Hypergraph Matching Problem Research Articles

Joint Object Detection and Multi-Object Tracking Based on Hypergraph Matching

Addressing the challenges in online multi-object tracking algorithms under complex scenarios, where the independence among feature extraction, object detection, and data association modules leads to both error accumulation and the difficulty of maintaining visual consistency for occluded objects, we have proposed an end-to-end multi-object tracking method based on hypergraph matching (JDTHM). Initially, a feature extraction and object detection module is introduced to achieve preliminary localization and description of the objects. Subsequently, a deep feature aggregation module is designed to extract temporal information from historical tracklets, amalgamating features from object detection and feature extraction to enhance the consistency between the current frame features and the tracklet features, thus preventing identity swaps and tracklet breaks caused by object detection loss or distortion. Finally, a data association module based on hypergraph matching is constructed, integrating with object detection and feature extraction into a unified network, transforming the data association problem into a hypergraph matching problem between the tracklet hypergraph and the detection hypergraph, thereby achieving end-to-end model optimization. The experimental results demonstrate that this method has yielded favorable qualitative and quantitative analysis results on three multi-object tracking datasets, thereby validating its effectiveness in enhancing the robustness and accuracy of multi-object tracking tasks.

Fit Entropy-Based Dynamic Communication Resource Slicing-Optimization Method in Smart Distribution Grids on the Medium-Low-Voltage Side.

With the rapid development of new energy and smart technology, the demand for inter-device communication in medium-low-voltage smart distribution grids has sharply increased, leading to a surge in the variety and quantity of communication services. To meet the needs of diverse and massive communication services, deploying service function chains to flexibly combine virtual resources has become crucial. This paper proposes an optimization method based on fit entropy and network utility to address the limited communication network resources in medium-low-voltage smart distribution grids. This was conducted by modeling the distribution grid as a three-domain model consisting of a service domain, a logical domain, and a physical domain and transforming it into a hierarchical bipartite hypergraph-matching problem, which is a complex combinatorial optimization problem. This paper introduces two matching optimization algorithms: "business domain-logic domain-physical domain integration" and "service domain-logic domain, logic domain-physical domain two-stage", which effectively address this problem based on fit entropy and utility. The simulation results demonstrate that these algorithms significantly improve service success rates and resource utilization, enhancing overall network utility.

Almost tight bounds for online hypergraph matching

In the online hypergraph matching problem, hyperedges of size k over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A naïve greedy algorithm for this problem achieves a competitive ratio of 1k. We show that no (randomized) online algorithm has competitive ratio better than 2+o(1)k. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio 1−o(1)ln⁡(k) and show that no online algorithm can have competitive ratio strictly better than 1+o(1)ln⁡(k). Lastly, we give a 1−o(1)ln⁡(k) competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.

SIGMA++: Improved Semantic-Complete Graph Matching for Domain Adaptive Object Detection.

Domain Adaptive Object Detection (DAOD) generalizes the object detector from an annotated domain to a label-free novel one. Recent works estimate prototypes (class centers) and minimize the corresponding distances to adapt the cross-domain class conditional distribution. However, this prototype-based paradigm 1) fails to capture the class variance with agnostic structural dependencies, and 2) ignores the domain-mismatched classes with a sub-optimal adaptation. To address these two challenges, we propose an improved SemantIc-complete Graph MAtching framework, dubbed SIGMA++, for DAOD, completing mismatched semantics and reformulating adaptation with hypergraph matching. Specifically, we propose a Hypergraphical Semantic Completion (HSC) module to generate hallucination graph nodes in mismatched classes. HSC builds a cross-image hypergraph to model class conditional distribution with high-order dependencies and learns a graph-guided memory bank to generate missing semantics. After representing the source and target batch with hypergraphs, we reformulate domain adaptation with a hypergraph matching problem, i.e., discovering well-matched nodes with homogeneous semantics to reduce the domain gap, which is solved with a Bipartite Hypergraph Matching (BHM) module. Graph nodes are used to estimate semantic-aware affinity, while edges serve as high-order structural constraints in a structure-aware matching loss, achieving fine-grained adaptation with hypergraph matching. The applicability of various object detectors verifies the generalization of SIGMA++, and extensive experiments on nine benchmarks show its state-of-the-art performance on both AP 50 and adaptation gains.

Game-theoretic hypergraph matching with density enhancement

Feature matching plays a fundamental role in computer vision and pattern recognition. As straightforward comparison of feature descriptors is not enough to provide reliable matching results in many situations, graph matching makes use of the pairwise relationship between features to improve matching accuracy. Hypergraph matching further employs the relationship among multiple features to provide more invariance between feature correspondences. Existing hypergraph matching algorithms usually solve an assignment problem, where outliers may result in a large number of false matches. In this paper we cast the hypergraph matching problem as a non-cooperative multi-player game, and obtain the matches by extracting the evolutionary stable strategies. Our algorithm exerts a strong constraint on the consistency of obtained matches, and false matches are excluded effectively. In order to increase the number of matches without increasing the computation load evidently, we present a density enhancement method to improve the matching results. We further propose two methods to enforce the one-to-one constraint, thereby removing false matches and maintaining a high matching accuracy. Experiments with both synthetic and real datasets validate the effectiveness of our algorithm.

On the complexity of stable hypergraph matching, stable multicommodity flow and related problems

In this paper we study stable multicommodity flows, stable hypergraph matchings and some related problems, like the Hospital-Resident-Couple (hrc) problem. We give a simpler proof of the existence of a stable multicommodity flow by reducing it to scarf. We show that finding a stable fractional solution of a hrc or a 3-dimensional stable matching (stable family) instance is PPAD-complete. Then we derive from these results that the Fractional hypergraph matching problem is PPAD-complete even if all hyperedge sizes and vertex degrees are at most 3. Furthermore, we prove that computing stable multicommodity flows is PPAD-complete even if the number of commodities is 3, the sum of in and out degrees is at most 3 at each vertex and all terminals and sources are the same. We also show that deciding if there exists a stable integer multicommodity flow is NP-complete even with only two commodities and similar restrictions.

Random cliques in random graphs and sharp thresholds for F$$ F $$‐factors

AbstractWe show that for each , in a density range extending up to, and slightly beyond, the threshold for a ‐factor, the copies of in the random graph are randomly distributed, in the (one‐sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for to contain a ‐factor. The case is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for ‐factors in random ‐uniform hypergraphs, as well as (in some cases weaker) generalizations replacing by certain other (hyper)graphs.

Technical Note—Online Hypergraph Matching with Delays

Scheduling High-Capacity Shared Rides via Hypergraph Matchings Sharing economies, such as Uber Pool and Lyft Shared Saver, are becoming ubiquitous in modern transportation services. “Online Hypergraph Matching with Delays” by M. Pavone, A. Saberi, M. Schiffer, and M. Tsao studies an online hypergraph matching problem inspired by on-demand scheduling of shared rides in ridehailing systems. They formulate a ridesharing problem through a weighted hypergraph in which riders are represented by vertices and hyperedges and the associated weights represent the utility of serving the associated vertices groups of riders with a single vehicle. A ridesharing algorithm is, thus, a matching algorithm for the hypergraph. To model the on-demand nature of ridesharing, riders arrive sequentially and must either be quickly and irrevocably matched or discarded. In this work, the authors present a polynomial-time algorithm based on randomized batching and prove that it has the optimal worst-case performance.

Random triangles in random graphs

AbstractIn a recent paper, Oliver Riordan shows that for and p up to and slightly larger than the threshold for a Kr‐factor, the hypergraph formed by the copies of Kr in G(n, p) contains a copy of the binomial random hypergraph with . For r = 3, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all . With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a Kr‐factor in G(n, p) for . In this note, we resolve the missing case r = 3 by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in G(n, p).

Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking

The likelihood determined by the distance between measurements and predicted states of targets is widely used in many filters for data association. However, if the actual motion model of targets is not coincided with the preset dynamic motion model, this criterion will lead to poor performance when close-space targets are tracked. For rigid target tracking task, the structure of rigid targets can be exploited to improve the data association performance. In this paper, the structure of the rigid target is represented as a hypergraph, and the problem of data association is formulated as a hypergraph matching problem. However, the performance of hypergraph matching degrades if there are missed detections and clutter. To overcome this limitation, we propose a joint probabilistic hypergraph matching labeled multi-Bernoulli (JPHGM-LMB) filter with all undetected cases being considered. In JPHGM-LMB, the likelihood is built based on group structure rather than the distance between predicted states and measurements. Consequently, the probability of each target associated with each measurement (joint association probabilities) can be obtained. Then, the structure information is integrated into LMB filter by revising each single target likelihood with joint association probabilities. However, because all undetected cases is considered, proposed approach is usable in real time only for a limited number of targets. Extensive simulations have demonstrated the significant performance improvement of our proposed method.

Deterministic Dynamic Matching in O(1) Update Time

We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2011) , Solomon (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2016) gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of (2+varepsilon ) and an amortized update time of O(log n/varepsilon ^2). Our result can be generalized to give a fully dynamic O(f^3)-approximate algorithm with O(f^2) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.

Intrinsic shape matching via tensor-based optimization

This paper presents a simple yet efficient framework for finding a set of sparse correspondences between two non-rigid shapes using a tensor-based optimization technique. To make the matching consistent, we propose to use third-order potentials to define the similarity tensor measure between triplets of feature points. Given two non-rigid 3D models, we first extract two sets of feature points residing in shape extremities, and then build the similarity tensor as a combination of the geodesic-based and prior-based similarities. The hyper-graph matching problem is formulated as the maximization of an objective function over all possible permutations of points, and it is solved by a tensor power iteration technique, which involves row/column normalization. Finally, a consistent set of discrete correspondences is automatically obtained. Various experimental results have demonstrated the superiority of our proposed method, compared with several state-of-the-art methods.

Learning Kinematic Structure Correspondences Using Multi-Order Similarities.

In this paper, we present a novel framework for finding the kinematic structure correspondences between two articulated objects in videos via hypergraph matching. In contrast to appearance and graph alignment based matching methods, which have been applied among two similar static images, the proposed method finds correspondences between two dynamic kinematic structures of heterogeneous objects in videos. Thus our method allows matching the structure of objects which have similar topologies or motions, or a combination of the two. Our main contributions can be summarised as follows: (i) casting the kinematic structure correspondence problem into a hypergraph matching problem by incorporating multi-order similarities with normalising weights, (ii) introducing a structural topology similarity measure by aggregating topology constrained subgraph isomorphisms, (iii) measuring kinematic correlations between pairwise nodes, and (iv) proposing a combinatorial local motion similarity measure using geodesic distance on the Riemannian manifold. We demonstrate the robustness and accuracy of our method through a number of experiments on synthetic and real data, outperforming various other state of the art methods. Our method is not limited to a specific application nor sensor, and can be used as building block in applications such as action recognition, human motion retargeting to robots, and articulated object manipulation amongst others.

A quadratic penalty method for hypergraph matching

Hypergraph matching is a fundamental problem in computer vision. Mathematically, it maximizes a polynomial objective function, subject to assignment constraints. In this paper, we reformulate the hypergraph matching problem as a sparse constrained optimization problem. By dropping the sparse constraint, we show that the resulting relaxation problem can recover the global minimizer of the original problem. This property heavily depends on the special structures of hypergraph matching. The critical step in solving the original problem is to identify the location of nonzero entries (referred to as the support set) in a global minimizer. Inspired by such observation, we apply the quadratic penalty method to solve the relaxation problem. Under reasonable assumptions, we show that the support set of the global minimizer in a hypergraph matching problem can be correctly identified when the number of iterations is sufficiently large. A projected gradient method is applied as a subsolver to solve the quadratic penalty subproblem. Numerical results demonstrate that the exact recovery of the support set indeed happens, and the proposed algorithm is efficient in terms of both accuracy and CPU time.

Probabilistic hypergraph matching based on affinity tensor updating

Graph matching is a fundamental problem in artificial intelligence and structural data processing. Hypergraph matching has recently become popular in the graph matching community. Existing hypergraph matching algorithms usually resort to the continuous methods, while the combinatorial nature of hypergraph matching is not well considered. Therefore in this paper, we propose a novel hypergraph matching algorithm by introducing the affinity tensor updating based graduated projection. Specifically, the hypergraph matching problem is first formulated as a combinatorial optimization problem in a high order polynomial form. Then this NP-hard problem is relaxed and interpreted in a probabilistic manner, which is approximately solved by iterative techniques. The updating of the affinity tensor is performed in each iteration, besides the updating of probabilistic assignment vector. Experimental results on both synthetic and real-world datasets witness the effectiveness of the proposed method.

A Note on $\mathtt{V}$-free 2-matchings

Motivated by a conjecture of Liang, we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.

Coverings and matchings in r‐partite hypergraphs

AbstractRyser's conjecture postulates that for r ‐partite hypergraphs, τ ≤ (r ‐ 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this article, we prove several results pertaining to matchings and coverings in r ‐partite intersecting hypergraphs. First, we prove that finding a minimum cardinality vertex cover for an r ‐partite intersecting hypergraph is NP‐hard. Second, we note Ryser's conjecture for intersecting hypergraphs is easily resolved if a given hypergraph does not contain a particular subhypergraph, which we call a “tornado.” We prove several bounds on the covering number of tornados. Finally, we prove the integrality gap for the standard integer linear programming formulation of the maximum cardinality r ‐partite hypergraph matching problem is at least r ‐ k where k is the smallest positive integer such that r ‐ k is a prime power. © 2012 Wiley Periodicals, Inc. NETWORKS, Vol. 2012

A Tensor-Based Algorithm for High-Order Graph Matching

This paper addresses the problem of establishing correspondences between two sets of visual features using higher order constraints instead of the unary or pairwise ones used in classical methods. Concretely, the corresponding hypergraph matching problem is formulated as the maximization of a multilinear objective function over all permutations of the features. This function is defined by a tensor representing the affinity between feature tuples. It is maximized using a generalization of spectral techniques where a relaxed problem is first solved by a multidimensional power method and the solution is then projected onto the closest assignment matrix. The proposed approach has been implemented, and it is compared to state-of-the-art algorithms on both synthetic and real data.

On linear and semidefinite programming relaxations for hypergraph matching

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following:

On distributed scheduling in wireless networks exploiting broadcast and network coding

In this paper, we consider cross-layer optimization in wireless networks with wireless broadcast advantage, focusing on the problem of distributed scheduling of broadcast links. The wireless broadcast advantage is most useful in multicast scenarios. As such, we include network coding in our design to exploit the throughput gain brought in by network coding for multicasting. We derive a subgradient algorithm for joint rate control, network coding and scheduling, which however requires centralized link scheduling. Under the primary interference model, link scheduling problem is equivalent to a maximum weighted hypergraph matching problem that is NP-complete. To solve the scheduling problem distributedly, locally greedy and randomized approximation algorithms are proposed and shown to have bounded worst-case performance. With random network coding, we obtain a fully distributed cross-layer design. Numerical results show promising throughput gain using the proposed algorithms, and surprisingly, in some cases even with less complexity than cross-layer design without broadcast advantage.