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Register a new userA Weighted Common Subgraph Matching Algorithm
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Xu Yang Dr.
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We propose a weighted common subgraph (WCS) matching algorithm to find the most similar subgraphs in two labeled weighted graphs. WCS matching, as a natural generalization of the equal-sized graph matching or subgraph matching, finds wide applications in many computer vision and machine learning tasks. In this paper, the WCS matching is first formulated as a combinatorial optimization problem over the set of partial permutation matrices. Then it is approximately solved by a recently proposed combinatorial optimization framework - Graduated NonConvexity and Concavity Procedure (GNCCP). Experimental comparisons on both synthetic graphs and real world images validate its robustness against noise level, problem size, outlier number, and edge density.
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This article identifies a key algorithmic ingredient in the edge-weighted online matching algorithm by Zadimoghaddam (2017) and presents a simplified algorithm and its analysis to demonstrate how it works in the unweighted case.
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pal for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to be applied in these models. Despite its simplicity and appeal, and despite being studied in multiple works, the greedy-based algorithm was not fully understood in any of the studied online models, and its actual performance remained an open question for more than a decade. We provide a thorough analysis of the greedy-based algorithm in several online models. For vertex arrivals in bipartite graphs, we characterize the exact competitive-ratio of this algorithm in the random-order model, for any arrival order of the vertices subsequent to the sampling phase (adversarial and random orders in particular). We use it to derive tight analysis in the recent adversarial-order model with a sample (AOS model) for any sample size, providing the first result in this model beyond the simple secretary problem. Then, we generalize and strengthen the black box method of converting results in the random-order model to single-sample prophet inequalities, and use it to derive the state-of-the-art single-sample prophet inequality for the problem. Finally, we use our new techniques to analyze the greedy-based algorithm for edge arrivals in general graphs and derive results in all the mentioned online models. In this case as well, we improve upon the state-of-the-art single-sample prophet inequality.
The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other hand, the problem is known to be ${bf NP}$-hard in vertex-labeled partial $11$-trees of bounded degree. We consider series-parallel graphs, i.e., partial $2$-trees. We show that the problem remains ${bf NP}$-hard in biconnected series-parallel graphs with all but one vertex of degree $3$ or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series-parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.
Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial $1/2$-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondrak (SODA 2013) that restricts beating a $1/2$ competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.
In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of textsc{Clique}, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by $k := text{vc}(G_1)+text{vc}(G_2)$ the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time $2^{O(k log k)}$. We complement this result by showing that, unless the ETH fails, it cannot be solved in time $2^{o(k log k)}$. This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by $k$, unless $NP subseteq mathsf{coNP}/poly$. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.
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