More Like this

1. Insight into Voting Problem Complexity Using Randomized Classes

   https://doi.org/10.24963/ijcai.2022/42  

   Fitzsimmons, Zack; Hemaspaandra, Edith (July 2022, International Joint Conference on Artificial Intelligence (IJCAI))

   The first step in classifying the complexity of an NP problem is typically showing the problem in P or NP-complete. This has been a successful first step for many problems, including voting problems. However, in this paper we show that this may not always be the best first step. We consider the problem of constructive control by replacing voters (CCRV) introduced by Loreggia et al. [2015, https://dl.acm.org/doi/10.5555/2772879.2773411] for the scoring rule First-Last, which is defined by (1, 0, ..., 0, -1). We show that this problem is equivalent to Exact Perfect Bipartite Matching, and so CCRV for First-Last can be determined in random polynomial time. So on the one hand, if CCRV for First-Last is NP-complete then RP = NP, which is extremely unlikely. On the other hand, showing that CCRV for First-Last is in P would also show that Exact Perfect Bipartite Matching is in P, which would solve a well-studied 40-year-old open problem.Considering RP as an option for classifying problems can also help classify problems that until now had escaped classification. For example, the sole open problem in the comprehensive table from Erdélyi et al. [2021, https://doi.org/10.1007/s10458-021-09523-9] is CCRV for 2-Approval. We show that this problem is in RP, and thus easy since it is widely assumed that P = RP. 

   more » « less
2. Vertex degree sums for perfect matchings in 3-uniform hypergraphs

   Zhang, Yi and (September 2018, The Electronic journal of combinatorics)

   We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without isolated vertex. Suppose that $$H$$ is a 3-uniform hypergraph whose order $$n$$ is sufficiently large and divisible by $$3$$. If $$H$$ contains no isolated vertex and $$\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$$ for any two vertices $$u$$ and $$v$$ that are contained in some edge of $$H$$, then $$H$$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different \emph{space barrier} from the one for the corresponding Dirac problem. 

   more » « less
3. Algorithms for Metric Learning via Contrastive Embeddings

   Ihara, Diego; Mohammadi, Neshat; Sidiropoulos, Anastasios (June 2019, Leibniz international proceedings in informatics)

   We study the problem of supervised learning a metric space under discriminative constraints. Given a universe X and sets S, D subset binom{X}{2} of similar and dissimilar pairs, we seek to find a mapping f:X -> Y, into some target metric space M=(Y,rho), such that similar objects are mapped to points at distance at most u, and dissimilar objects are mapped to points at distance at least l. More generally, the goal is to find a mapping of maximum accuracy (that is, fraction of correctly classified pairs). We propose approximation algorithms for various versions of this problem, for the cases of Euclidean and tree metric spaces. For both of these target spaces, we obtain fully polynomial-time approximation schemes (FPTAS) for the case of perfect information. In the presence of imperfect information we present approximation algorithms that run in quasi-polynomial time (QPTAS). We also present an exact algorithm for learning line metric spaces with perfect information in polynomial time. Our algorithms use a combination of tools from metric embeddings and graph partitioning, that could be of independent interest. 

   more » « less
4. Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability

   Trobst, Thorben; Vazirani, Vijay V (July 2024, ACM EC)

   Unlike ordinal-utility matching markets, which are well-developed from the viewpoint of both theory and practice, recent insights from a computer science perspective have left cardinal-utility matching markets in a state of flux. The celebrated pricing-based mechanism for one-sided cardinal-utility matching markets due to Hylland and Zeckhauser [1979](HZ), which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete (Vazirani and Yannakakis [2021], Chen et al. [2022]). This led us to ask the question: is there an alternative, polynomial time, mechanism for one-sided cardinal-utility matching markets which achieves the desirable properties of HZ, i.e. (ex-ante) envyfreeness (EF) and Pareto-optimality (PO)? We show that the problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is by itself already PPAD-complete. However, a (2 + ǫ)-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani [2022]. Moreover, the mechanism is also (2+ǫ)-approximately incentive compatible. We also present several results on two-sided cardinal-utility matching markets, including non-existence of EF+PO lotteries as well as existence of justified-envy-free and weak Pareto-optimal lotteries. 

   more » « less
5. Seeded graph matching via large neighborhood statistics

   https://doi.org/10.1002/rsa.20934  

   Mossel, Elchanan; Xu, Jiaming (June 2020, Random Structures & Algorithms)

   We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of verticesn. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low asin the sparse graph regime (with the average degree smaller than) andin the dense graph regime, for a small positive constant. Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds. 

   more » « less