We propose a new and strengthened Branch-and-Bound (BnB) algorithm for the maximum common (connected) induced subgraph problem based on two new operators, Long-Short Memory (LSM) and Leaf vertex Union Match (LUM). Given two graphs for which we search for the maximum common (connected) induced subgraph, the first operator of LSM maintains a score for the branching node using the short-term reward of each vertex of the first graph and the long-term reward of each vertex pair of the two graphs. In this way, the BnB process learns to reduce the search tree size significantly and improve the algorithm performance. The second operator of LUM further improves the performance by simultaneously matching the leaf vertices connected to the current matched vertices, and allows the algorithm to match multiple vertex pairs without affecting the solution optimality. We incorporate the two operators into the state-of-the-art BnB algorithm McSplit, and denote the resulting algorithm as McSplit+LL. Experiments show that McSplit+LL outperforms McSplit+RL, a more recent variant of McSplit using reinforcement learning that is superior than McSplit.
It is $\mathsf{NP}$-hard to determine the minimum number of branching vertices needed in a single-source distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is an interval graph. In earlier work, it was shown that every interval graph with $k$ terminal vertices admits an all-pairs distance-preserving subgraph with $O(k\log k)$ branching vertices. We consider graphs that can be expressed as the strong product of two interval graphs, and present a polynomial time algorithm that takes such a graph with $k$ terminals as…
We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with $k$ terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is $\mathsf{NP}$-hard for general graphs. Then, we show that every interval graph admits a subgraph with $O(k)$ branching vertices that approximates pairwise terminal distances up to an additive term of $+1$. We also present an interval graph $G_{\mathrm{int}}$ for which the $+1$ approximation is necessary to obtain the $O(k)$ upper bound on the number of branching vertices. In pa…
The Maximum Balanced Subgraph Problem (MBSP) is the problem of finding a subgraph of a signed graph that is balanced and maximizes the cardinality of its vertex set. We are interested in the exact solution of the problem: an improved version of a branch-and-cut algorithm is proposed. Extensive computational experiments are carried out on a set of instances from three applications previously discussed in the literature as well as on a set of random instances.
In the Vertex Triangle 2-Club problem, we are given an undirected graph $G$ and aim to find a maximum-vertex subgraph of $G$ that has diameter at most 2 and in which every vertex is contained in at least $\ell$ triangles in the subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies on an ILP formulation [Almeida and Brás, Comput. Oper. Res. 2019]. In this work, we develop a combinatorial branch-and-bound algorithm that, coupled with a set of data reduction rules, outperforms the existing implementation and is able to find optimal solutions on sparse real-world graphs wi…
Given an undirected node-weighted graph, the Maximum-Weight Connected Subgraph problem (MWCS) is to identify a subset of nodes of maximalsum of weights that induce a connected subgraph. MWCS is closely related to the well-studied Prize Collecting Steiner Tree problem and has many applications in different areas, including computational biology, network design and computer vision. The problem is NP-hard and even hard to approximate within a constant factor. In this work we describe an algorithmic scheme for solving MWCS to provable optimality, which is based on preprocessing rules, new results …
This paper introduces the Simultaneous assignment problem. Let us given a graph with a weight and a capacity function on its edges, and a set of its subgraphs along with a degree upper bound function for each of them. We are also given a laminar system on the node set with an upper bound on the degree-sum in each member of the system. Our goal is to assign each edge a non-negative integer below its capacity such that the total weight is maximized, the degrees in each subgraph are below the associated degree upper bound, and the degree-sum bound is respected in each member of the laminar system…
The Maximum Common Subgraph is a computationally challenging problem with countless practical applications. Even if it has been long proven NP-hard, its importance still motivates searching for exact solutions. This work starts by discussing the possibility to extend an existing, very effective branch-and-bound procedure on parallel multi-core and many-core architectures. We analyze a parallel multi-core implementation that exploits a divide-and-conquer approach based on a thread-pool, which does not deteriorate the original algorithmic efficiency and it is not memory bound. We extend the algo…
Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of $P_4$-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the tri…
We study the \emph{interval completion} problem, which asks for the insertion of a set of at most $k$ edges to make a graph of $n$ vertices into an interval graph. We focus on chordal graphs with no small obstructions, where every remaining obstruction is known to have a shallow property. From such a shallow obstruction we single out a subset 6 or 7 vertices, called the frame, and 5 missed edges in the subgraph induced by the frame. We show that if none of these edges is inserted, then the frame cannot be altered at all, and the whole obstruction is also fixed, by and large, in the sense that …
We study the \textsc{Max Partial $H$-Coloring} problem: given a graph $G$, find the largest induced subgraph of $G$ that admits a homomorphism into $H$, where $H$ is a fixed pattern graph without loops. Note that when $H$ is a complete graph on $k$ vertices, the problem reduces to finding the largest induced $k$-colorable subgraph, which for $k=2$ is equivalent (by complementation) to \textsc{Odd Cycle Transversal}. We prove that for every fixed pattern graph $H$ without loops, \textsc{Max Partial $H$-Coloring} can be solved: $\bullet$ in $\{P_5,F\}$-free graphs in polynomial time, whenever $F…
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that…
Given a graph $G$, the Connected Vertex Cover problem (CVC) asks to find a minimum cardinality vertex cover of $G$ that induces a connected subgraph. In this paper we describe some approaches to solve the CVC problem exactly. First, we give compact mixed-integer extended formulations for CVC: these are the first formulations proposed for this problem, and can be easily adapted to variations of the problem such as Tree Cover. Second, we describe a simple branch and bound algorithm for the CVC problem. Finally, we implement our algorithm and compare its performance against our best formulation: …
This paper introduces the Packing While Traveling problem as a new non-linear knapsack problem. Given are a set of cities that have a set of items of distinct profits and weights and a vehicle that may collect the items when visiting all the cities in a fixed order. Each selected item contributes its profit, but produces a transportation cost relative to its weight. The problem asks to find a subset of the items such that the total gain is maximized. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. We propose a pre-processing scheme that decr…
Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in K-dimensional space such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes…
An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and out-branchings of $D$, respectively. We give fixed parameter tractable algorithms for deciding whether $l_s(D)\geq k$ and whether $l(D)\geq k$ for a digraph $D$ on $n$ vertices, both with time complexity $2^{O(k\log k)} \cdot n^{O(1)}$. This improves on previous algorithms with complexity $2^{O(k^3\log k)} \cdot n^{O(1)}$ and $2^{O(k\log^2 k)} \cdot n^{O(1)}$…
A subgraph $T$ of a digraph $D$ is an {\em out-branching} if $T$ is an oriented spanning tree with only one vertex of in-degree zero (called the {\em root}). The vertices of $T$ of out-degree zero are {\em leaves}. In the {\sc Directed $k$-Leaf} Problem, we are given a digraph $D$ and an integral parameter $k$, and we are to decide whether $D$ has an out-branching with at least $k$ leaves. Recently, Kneis et al. (2008) obtained an algorithm for the problem of running time $4^{k}\cdot n^{O(1)}$. We describe a new algorithm for the problem of running time $3.72^{k}\cdot n^{O(1)}$. In {\sc Rooted…
Finding all maximal $k$-plexes on networks is a fundamental research problem in graph analysis due to many important applications, such as community detection, biological graph analysis, and so on. A $k$-plex is a subgraph in which every vertex is adjacent to all but at most $k$ vertices within the subgraph. In this paper, we study the problem of enumerating all large maximal $k$-plexes of a graph and develop several new and efficient techniques to solve the problem. Specifically, we first propose several novel upper-bounding techniques to prune unnecessary computations during the enumeration …
Quasi-threshold graphs are $\{C_4, P_4\}$-free graphs, i.e., they do not contain any cycle or path of four nodes as an induced subgraph. We study the $\{C_4, P_4\}$-free editing problem, which is the problem of finding a minimum number of edge insertions or deletions to transform an input graph into a quasi-threshold graph. This problem is NP-hard but fixed-parameter tractable (FPT) in the number of edits by using a branch-and-bound algorithm and admits a simple integer linear programming formulation (ILP). Both methods are also applicable to the general $F$-free editing problem for any finite…
Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates
Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）；Princeton University, Operations Research and Financial Engineering（普林斯顿大学运筹与金融工程学系）
We consider a variety of NP-Complete network connectivity problems. We introduce a novel dual-based approach to approximating network design problems with cut-based linear programming relaxations. This approach gives a $3/2$-approximation to Minimum 2-Edge-Connected Spanning Subgraph that is equivalent to a previously proposed algorithm. One well-studied branch of network design models ad hoc networks where each node can either operate at high or low power. If we allow unidirectional links, we can formalize this into the problem Dual Power Assignment (DPA). Our dual-based approach gives a $3/2…
We give alternative definitions for maximum matching width, e.g. a graph $G$ has $\operatorname{mmw}(G) \leq k$ if and only if it is a subgraph of a chordal graph $H$ and for every maximal clique $X$ of $H$ there exists $A,B,C \subseteq X$ with $A \cup B \cup C=X$ and $|A|,|B|,|C| \leq k$ such that any subset of $X$ that is a minimal separator of $H$ is a subset of either $A, B$ or $C$. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing…
