On Robust Algorithms for the Maximum Weight Stable Set Problem

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Some observations on maximum weight stable sets in certain P5-free graphs

As usual, () denotes the path on vertices, and () denotes the cycle on vertices. For a family of graphs, we say that a graph is ‐free if no induced subgraph of is isomorphic to any graph in . We present a decomposition theorem for the class of ‐free graphs; in fact, we give a complete structural characterization of ‐free graphs that do not admit a clique‐cutset. We use this decomposition theorem to show that the class of ‐free graphs is ‐bounded by a linear function (more precisely, every ‐free graph satisfies ). We also use the decomposition theorem to construct an algorithm for the minimum coloring problem, an algorithm for the maximum weight stable set problem, and an algorithm for the maximum weight clique problem for this class, where denotes the number of vertices and the number of edges of the input graph.

Stable sets of maximum weight in ( [formula omitted], banner)-free graphs

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a message-passing distributed computing model for graph processing and present lower bounds and algorithms for several graph problems. This work is inspired by recent large-scale graph processing systems (e.g., Pregel and Giraph) which are designed based on the message-passing model of distributed computing. Our model consists of a point-to-point communication network of $k$ machines interconnected by bandwidth-restricted links. Communicating data between the machines is the costly operation (as opposed to local computation). The network is used to process an arbitrary $n$-node input graph (typically $n \gg k > 1$) that is randomly partitioned among the $k$ machines (a common implementation in many real world systems). Our goal is to study fundamental complexity bounds for solving graph problems in this model. We present techniques for obtaining lower bounds on the distributed time complexity. Our lower bounds develop and use new bounds in random-partition communication complexity. We first show a lower bound of $\Omega(n/k)$ rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST). We also show an $\Omega(n/k^2)$ lower bound for connectivity, ST verification and other related problems. We give algorithms for various fundamental graph problems in our model. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in $\tilde{O}(n/k)$ time, whereas for shortest paths, we present algorithms that run in $\tilde{O}(n/\sqrt{k})$ time (for $(1+\epsilon)$-factor approx.) and in $\tilde{O}(n/k)$ time (for $O(\log n)$-factor approx.) respectively.

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.

Stable sets and polynomials

A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.KeywordsNegative CycleClique InequalityHeavy EdgeRational PolyhedronLight EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Graph properties checkable in linear time in the number of vertices

Context-free language reachability (CFL-reachability) is a fundamental framework for program analysis. A large variety of static analyses can be formulated as CFL-reachability problems, which determines whether specific source-sink pairs in an edge-labeled graph are connected by a reachable path, i.e., a path whose edge labels form a string accepted by the given CFL. Computing CFL-reachability is expensive. The fastest algorithm exhibits a slightly subcubic time complexity with respect to the input graph size. Improving the scalability of CFL-reachability is of practical interest, but reducing the time complexity is inherently difficult. In this paper, we focus on improving the scalability of CFL-reachability from a more practical perspective---reducing the input graph size. Our idea arises from the existence of trivial edges, i.e., edges that do not affect any reachable path in CFL-reachability. We observe that two nodes joined by trivial edges can be folded---by merging the two nodes with all the edges joining them removed---without affecting the CFL-reachability result. By studying the characteristic of the recursive state machines (RSMs), an alternative form of CFLs, we propose an approach to identify foldable node pairs without the need to verify the underlying reachable paths (which is equivalent to solving the CFL-reachability problem). In particular, given a CFL-reachability problem instance with an input graph G and an RSM, based on the correspondence between paths in G and state transitions in RSM, we propose a graph folding principle, which can determine whether two adjacent nodes are foldable by examining only their incoming and outgoing edges. On top of the graph folding principle, we propose an efficient graph folding algorithm GF. The time complexity of GF is linear with respect to the number of nodes in the input graph. Our evaluations on two clients (alias analysis and value-flow analysis) show that GF significantly accelerates RSM/CFL-reachability by reducing the input graph size. On average, for value-flow analysis, GF reduces 60.96% of nodes and 42.67% of edges of the input graphs, obtaining a speedup of 4.65× and a memory usage reduction of 57.35%. For alias analysis, GF reduces 38.93% of nodes and 35.61% of edges of the input graphs, obtaining a speedup of 3.21× and a memory usage reduction of 65.19%.