Graph Complexity and Slice Functions

Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, Rödl, and Savický [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity. A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of {2-slice functions} would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. We prove, for instance, that a lower bound of n ^1+Ω(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 ^Ω(?) lower bound on the formula size over a complete basis of another explicit function g on l variables, where l=Θ( log n) . We also consider lower bound questions for depth-3 bipartite graph complexity. We prove a weak lower bound on this measure using algebraic methods. For instance, our result gives a lower bound of Ω(( log n) ³ / ( log log n) ⁵ ) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. Lower bounds for depth-3 bipartite graph complexity are motivated by two significant applications: (i) a lower bound of n ^Ω(1) on the depth-3 complexity of an explicit n -vertex bipartite graph would yield superlinear size lower bounds on log-depth Boolean circuits for an explicit function, and (ii) a lower bound of $\exp((\log \log n)^{\omega(1)})$ would give an explicit language outside the class Σ ₂ ^cc of the two-party communication complexity as defined by Babai, Frankl, and Simon [BFS]. Our lower bound proof is based on sign-representing polynomials for DNFs and lower bounds on ranks of ±1 matrices even after being subjected to sign-preserving changes to their entries. For the former, we use a result of Nisan and Szegedy [NS] and an idea from a recent result of Klivans and Servedio [KS]. For the latter, we use a recent remarkable lower bound due to Forster [F1].     

Diversified top-<Emphasis Type="Italic">k</Emphasis> clique search

Maximal clique enumeration is a fundamental problem in graph theory and has been extensively studied. However, maximal clique enumeration is time-consuming in large graphs and always returns enormous cliques with large overlaps. Motivated by this, in this paper, we study the diversified top-k clique search problem which is to find top-k cliques that can cover most number of nodes in the graph. Diversified top-k clique search can be widely used in a lot of applications including community search, motif discovery, and anomaly detection in large graphs. A naive solution for diversified top-k clique search is to keep all maximal cliques in memory and then find k of them that cover most nodes in the graph by using the approximate greedy max k-cover algorithm. However, such a solution is impractical when the graph is large. In this paper, instead of keeping all maximal cliques in memory, we devise an algorithm to maintain k candidates in the process of maximal clique enumeration. Our algorithm has limited memory footprint and can achieve a guaranteed approximation ratio. We also introduce a novel light-weight \(\mathsf {PNP}\)-\(\mathsf {Index}\), based on which we design an optimal maximal clique maintenance algorithm. We further explore three optimization strategies to avoid enumerating all maximal cliques and thus largely reduce the computational cost. Besides, for the massive input graph, we develop an I/O efficient algorithm to tackle the problem when the input graph cannot fit in main memory. We conduct extensive performance studies on real graphs and synthetic graphs. One of the real graphs contains 1.02 billion edges. The results demonstrate the high efficiency and effectiveness of our approach.     

Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.     

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths that either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) it computes a clique partition having size that is guaranteed to be within (1+ε) of the optimum size if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG’s is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(\frac{\log^{*} n}{{\varepsilon}^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex-weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS break down. Yet, surprisingly, it admits a (2+ε)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.     

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n−18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.     

Clique-perfectness and balancedness of some graph classes

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness of P₄-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for any P₄-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced.     

On Cutwidth Parameterized by Vertex Cover

We study the Cutwidth problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 ^k n ^O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 ^n/2 n ^O(1)) time algorithm for Cutwidth on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for Cutwidth on a graph class where the problem remains NP-complete. Additionally, we show that Cutwidth parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP?coNP/poly. Our kernelization lower bound contrasts with the recent results of Bodlaender et al. (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both Treewidth and Pathwidth parameterized by vertex cover do admit polynomial kernels.     

Stable sets in k-colorable -free graphs

We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P₅-free graphs, or, more generally, to the class of P₅-free graphs that contain no clique of size k+1. This is based on the following structural result: every connected k-colorable P₅-free graph has a vertex whose non-neighbors induce a (k−1)-colorable subgraph.     

Super-connected but not super edge-connected graphs

A connected graph G is super-connected (resp. super edge-connected) if every minimum vertex-cut (resp. edge-cut) isolates a vertex of G. In [Super connectivity of line graphs, Inform. Process. Lett. 94 (2005) 191-195], Xu et al. shows that a super-connected graph with minimum degree at least 4 is also super edge-connected. In this paper, a characterization of all super-connected but not super edge-connected graphs is given. It follows from this result that there is a unique super-connected but not super edge-connected graph with minimum degree 3, that is, the Ladder graph L₃ of order 6, and that there are infinitely many super-connected but not super edge-connected graphs with minimum degree 1 or 2.     

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K_1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general.The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K_i,j-free graphs, which exclude, for some fixed i,j∈N, the complete bipartite graph K_i,j as a subgraph. For i,j≥2, the class of claw-free graphs and any class of K_i,j-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable.We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t≥4, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K_1,t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes.Finally, we show that for any t∈N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.     

On the bi-enhancement of chordal-bipartite probe graphs

Given a class C of graphs, a graph G=(V,E) is said to be a C-probe graph if there exists a stable (i.e., independent) set of vertices X⊆V and a set F of pairs of vertices of X such that the graph G^′=(V,E∪F) is in the class C. Recently, there has been increasing interest and research on a variety of C-probe graph classes, such as interval probe graphs, chordal probe graphs and chain probe graphs.In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B^∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in Zhang (1994) [18] and to one on chordal probe graphs in Golumbic and Lipshteyn (2004) [11].     

Using graph coloring in an algebraic compiler

An algebraic compiler allows incremental development of the source program and builds its target image by composing the target images of the program components. In this paper we describe the general structure of an algebraic compiler focusing on compositional code generation. We show that the mathematical model for register management by an algebraic compiler is a graph coloring problem in which an optimally colored graph is obtained by composing optimally colored subgraphs. More precisely, we define the clique-composition of graphs and as the graph obtained by joining all the vertices in a clique in with all the vertices in a clique in and show that optimal register management by an algebraic compiler is achieved by performing clique-composition operations. Thus, an algebraic compiler provides automatically adequate clique separation of the global register management graph. We present a linear-time algorithm that takes as input optimally colored graphs and and constructs an optimal coloring of any clique-composition of and . Motivated by the operation of clique-composition, we define the class of clique-composable graphs as those graphs that can be iteratively built from single vertices using the clique-composition operation. We show that the class of clique-composable graphs coincides with the well-known class of chordal graphs. Received: 11 May 1995 / 30 November 1995     

New applications of clique separator decomposition for the Maximum Weight Stable Set problem

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P₅-free (the P₅ is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of P₅-free graphs is still unknown.     

Distributed discovery of large near-cliques

Given an undirected graph and 0 ￡ e ￡ 1{0\le\epsilon\le1}, a set of nodes is called an e{\epsilon}-near clique if all but an e{\epsilon} fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size e{\epsilon}-near clique if there exists an e³{\epsilon^3}-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n ^−Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) ^α ) for some a ? (0,1){\alpha \in (0,1)}. Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.     

On the cubicity of certain graphs

The boxicity of a graph G is the minimum dimension b such that G is representable as the intersection graph of axis-parallel boxes in the b-dimensional space. When the boxes are restricted to be axis-parallel b-dimensional cubes, the minimum dimension b required to represent G is called the cubicity of G. In this paper we show that cubicity(Hd)?2d, where Hd is the d-dimensional hypercube. (The d-dimensional hypercube is the graph on d² vertices which corresponds to the d²d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate.) We also show that cubicity(Hd)?(d−1)/(logd). We also show that (1) cubicity(G)?(logα)/(log(D+1)), (2) cubicity(G)?(logn−logω)/(logD), where α,ω,D and n denote the stability number, the clique number, the diameter and the number of vertices of G. As consequences of these lower bounds we provide lower bounds for the cubicity of planar graphs, bipartite graphs, triangle-free graphs, etc., in terms of their diameter and the number of vertices.     

On Independent Vertex Sets in Subclasses of Apple-Free Graphs

In a finite undirected graph, an apple consists of a chordless cycle of length at least 4, and an additional vertex which is not in the cycle and sees exactly one of the cycle vertices. A graph is apple-free if it contains no induced subgraph isomorphic to an apple. Apple-free graphs are a common generalization of chordal graphs, claw-free graphs and cographs and occur in various papers. The Maximum Weight Independent Set (MWS) problem is efficiently solvable on chordal graphs, on cographs as well as on claw-free graphs. In this paper, we obtain partial results on some subclasses of apple-free graphs where our results show that the MWS problem is solvable in polynomial time. The main tool is a combination of clique separators with modular decomposition. Our algorithms are robust in the sense that there is no need to recognize whether the input graph is in the given graph class; the algorithm either solves the MWS problem correctly or detects that the input graph is not in the given class.     

Cliques with maximum/minimum edge neighborhood and neighborhood density

Given a graph G=(V,E) and a clique C of G, the edge neighborhood of C can be defined as the total number of edges running between C and V?C, being denoted by N(C). The density of the edge neighborhood N(C) can be set as the ratio (|N(C)|/(|C|·|V?C|)).This paper addresses maximum/minimum edge neighborhood and neighborhood density cliques in G. Two versions will be undertaken, by fixing, or not, the size of the cliques.From an optimization point of view these problems do not bring much novelty, as they can be seen as particular or special versions of weighted clique problems. However, from a practical point of view, they concentrate on certain kinds of properties of cliques, rather than their size, revealing clique's engagement in the graph. In fact, a maximum edge neighborhood clique should be strongly embraced in the graph, while a minimum edge neighborhood clique should reveal an almost isolated strong component. In particular, special versions of the new problems allow to distinguish among cliques of the same size, namely among possible tied maximum cliques in the graph.We propose node based formulations for these edge based clique related problems. Using these models, we present computational results and suggest applications where the new problems can bring additional insights.     

Perfectly colorable graphs

We define a perfect coloring of a graph G as a proper coloring of G such that every connected induced subgraph H of G uses exactly ω(H) many colors where ω(H) is the clique number of H. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.     

All normalized anti-monotonic overlap graph measures are bounded

In graph mining, a frequency measure for graphs is anti-monotonic if the frequency of a pattern never exceeds the frequency of a subpattern. The efficiency and correctness of most graph pattern miners relies critically on this property. We study the case where frequent subgraphs have to be found in one graph. Vanetik et al. (Data Min Knowl Disc 13(2):243?C260, 2006) already gave sufficient and necessary conditions for anti-monotonicity of graph measures depending only on the edge-overlaps between the instances of the pattern in a labeled graph. We extend these results to homomorphisms, isomorphisms and homeomorphisms on both labeled and unlabeled, directed and undirected graphs, for vertex- and edge-overlap. We show a set of reductions between the different morphisms that preserve overlap. As a secondary contribution, we prove that the popular maximum independent set measure assigns the minimal possible normalized frequency and we introduce a new measure based on the minimum clique partition that assigns the maximum possible normalized frequency. In that way, we obtain that all normalized anti-monotonic overlap graph measures are bounded from above and below. We also introduce a new measure sandwiched between the former two based on the polynomial time computable Lovász ??-function.