The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted version, each vertex in the graph is associated with a weight and we are going to find an independent set of maximum total vertex weight. In this paper, we design several reduction rules and a fast exact algorithm for the maximum weighted independent set problem, and use the measure-and-conquer technique to analyze the running time bound of the algorithm. Our algorithm works on general weighted graphs and it has a good running time bound on sparse graphs. If the graph has an average degree at most 3, our algorithm runs in $O^*(1.1443^n)$ time and polynomial space, improving previous running time bounds for the problem in cubic graphs using polynomial space.
We give a parameterized algorithm for weighted vertex cover on graphs with maximum degree 3 whose time complexity is $O^*(1.402^t)$, where $t$ is the minimum size of a vertex cover of the input graph.
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-time algorithms on $n$-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prov…
Several papers have achieved time $O(\sqrt n m)$ for cardinality matching, starting from first principles. This results in a long derivation. We simplify the task by employing well-known concepts for maximum weight matching. We use Edmonds' algorithm to derive the structure of shortest augmenting paths. We extend this to a complete algorithm for maximum cardinality matching in time $O(\sqrt n m)$.
We propose a new greedy algorithm for the maximum cardinality matching problem. We give experimental evidence that this algorithm is likely to find a maximum matching in random graphs with constant expected degree c>0, independent of the value of c. This is contrary to the behavior of commonly used greedy matching heuristics which are known to have some range of c where they probably fail to compute a maximum matching.
We present a fully dynamic algorithm for maintaining approximate maximum weight matching in general weighted graphs. The algorithm maintains a matching ${\cal M}$ whose weight is at least $1/8 M^{*}$ where $M^{*}$ is the weight of the maximum weight matching. The algorithm achieves an expected amortized $O(\log n \log \mathcal C)$ time per edge insertion or deletion, where $\mathcal C$ is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph. Using a simple randomized scaling technique, we are able to obtain a matching whith expected approximation r…
We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop …
The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown $NP$-hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. The corresponding approximation problems are shown $NP$-hard for certain approximation performance bounds. Moreover, some natural parameterized variants of the problem are shown $W[P]$-hard. The results are in contrast with the maximum matroid-matroid intersection which is solvable in polynomial time by an old result of Edmonds. We also prove that it is $NP$-hard to approxim…
Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the cases of c=1 and 2. The time complexities of the algorithms are O((k+m)log n+n) for c=1 and O(k^3+k^2n+m(m+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|…
In this paper, we consider the problem of planning a path for a robot to monitor a known set of features of interest in an environment. We represent the environment as a graph with vertex weights and edge lengths. The vertices represent regions of interest, edge lengths give travel times between regions, and the vertex weights give the importance of each region. As the robot repeatedly performs a closed walk on the graph, we define the weighted latency of a vertex to be the maximum time between visits to that vertex, weighted by the importance (vertex weight) of that vertex. Our goal is to fin…
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze t…
Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, a larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there wer…
In mobile wireless sensor networks (MWSNs), each sensor has the ability not only to sense and transmit data but also to move to some specific location. Because the movement of sensors consumes much more power than that in sensing and communication, the problem of scheduling mobile sensors to cover all targets and maintain network connectivity such that the total movement distance of mobile sensors is minimized has received a great deal of attention. However, in reality, due to a limited budget or numerous targets, mobile sensors may be not enough to cover all targets or form a connected networ…
Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such "flipping" of the objective function was done for many classical optimization problems. For example, Minimum Vertex Cover becomes Maximum Minimal Vertex Cover, Maximum Independent Set becomes Minimum Maximal Independent Set and so on. In this paper, we propose to study the weighted version of Maximum Minimal Edge Cover called Upper Edge Cover, a problem having ap…
In a network, a $k$-plex represents a subset of $n$ vertices where the degree of each vertex in the subnetwork induced by this subset is at least $n-k$. The maximum edge-weight $k$-plex partitioning problem (Max-EkPP) is to find the $k$-plex partitioning in edge-weighted network, such that the sum of edge weights is maximal. The Max-EkPP has an important role in discovering new information in large sparse biological networks. We propose a variable neighborhood search (VNS) algorithm for solving Max-EkPP. The VNS implements a local search based on the 1-swap first improvement strategy and the o…
A maximum weighted matching for bipartite graphs $G=(A \cup B,E)$ can be found by using the algorithm of Edmonds and Karp with a Fibonacci Heap and a modified Dijkstra in $O(nm + n^2 \log{n})$ time where n is the number of nodes and m the number of edges. For the case that $|A|=|B|$ the number of edges is $n^2$ and therefore the complexity is $O(n^3)$. In this paper we want to present a simple heuristic method to reduce the number of edges of complete bipartite graphs $G=(A \cup B,E)$ with $|A|=|B|$ such that $m = n\log{n}$ and therefore the complexity of such that $m = n\log{n}$ and therefore…
This paper gives poly-logarithmic-round, distributed D-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio D is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with D=2). Via duality, the paper also gives poly-logarithmic-round, distributed D-approximation algorithms for Fractional Packing linear programs (where D is the maximum number of constraints in which any variable occurs), and for Max Wei…
Given a graph $G$, a non-negative integer $k$, and a weight function that maps each vertex in $G$ to a positive real number, the \emph{Maximum Weighted Budgeted Independent Set (MWBIS) problem} is about finding a maximum weighted independent set in $G$ of cardinality at most $k$. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in $G$, is called the \emph{Maximum Independent Vertex Coverage (MIVC)} problem. In other words, the MIVC problem is about finding an independent set of cardinality at most $k$ with maximum coverage. Since it is a generalization of…
The maximum clique problem is a well known NP-Hard problem with applications in data mining, network analysis, informatics, and many other areas. Although there exist several algorithms with acceptable runtimes for certain classes of graphs, many of them are infeasible for massive graphs. We present a new exact algorithm that employs novel pruning techniques to very quickly find maximum cliques in large sparse graphs. Extensive experiments on several types of synthetic and real-world graphs show that our new algorithm is up to several orders of magnitude faster than existing algorithms for mos…
University of Technology Sydney（悉尼科技大学）；University of Southern California（南加州大学）
In this paper, we aim to design sparse D-optimal (determinantoptimal) pose-graph SLAM problems through the synthesis of sparse graphs with the maximum weighted number of spanning trees. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, several new theoretical results are established in this paper, including the monotone log-submodularity of the weighted number of spanning trees. By exploiting these structures, we design a complementary pair of near-optimal efficient approximation algorithms with provable guarantees. Our theor…
In this paper, we reduce the maximum weighted matching problem to the largest cardinality matching in {\bf CONGEST}. The paper presents two technical contributions. The first of them is a simple $poly(\log n, \frac{1}{\varepsilon}, t, \ln w_t)$-round {\bf CONGEST} algorithm for reducing the maximum weighted matching problem to the largest cardinality matching problem. This is achieved under the assumption that all vertices know all edge-weights $\{w_1,....,w_t\}$ (in particular, they know $t$, the number of different edge-weights), though a particular vertex may not know the weight of a partic…
