is achieved by an independent clique solution (Section 4.3).
4.1 Independent Clique Solutions
First, we note the following definition:
Two cliques in a graph are said to be independent if no vertex of one clique has any vertex of the other clique as its neighbor.
Note that the above definition is a generalization of the definition of independence of vertices which are in fact ’s.
Let be a graph and . An independent clique solution (ICS) is a solution of whose support is a union of independent cliques.
We denote the set of independent clique solutions of LCP as ICS. Consider the following optimization problem which gives us the norm maximizing solution amongst ICS:
We now show that for any graph an ICS exists if is greater than a threshold . We give a constructive proof. Algorithm 1 gives a method to construct an ICS using maximum independent sets in any graph. The justification of why this algorithm gives an ICS under suitable conditions on is given in Theorem 2.
We will show that this algorithm returns which is an ICS supported on for a maximum independent set in . Below we argue in Lemma 6 that these sets are independent cliques. In Theorem 2, we show that the returned solves the LCP. We first note a few remarks.
In Algorithm 1, for each , . To see this, note that . Since , . Since , . Thus, no vertex in is in .
In Algorithm 1, at the return step, , i.e., we eventually remove all vertices from the graph. To see this, the set of removed nodes in each step, say , is such that . Taking union on both sides from to , we get . being a maximal independent set, and hence .
Now, we prove a few lemmas about the algorithm that help us prove its validity. First, we show that after each iteration, the updated values and defined in creftype 9 and creftype 8 satisfy the conditions satisfied by and respectively before the iteration.
[label = ()]
It suffices to show that with and as defined in creftype 8 and creftype 9. By definition, we have . To show that , we need to show that in the vertices we removed from to form , there is no member of . Suppose we have executed creftype 11 of the iteration of the algorithm for some . The set of vertices removed from at this stage to form is , and we need to show that . Now, for each , contains which is a member of but not . The set also has neighbors of which can not be in since and is an independent set. Lastly, has neighbors of (defined on creftype 4 of Algorithm 1) that are also not in since is such that and hence . Since this is true for all , , and hence .
We show this by induction on the iteration number. At the start of the first iteration of the for loop (creftype 3 to creftype 12 in Algorithm 1), is a maximum independent set of . Assume that is a maximum independent set of after iterations (i.e., at the start of iteration ). To show that the claim is true at the start of the iteration, it suffices to show that is a maximum independent set of at the end of iteration. Note that , since only one element () is removed from to form . Suppose is not a maximum independent set in . Then there exists an independent set of such that and hence . Now note that and since there is no edge between any vertex in and . But then is an independent set of with , which is a contradiction. Thus, is a maximum independent set in .
We now come to the first step of showing that Algorithm 1 returns an ICS, namely, showing are independent cliques.
First we prove that forms a clique for all . At the iteration of Algorithm 1 let be as defined in creftype 4 and let be a maximum independent set of with . To show that forms a clique, it suffices to show that forms a clique since all vertices of are neighbors of . We prove this by contradiction. Suppose is not a clique. Then choose an independent set of such that it has at least two vertices. Consider . We claim that is an independent set. To prove this, it suffices to show that is not an edge in . Now, whereby each vertex in has only one neighbor in . Moreover, , whereby for all vertices in , their only neighbor in is . Consequently, is an independent set. Moreover, , which is a contradiction since must be a maximum independent set of by definition. Hence, and therefore must be a clique. Clearly, this holds for .
Now, consider two cliques and for . Without loss of generality, we assume . Then, . Thus, can have no neighbors of as its elements. Hence, and are independent.
To prove that Algorithm 1 outputs an ICS, we need to show that returned at the end of the algorithm satisfies LCP conditions Eqs. 4, 3, and 2 for all and that is a union of independent cliques. The fact that is a union of independent cliques is evident from Lemma 6. To show that the LCP conditions Eqs. 4, 3, and 2 are satisfied, we show for the generated when we exit from the algorithm, these conditions are satisfied for . We do this in Theorem 2.
Before we discuss Theorem 2, we have a lemma describing the conditions on the neighborhood of a vertex so that with such a neighborhood and corresponding discounted sum of closed neighborhood, for the LCP conditions Eqs. 4, 3, and 2 to be satisfied for .
[Two Clique Lemma] Consider a vertex and cliques and of graph such that and there exists another clique for some with such that . Let such that
and represents the size of the largest clique in the graph .
See Section B.1.
Note that for any graph with at least one edge, the value of is at least and hence the value of is at least , which is the golden ratio. For trees , We do not know of any deeper significance or interpretations of the appearance of the golden ratio in this problem.
We say that vertex is fully connected to a clique if all the vertices of are neighbors of , i.e. .
For LCP with