572 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
• When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard.
• When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
2-Approximation for Prize-Collecting Steiner Forest
Approximation algorithms for the prize-collecting Steiner forest problem
(PCSF) have been a subject of research for over three decades, starting with
the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on
Steiner forest and prize-collecting problems. In this paper, we propose and
analyze a natural deterministic algorithm for PCSF that achieves a
-approximate solution in polynomial time. This represents a significant
improvement compared to the previously best known algorithm with a
-approximation factor developed by Hajiaghayi and Jain in 2006.
Furthermore, K{\"{o}}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have
established an integrality gap of at least for the natural LP relaxation
for PCSF. However, we surpass this gap through the utilization of a
combinatorial algorithm and a novel analysis technique. Since is the best
known approximation guarantee for Steiner forest problem, which is a special
case of PCSF, our result matches this factor and closes the gap between the
Steiner forest problem and its generalized version, PCSF
Constrained Submodular Maximization via New Bounds for DR-Submodular Functions
Submodular maximization under various constraints is a fundamental problem
studied continuously, in both computer science and operations research, since
the late 's. A central technique in this field is to approximately
optimize the multilinear extension of the submodular objective, and then round
the solution. The use of this technique requires a solver able to approximately
maximize multilinear extensions. Following a long line of work, Buchbinder and
Feldman (2019) described such a solver guaranteeing -approximation for
down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no
solver can guarantee better than -approximation. In this paper, we
present a solver guaranteeing -approximation, which significantly
reduces the gap between the best known solver and the inapproximability result.
The design and analysis of our solver are based on a novel bound that we prove
for DR-submodular functions. This bound improves over a previous bound due to
Feldman et al. (2011) that is used by essentially all state-of-the-art results
for constrained maximization of general submodular/DR-submodular functions.
Hence, we believe that our new bound is likely to find many additional
applications in related problems, and to be a key component for further
improvement.Comment: 48 page
LP-based approximation algorithms for partial-ordered scheduling and matroid augmentation
In this thesis, we study two NP-hard problems from Combinatorial Optimization, from the perspective of approximation algorithms. The first problem we study is called Partial-Order Scheduling on Parallel Machines, which we abbreviate to PO Scheduling. Here, we are given a partially ordered set of jobs which we want to schedule to a set of machines. Each job has some weight and some processing time associated to it. On each machine, the order of the jobs scheduled to it must agree with the given partial order, i.e., a job can only be started once all its predecessors scheduled to the same machine have been completed. However, two jobs scheduled to different machines are not constrained in any way. Thus, PO Scheduling deviates from the well-studied problem of precedence-constrained scheduling in this regard. The goal of PO Scheduling is to find a feasible schedule which minimizes the sum of weighted completion times of the jobs. PO Scheduling generalizes an already NP-hard version of scheduling introduced by Bosman, Frascaria, Olver, Sitters and Stougie [3], where they ask the same question as in PO Scheduling for the case where the jobs are totally ordered. The authors above present a constant-factor approximation algorithm for their problem. We conjecture that there is a constant-factor approximation algorithm for PO Scheduling as well. While we do not solve the problem, we give approximation algorithms for the special case that the partial order consists of disjoint totally ordered chains of linearly bounded length. Additionally, we give a structural result for optimal schedules in the case that the partial order consists of disjoint, backwardly ordered (with regard to the Smith-ratio) chains. We point towards some potential research directions. For the Weighted Tree Augmentation Problem, we are given a graph with a distinguished spanning tree. Each non tree-edge has a cost associated to it. The goal is to find a cost-minimal set of edges such that when we add them to the tree-edges, the resulting graph is 2-edge-connected. Weighted tree augmentation is NP-hard. There has been recent progress in decreasing the best-known approximation factor for the problem by Traub and Zenklusen to (1.5 + ε) [51, 52]. We study a generalization of weighted tree augmentation, called the Weighted Matroid Augmentation Problem, which we abbreviate to WMAP. In WMAP, we consider a matroid with a distinguished basis and a cost function on the non-basis elements. The goal is to find a cost-minimal set such that the union of the fundamental circuits of the elements in the set with regard to the distinguished basis cover that basis. We conjecture that there is a 2-approximation algorithm for the problem in the case that the matroid is regular. While we do not solve the problem, we give an approximation algorithm for the special case of the cographic matroid and show that there is no constant-factor approximation algorithm for WMAP for representable matroids unless P = NP
Analysing trajectory similarity and improving graph dilation
In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs
A linearly ordered (LO) -colouring of a hypergraph is a colouring of its vertices with colours such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO -colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO -colourable, and the case that it is not even LO -colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023)
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