156,098 research outputs found
Implicit branching and parameterized partial cover problems
AbstractCovering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems has been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
Distributed Computation of 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 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 -node input graph (typically )
that is randomly partitioned among the 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 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 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 time, whereas for shortest paths, we present
algorithms that run in time (for -factor
approx.) and in time (for -factor approx.)
respectively.Comment: In Proceedings of SODA 201
Temporal vertex cover with a sliding time window.
Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other “path-related” temporal notions, only few attempts have been made to investigate “non-path” temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions
Construction of near-optimal vertex clique covering for real-world networks
We propose a method based on combining a constructive and a bounding heuristic to solve the vertex clique covering problem (CCP), where the aim is to partition the vertices of a graph into the smallest number of classes, which induce cliques. Searching for the solution to CCP is highly motivated by analysis of social and other real-world networks, applications in graph mining, as well as by the fact that CCP is one of the classical NP-hard problems. Combining the construction and the bounding heuristic helped us not only to find high-quality clique coverings but also to determine that in the domain of real-world networks, many of the obtained solutions are optimal, while the rest of them are near-optimal. In addition, the method has a polynomial time complexity and shows much promise for its practical use. Experimental results are presented for a fairly representative benchmark of real-world data. Our test graphs include extracts of web-based social networks, including some very large ones, several well-known graphs from network science, as well as coappearance networks of literary works' characters from the DIMACS graph coloring benchmark. We also present results for synthetic pseudorandom graphs structured according to the Erdös-Renyi model and Leighton's model
Hitting Meets Packing: How Hard Can it Be?
We study a general family of problems that form a common generalization of
classic hitting (also referred to as covering or transversal) and packing
problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of
a graph G prevent us from packing vertex-disjoint objects of type X?
This problem captures a spectrum of problems with standard hitting and packing
on opposite ends. Our main motivating question is whether the combination
X-HitPack can be significantly harder than these two base problems. Already for
a particular choice of X, this question can be posed for many different
complexity notions, leading to a large, so-far unexplored domain in the
intersection of the areas of hitting and packing problems.
On a high-level, we present two case studies: (1) X being all cycles, and (2)
X being all copies of a fixed graph H. In each, we explore the classical
complexity, as well as the parameterized complexity with the natural parameters
k+l and treewidth. We observe that the combined problem can be drastically
harder than the base problems: for cycles or for H being a connected graph with
at least 3 vertices, the problem is \Sigma_2^P-complete and requires
double-exponential dependence on the treewidth of the graph (assuming the
Exponential-Time Hypothesis). In contrast, the combined problem admits
qualitatively similar running times as the base problems in some cases,
although significant novel ideas are required. For example, for X being all
cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching
method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover
and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on
graphs of treewidth tw. The key step enabling this running time relies on a
combinatorial bound obtained from an algebraic (linear delta-matroid)
representation of possible matchings
Can Language Models Solve Graph Problems in Natural Language?
Large language models (LLMs) are increasingly adopted for a variety of tasks
with implicit graphical structures, such as planning in robotics, multi-hop
question answering or knowledge probing, structured commonsense reasoning, and
more. While LLMs have advanced the state-of-the-art on these tasks with
structure implications, whether LLMs could explicitly process textual
descriptions of graphs and structures, map them to grounded conceptual spaces,
and perform structured operations remains underexplored. To this end, we
propose NLGraph (Natural Language Graph), a comprehensive benchmark of
graph-based problem solving designed in natural language. NLGraph contains
29,370 problems, covering eight graph reasoning tasks with varying complexity
from simple tasks such as connectivity and shortest path up to complex problems
such as maximum flow and simulating graph neural networks. We evaluate LLMs
(GPT-3/4) with various prompting approaches on the NLGraph benchmark and find
that 1) language models do demonstrate preliminary graph reasoning abilities,
2) the benefit of advanced prompting and in-context learning diminishes on more
complex graph problems, while 3) LLMs are also (un)surprisingly brittle in the
face of spurious correlations in graph and problem settings. We then propose
Build-a-Graph Prompting and Algorithmic Prompting, two instruction-based
approaches to enhance LLMs in solving natural language graph problems.
Build-a-Graph and Algorithmic prompting improve the performance of LLMs on
NLGraph by 3.07% to 16.85% across multiple tasks and settings, while how to
solve the most complicated graph reasoning tasks in our setup with language
models remains an open research question. The NLGraph benchmark and evaluation
code are available at https://github.com/Arthur-Heng/NLGraph.Comment: NeurIPS 2023 Spotligh
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