IA Scholar Query: The Parameterized Complexity of k-Edge Induced Subgraphs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 06 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Construction of Fully Faithful Tropicalizations for Curves in Ambient Dimension 3
https://scholar.archive.org/work/azx26kbvwzhsjcodu6sqarccza
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the tropicalization has nice properties. We construct, for any Mumford curve X, a map to a three-dimensional torus, such that the tropicalization is isometric to a subgraph of the Berkovich space X^ an, called the extended skeleton. In this case, we say the tropicalization is "fully faithful." Additionally, given a map X to a toric variety Y, which induces a fully faithful tropicalization, we show that we can extend the map to X → Y × (𝐏^1)^n such that the new tropicalization is smooth and fully faithful.Trevor Gunn, Philipp Jellwork_azx26kbvwzhsjcodu6sqarcczaTue, 06 Dec 2022 00:00:00 GMTStars: Tera-Scale Graph Building for Clustering and Graph Learning
https://scholar.archive.org/work/cgrr3xvm65hfrngoynyt6ul6o4
A fundamental procedure in the analysis of massive datasets is the construction of similarity graphs. Such graphs play a key role for many downstream tasks, including clustering, classification, graph learning, and nearest neighbor search. For these tasks, it is critical to build graphs which are sparse yet still representative of the underlying data. The benefits of sparsity are twofold: firstly, constructing dense graphs is infeasible in practice for large datasets, and secondly, the runtime of downstream tasks is directly influenced by the sparsity of the similarity graph. In this work, we present Stars: a highly scalable method for building extremely sparse graphs via two-hop spanners, which are graphs where similar points are connected by a path of length at most two. Stars can construct two-hop spanners with significantly fewer similarity comparisons, which are a major bottleneck for learning based models where comparisons are expensive to evaluate. Theoretically, we demonstrate that Stars builds a graph in nearly-linear time, where approximate nearest neighbors are contained within two-hop neighborhoods. In practice, we have deployed Stars for multiple data sets allowing for graph building at the Tera-Scale, i.e., for graphs with tens of trillions of edges. We evaluate the performance of Stars for clustering and graph learning, and demonstrate 10 1000-fold improvements in pairwise similarity comparisons compared to different baselines, and 2 10-fold improvement in running time without quality loss.CJ Carey, Jonathan Halcrow, Rajesh Jayaram, Vahab Mirrokni, Warren Schudy, Peilin Zhongwork_cgrr3xvm65hfrngoynyt6ul6o4Mon, 05 Dec 2022 00:00:00 GMTHyperbolic Curvature Graph Neural Network
https://scholar.archive.org/work/fnzmhgbykbhsvkd5z3djpv2jda
Hyperbolic space is emerging as a promising learning space for representation learning, owning to its exponential growth volume. Compared with the flat Euclidean space, the curved hyperbolic space is far more ambient and embeddable, particularly for datasets with implicit tree-like architectures, such as hierarchies and power-law distributions. On the other hand, the structure of a real-world network is usually intricate, with some regions being tree-like, some being flat, and others being circular. Directly embedding heterogeneous structural networks into a homogeneous embedding space unavoidably brings inductive biases and distortions. Inspiringly, the discrete curvature can well describe the local structure of a node and its surroundings, which motivates us to investigate the information conveyed by the network topology explicitly in improving geometric learning. To this end, we explore the properties of the local discrete curvature of graph topology and the continuous global curvature of embedding space. Besides, a Hyperbolic Curvature-aware Graph Neural Network, HCGNN, is further proposed. In particular, HCGNN utilizes the discrete curvature to lead message passing of the surroundings and adaptively adjust the continuous curvature simultaneously. Extensive experiments on node classification and link prediction tasks show that the proposed method outperforms various competitive models by a large margin in both high and low hyperbolic graph data. Case studies further illustrate the efficacy of discrete curvature in finding local clusters and alleviating the distortion caused by hyperbolic geometry.Menglin Yang, Min Zhou, Lujia Pan, Irwin Kingwork_fnzmhgbykbhsvkd5z3djpv2jdaSun, 04 Dec 2022 00:00:00 GMTParameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments
https://scholar.archive.org/work/7wd6y3y765dmjcohjx4r74cfie
In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are given a weighted set of n axis-parallel rectangles in the plane. The task is to find a subset of pairwise non-overlapping rectangles with the maximum possible total weight. This problem is NP-hard and the best-known polynomial-time approximation algorithm, due to by Chalermsook and Walczak (SODA 2021), achieves approximation factor O(loglog n ). While in the unweighted setting, constant factor approximation algorithms are known, due to Mitchell (FOCS 2021) and to Gálvez et al. (SODA 2022), it remains open to extend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterized approximation. Grandoni et al. (ESA 2019) gave a (1-ϵ)-approximation algorithm with running time k^O(k/ϵ^8) n^O(1/ϵ^8) time, where k is the number of rectangles in an optimum solution. Unfortunately, their algorithm works only in the unweighted setting and they left it as an open problem to give a parameterized approximation scheme in the weighted setting. Our contribution is a partial answer to the open question of Grandoni et al. (ESA 2019). We give a parameterized approximation algorithm for MWISR that given a parameter k, finds a set of non-overlapping rectangles of weight at least (1-ϵ) opt_k in 2^O(k log(k/ϵ)) n^O(1/ϵ) time, where opt_k is the maximum weight of a solution of cardinality at most k. Note that thus, our algorithm may return a solution consisting of more than k rectangles. To complement this apparent weakness, we also propose a parameterized approximation scheme with running time 2^O(k^2 log(k/ϵ)) n^O(1) that finds a solution with cardinality at most k and total weight at least (1-ϵ)opt_k for the special case of axis-parallel segments.Jana Cslovjecsek and Michał Pilipczuk and Karol Węgrzyckiwork_7wd6y3y765dmjcohjx4r74cfieSat, 03 Dec 2022 00:00:00 GMTLearning to Reason with a Scalable Probabilistic Logic
https://scholar.archive.org/work/yunqmaoql5cavgw7hdogwqzvsa
Learning to reason and understand the world's knowledge is a fundamental problem in Artificial Intelligence (AI). Traditional symbolic AI methods were popular in the 1980s, when first-order logic rules were mostly handwritten, and reasoning algorithms were built on top of them. In the 90s, more and more researchers became interested in statistical methods that deal with the uncertainty of the data, using probabilistic models. While it is always hypothesized that both the symbolic and statistical approaches are necessary for building intelligent systems, in practice, bridging the two in a combined framework often brings intractability—most probabilistic first-order logics are simply not efficient enough for real-world sized tasks. For example, Markov Logics [83] integrate first-order logics with Markov random field theory, but when mapping the entities in a knowledge base (KB) to the propositional theory (i.e., grounding), the size of the network depends on the number of facts in the KB—i.e., O(nk ) where k is the arity of the predicate, and n is the number of KB constants. In this thesis, we design a new probabilistic logic programming paradigm to address various scalability issues in probabilistic logics. We propose a group of scalable methods for inference, learning, and inducing the structure of probabilistic logics. More specifically, we propose a scalable probabilistic logic called ProPPR [105] to combine the best of the symbolic and statistical worlds. ProPPR can be viewed as a probabilistic version of Prolog, and we associate a feature vector for each clause to learn weights from data. The learned weights are used to control search during inference. ProPPR's inference scheme is very special: instead of performing potentially intractable global inference, ProPPR uses a provably-correct approximate personalized PageRank to conduct local grounding, whose inference time is independent of the size of the KB. To test ProPPR for large, real-world relational learning [...]William Yang Wangwork_yunqmaoql5cavgw7hdogwqzvsaFri, 02 Dec 2022 00:00:00 GMTNatural and Artificial Dynamics in Graphs: Concept, Progress, and Future
https://scholar.archive.org/work/7jarjqraobdr7ojfxkt3r7flbe
Graph structures have attracted much research attention for carrying complex relational information. Based on graphs, many algorithms and tools are proposed and developed for dealing with real-world tasks such as recommendation, fraud detection, molecule design, etc. In this paper, we first discuss three topics of graph research, i.e., graph mining, graph representations, and graph neural networks (GNNs). Then, we introduce the definitions of natural dynamics and artificial dynamics in graphs, and the related works of natural and artificial dynamics about how they boost the aforementioned graph research topics, where we also discuss the current limitation and future opportunities.Dongqi Fu, Jingrui Hework_7jarjqraobdr7ojfxkt3r7flbeFri, 02 Dec 2022 00:00:00 GMTRelation-aware Language-Graph Transformer for Question Answering
https://scholar.archive.org/work/3e45wgrrezburkgpgq44nfsyym
Question Answering (QA) is a task that entails reasoning over natural language contexts, and many relevant works augment language models (LMs) with graph neural networks (GNNs) to encode the Knowledge Graph (KG) information. However, most existing GNN-based modules for QA do not take advantage of rich relational information of KGs and depend on limited information interaction between the LM and the KG. To address these issues, we propose Question Answering Transformer (QAT), which is designed to jointly reason over language and graphs with respect to entity relations in a unified manner. Specifically, QAT constructs Meta-Path tokens, which learn relation-centric embeddings based on diverse structural and semantic relations. Then, our Relation-Aware Self-Attention module comprehensively integrates different modalities via the Cross-Modal Relative Position Bias, which guides information exchange between relevant entities of different modalities. We validate the effectiveness of QAT on commonsense question answering datasets like CommonsenseQA and OpenBookQA, and on a medical question answering dataset, MedQA-USMLE. On all the datasets, our method achieves state-of-the-art performance. Our code is available at http://github.com/mlvlab/QAT.Jinyoung Park, Hyeong Kyu Choi, Juyeon Ko, Hyeonjin Park, Ji-Hoon Kim, Jisu Jeong, Kyungmin Kim, Hyunwoo J. Kimwork_3e45wgrrezburkgpgq44nfsyymFri, 02 Dec 2022 00:00:00 GMTBatch Exchanges with Constant Function Market Makers: Axioms, Equilibria, and Computation
https://scholar.archive.org/work/3syya4jtjzg5hbc4zai2cml25u
Batch trading systems and constant function market makers (CFMMs) are two distinct market design innovations that have recently come to prominence as ways to address some of the shortcomings of decentralized trading systems. However, different deployments have chosen substantially different methods for integrating the two innovations. We show here from a minimal set of axioms describing the beneficial properties of each innovation that there is in fact only one, unique method for integrating CFMMs into batch trading schemes that preserves all the beneficial properties of both. Deployment of a batch trading schemes trading many assets simultaneously requires a reliable algorithm for approximating equilibria in Arrow-Debreu exchange markets. We study this problem when batches contain limit orders and CFMMs. Specifically, we find that CFMM design affects the asymptotic complexity of the problem, give an easily-checkable criterion to validate that a user-submitted CFMM is computationally tractable in a batch, and give a convex program that computes equilibria on batches of limit orders and CFMMs. Equivalently, this convex program computes equilibria of Arrow-Debreu exchange markets when every agent's demand response satisfies weak gross substitutability and every agent has utility for only two types of assets. This convex program has rational solutions when run on many (but not all) natural classes of widely-deployed CFMMs.Geoffrey Ramseyer, Mohak Goyal, Ashish Goel, David Mazièreswork_3syya4jtjzg5hbc4zai2cml25uFri, 02 Dec 2022 00:00:00 GMTParsing and Generation for the Abstract Meaning Representation
https://scholar.archive.org/work/uv6k4fkzdzdb5b7cudyhqmb4jy
A key task in intelligent language processing is obtaining semantic representations that abstract away from surface lexical and syntactic decisions. The Abstract Meaning Representation (AMR) is one such representation, which represents the meaning of a sentence as labeled nodes in a graph (concepts) and labeled, directed edges between them (relations). Two traditional problems of semantic representations are producing them from natural language (parsing) as well as producing natural language from them (generation). In this thesis, I present algorithms for parsing and generation for AMR. In the first part of the thesis, I present a parsing algorithm for AMR that produces graphs that satisfy semantic well-formedness constraints. The parsing algorithm uses Lagrangian relaxation combined with an exact algorithm for finding the maximum, spanning, connected subgraph of a graph to produce AMR graphs that satisfy these constraints. In the second part of the thesis, I present a generation algorithm for AMR. The algorithm uses a tree-transducer that operates on a spanning-tree of the input AMR graph to produce output natural language sentences. Datasparsity of the training data is an issue for AMR generation, which we overcome by including synthetic rules in the tree-transducer.Jeffrey Flaniganwork_uv6k4fkzdzdb5b7cudyhqmb4jyFri, 02 Dec 2022 00:00:00 GMTFeasibility, Efficiency, and Robustness of Secure Computation
https://scholar.archive.org/work/y23vxqpc2fey5p373kgule6f24
Secure computation allows mutually distrusting parties to compute over private data. Such collaborations have widespread applications in social, scientific, commercial, and security domains. However, the overhead of achieving security is a major bottleneck to the adoption of such technologies. In this context, this thesis aims to design the most secure protocol within budgeted computational or network resources by mathematically formulating it as an optimization problem. With the rise in CPU power and cheap RAM, the offline-online model for secure computation has become the prominent model for real-world security systems. This thesis investigates the above-mentioned optimization problem in the information-theoretic offline-online model. In particular, this thesis presents the following selected sample of our research in greater detail. Round and Communication Complexity: Chor-Kushilevitz-Beaver characterized the round and communication complexity of secure two-party computation. Since then, the case of functions with randomized output remained unexplored. We proved the decidability of determining these complexities. Next, if such a protocol exists, we construct the optimal protocol; otherwise, we present an obstruction to achieving security. Rate and Capacity of secure computation: The efficiency of converting the offline samples into secure computation during the online phase is essential. However, investigating this "production rate" for general secure computations seems analytically intractable. Towards this objective, we introduce a new model of secure computation -- one without any communication -- that has several practical applications. We lay the mathematical foundations of formulating rate and capacity questions in this framework. Our research identifies the first tight rate and capacity results (a la Shannon) in secure computation. Reverse multiplication embedding: We identify a new problem in algebraic complexity theory that unifies several efficiency objectives in cryptography. Reverse multiplicatio [...]Hai H Nguyenwork_y23vxqpc2fey5p373kgule6f24Fri, 02 Dec 2022 00:00:00 GMTOn The Complexity of Distance-d Independent Set Reconfiguration
https://scholar.archive.org/work/q5narcdlkzaetkmjq4uw7bzc3y
For a fixed positive integer d ≥ 2, a distance-d independent set (DdIS) of a graph is a vertex subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (𝖳𝖲) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (𝖳𝖩), the target vertex needs not to be adjacent to the original one. The Distance-d Independent Set Reconfiguration (DdISR) problem under 𝖳𝖲/𝖳𝖩 asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d = 2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under 𝖳𝖲 and 𝖳𝖩 for any fixed d ≥ 3. On chordal graphs, we show that DdISR under 𝖳𝖩 is in 𝙿 when d is even and 𝙿𝚂𝙿𝙰𝙲𝙴-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is 𝙿𝚂𝙿𝙰𝙲𝙴-complete for d = 2 but in 𝙿 for d=3 under 𝖳𝖲, while under 𝖳𝖩 it is in 𝙿 for d = 2 but 𝙿𝚂𝙿𝙰𝙲𝙴-complete for d = 3. Additionally, certain well-known hardness results for d = 2 on general graphs, perfect graphs, planar graphs of maximum degree three and bounded bandwidth can be extended for d ≥ 3.Duc A. Hoangwork_q5narcdlkzaetkmjq4uw7bzc3yFri, 02 Dec 2022 00:00:00 GMTFacets and facet subgraphs of symmetric edge polytopes
https://scholar.archive.org/work/unwnmwegyve2rnuio6ufq5dgli
Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, "facet/face subgraphs" have emerged as one of the central concepts in describing this symmetry. Continuing along this line of inquiry we provide a complete description of the correspondence between facets/faces of a symmetric edge polytope and maximal bipartite subgraphs of the underlying connected graph.Tianran Chen, Robert Davis, Evgeniia Korchevskaiawork_unwnmwegyve2rnuio6ufq5dgliThu, 01 Dec 2022 00:00:00 GMTAn Improved Time-Efficient Approximate Kernelization for Connected Treedepth Deletion Set
https://scholar.archive.org/work/3ddh3hd5avfpvkek4cs5ozbmxy
We study the CONNECTED η-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S ⊆V(G) of at most k vertices such that G - S has treedepth at most ηand G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED η-TREEDEPTH DELETION does not admit polynomial kernel unless NP ⊆coNP/poly. This motivates us to design an approximate kernel of polynomial size for this problem. In this paper, we show that for every 0 < ϵ<= 1, CONNECTED η-TREEDEPTH DELETION SET admits a (1+ϵ)-approximate kernel with O(k^2^η+ 1/ϵ) vertices, i.e. a polynomial-sized approximate kernelization scheme (PSAKS).Eduard Eiben, Diptapriyo Majumdar, M. S. Ramanujanwork_3ddh3hd5avfpvkek4cs5ozbmxyThu, 01 Dec 2022 00:00:00 GMTSub-quadratic Algorithms for Kernel Matrices via Kernel Density Estimation
https://scholar.archive.org/work/bbhmtjx4krhojndibhoan5ms7q
Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency – given n input points, most kernel-based algorithms need to materialize the full n × n kernel matrix before performing any subsequent computation, thus incurring Ω(n^2) runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain subquadratic time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving linear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recent Kernel Density Estimation framework, which (after preprocessing in time subquadratic in n) can return estimates of row/column sums of the kernel matrix. In particular, we develop efficient reductions from weighted vertex and weighted edge sampling on kernel graphs, simulating random walks on kernel graphs, and importance sampling on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in sublinear (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsification, where we observe a 9x decrease in the number of kernel evaluations over baselines for LRA and a 41x reduction in the graph size for spectral sparsification.Ainesh Bakshi, Piotr Indyk, Praneeth Kacham, Sandeep Silwal, Samson Zhouwork_bbhmtjx4krhojndibhoan5ms7qThu, 01 Dec 2022 00:00:00 GMTConjunctive queries for logic-based information extraction
https://scholar.archive.org/work/wd2pb3qomzeb7lqcc3av3fepqq
This thesis offers two logic-based approaches to conjunctive queries in the context of information extraction. The first and main approach is the introduction of conjunctive query fragments of the logics FC and FC[REG], denoted as FC-CQ and FC[REG]-CQ respectively. FC is a first-order logic based on word equations, where the semantics are defined by limiting the universe to the factors of some finite input word. FC[REG] is FC extended with regular constraints. Our first results consider the comparative expressive power of FC[REG]-CQ in relation to document spanners (a formal framework for the query language AQL), and various fragments of FC[REG]-CQ – some of which coincide with well-known language generators, such as patterns and regular expressions. Then, we look at decision problems. We show that many decision problems for FC-CQ and FC[REG]-CQ (such as equivalence and regularity) are undecidable. The model checking problem for FC-CQ and FC[REG]-CQ is NP-complete even if the FC-CQ is acyclic – under the definition of acyclicity where each word equation in an FC-CQ is an atom. This leads us to look at the "decomposition" of an FC word equation into a conjunction of binary word equations (i.e., of the form x =˙ y · z). If a query consists of only binary word equations and the query is acyclic, then model checking is tractable and we can enumerate results efficiently. We give an algorithm that decomposes an FC-CQ into an acyclic FC-CQ consisting of binary word equations in polynomial time, or determines that this is not possible. The second approach is to consider the dynamic complexity of FC. This uses the common way of encoding words in a relational structure using a universe with a linear order along with symbol predicates. Then, each element of the universe can carry a symbol if the predicate for said symbol holds for that element. Instead of the "usual way" (looking at first-order logic over these structures), we study the dynamic complexity, where symbols can be modified. As each of these modifications only c [...]Sam M Thompsonwork_wd2pb3qomzeb7lqcc3av3fepqqWed, 30 Nov 2022 00:00:00 GMTWeisfeiler and Leman Go Relational
https://scholar.archive.org/work/bgpl4xfjcvfffp5bewjcohn4bu
Knowledge graphs, modeling multi-relational data, improve numerous applications such as question answering or graph logical reasoning. Many graph neural networks for such data emerged recently, often outperforming shallow architectures. However, the design of such multi-relational graph neural networks is ad-hoc, driven mainly by intuition and empirical insights. Up to now, their expressivity, their relation to each other, and their (practical) learning performance is poorly understood. Here, we initiate the study of deriving a more principled understanding of multi-relational graph neural networks. Namely, we investigate the limitations in the expressive power of the well-known Relational GCN and Compositional GCN architectures and shed some light on their practical learning performance. By aligning both architectures with a suitable version of the Weisfeiler-Leman test, we establish under which conditions both models have the same expressive power in distinguishing non-isomorphic (multi-relational) graphs or vertices with different structural roles. Further, by leveraging recent progress in designing expressive graph neural networks, we introduce the k-RN architecture that provably overcomes the expressiveness limitations of the above two architectures. Empirically, we confirm our theoretical findings in a vertex classification setting over small and large multi-relational graphs.Pablo Barcelo, Mikhail Galkin, Christopher Morris, Miguel Romero Orthwork_bgpl4xfjcvfffp5bewjcohn4buWed, 30 Nov 2022 00:00:00 GMTThe Smoothed Complexity of Policy Iteration for Markov Decision Processes
https://scholar.archive.org/work/3cglivb72rcdzohztggk3x2g7u
We show subexponential lower bounds (i.e., 2^Ω (n^c)) on the smoothed complexity of the classical Howard's Policy Iteration algorithm for Markov Decision Processes. The bounds hold for the total reward and the average reward criteria. The constructions are robust in the sense that the subexponential bound holds not only on the average for independent random perturbations of the MDP parameters (transition probabilities and rewards), but for all arbitrary perturbations within an inverse polynomial range. We show also an exponential lower bound on the worst-case complexity for the simple reachability objective.Miranda Christ, Mihalis Yannakakiswork_3cglivb72rcdzohztggk3x2g7uWed, 30 Nov 2022 00:00:00 GMTMaximum cut on interval graphs of interval count four is NP-complete
https://scholar.archive.org/work/s24r6zqdebdv7gmbhl5cfktyju
The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80's, being one of the problems proposed by Johnson on his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.Celina M. H. de Figueiredo, Alexsander A. de Melo, Fabiano S. Oliveira, Ana Silvawork_s24r6zqdebdv7gmbhl5cfktyjuTue, 29 Nov 2022 00:00:00 GMTAn algorithm to compute fundamental classes of spin components of strata of differentials
https://scholar.archive.org/work/55u4a3rk4rfojomfmvkysvtc64
We construct an algorithm for computing the cycle classes of the spin components of a stratum of differentials in the moduli space of stable curves ℳ_g,n. In addition, we implement it within the Sage package admcycles. Our main strategy is to reconstruct these cycles by their restrictions to the boundary of ℳ_g,n via clutching maps. These restrictions can be computed recursively by smaller dimensional spin classes and determine the original class via a certain system of linear equations. To study the spin parities on the boundary of a stratum of differentials of even type, we make use of the moduli space of multi-scale differentials introduced in [BCCGM19]. As an application of our algorithm, one can verify a conjecture on spin double ramification cycles stated in [CSS21] in many examples, by using the results computed by our algorithm.Yiu Man Wongwork_55u4a3rk4rfojomfmvkysvtc64Tue, 29 Nov 2022 00:00:00 GMTThe Complexity of Infinite-Horizon General-Sum Stochastic Games
https://scholar.archive.org/work/rzuknccxqfgzlntwc7xkouqibi
We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that under further structural assumptions on the rewards computing NE in such turn-based games is possible in polynomial time. Towards achieving these results we establish structural facts about stochastic games of broader utility, including monotonicity of utilities under single-state single-action changes and reductions to settings where each player controls a single state.Yujia Jin, Vidya Muthukumar, Aaron Sidfordwork_rzuknccxqfgzlntwc7xkouqibiTue, 29 Nov 2022 00:00:00 GMT