Popular Matchings and Limits to Tractability

We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if $G$ admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph: - we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; - we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth.Comment: arXiv admin note: text overlap with arXiv:1804.0014

Reflexive polytopes arising from partially ordered sets and perfect graphs

Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gr\"obner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $\delta$-polynomials of these reflexive polytopes.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1703.0441

Selfish peering and routing in the Internet

The Internet is a loose amalgamation of independent service providers acting in their own self-interest. We examine the implications of this economic reality on peering relationships. Specifically, we consider how the incentives of the providers might determine where they choose to interconnect with each other. We consider a game where two selfish network providers must establish peering points between their respective network graphs, given knowledge of traffic conditions and a nearest-exit routing policy for out-going traffic, as well as costs based on congestion and peering connectivity. We focus on the pairwise stability equilibrium concept and use a stochastic procedure to solve for the stochastically pairwise stable configurations. Stochastically stable networks are selected for their robustness to deviations in strategy and are therefore posited as the more likely networks to emerge in a dynamic setting. We note a paucity of stochastically stable peering configurations under asymmetric conditions, particularly to unequal interdomain traffic flow, with adverse effects on system-wide efficiency. Under bilateral flow conditions, we find that as the cost associated with the establishment of peering links approaches zero, the variance in the number of peering links of stochastically pairwise stable equilibria increases dramatically.Comment: Contribution to the Proceedings of the Complex Systems Summer School 2004, organized by the Santa Fe Institute (6 pages comprising 4 figures

A note on hitting maximum and maximal cliques with a stable set

It was recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $\omega \geq \frac 23(\Delta+1)$ unless the graph is the strong product of $K_{\omega/2}$ and an odd hole. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.Comment: 7 pages, two figures, accepted to J. Graph Theor

HoloNets: Spectral Convolutions do extend to Directed Graphs

Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and -- making use of certain advanced tools from complex analysis and spectral theory -- extend spectral convolutions to directed graphs. We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and -- as opposed to baselines -- may be rendered stable to resolution-scale varying topological perturbations.Comment: arXiv admin note: text overlap with arXiv:2310.0043

Coloring graph classes with no induced fork via perfect divisibility

For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$ if $\chi(G) \leq f(\omega(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $\chi$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork,$F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $\chi(G)\leq \omega(G)^2$. We also note that the class $\cal G$ does not admit a linear $\chi$-binding function.Comment: 16 page

Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis

Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ~ 10-20), the number of equilibria is already more than 100,000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to popular conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.Comment: 6 pages, 12 figures. Added a reference and corrected typo

Local sign stability and its implications for spectra of sparse random graphs and stability of ecosystems

We study the spectral properties of sparse random graphs with different topologies and type of interactions, and their implications on the stability of complex systems, with particular attention to ecosystems. Specifically, we focus on the behaviour of the leading eigenvalue in different type of random matrices (including interaction matrices and Jacobian-like matrices), relevant for the assessment of different types of dynamical stability. By comparing the results on Erdos-Renyi and Husimi graphs with sign-antisymmetric interactions or mixed sign patterns, we introduce a sufficient criterion, called strong local sign stability, for stability not to be affected by system size, as traditionally implied by the complexity-stability trade-off in conventional models of random matrices. The criterion requires sign-antisymmetric or unidirectional interactions and a local structure of the graph such that the number of cycles of finite length do not increase with the system size. Note that the last requirement is stronger than the classical local tree-like condition, which we associate to the less stringent definition of local sign stability, also defined in the paper. In addition, for strong local sign stable graphs which show stability to linear perturbations irrespectively of system size, we observe that the leading eigenvalue can undergo a transition from being real to acquiring a nonnull imaginary part, which implies a dynamical transition from nonoscillatory to oscillatory linear response to perturbations. Lastly, we ascertain the discontinuous nature of this transition.Comment: 55 pages, 17 figure

The Investment Management Game: Extending the Scope of the Notion of Core

The core is a dominant solution concept in economics and cooperative game theory; it is predominantly used for profit, equivalently cost or utility, sharing. This paper demonstrates the versatility of this notion by proposing a completely different use: in a so-called investment management game, which is a game against nature rather than a cooperative game. This game has only one agent whose strategy set is all possible ways of distributing her money among investment firms. The agent wants to pick a strategy such that in each of exponentially many future scenarios, sufficient money is available in the right firms so she can buy an optimal investment for that scenario. Such a strategy constitutes a core imputation under a broad interpretation, though traditional formal framework, of the core. Our game is defined on perfect graphs, since the maximum stable set problem can be solved in polynomial time for such graphs. We completely characterize the core of this game, analogous to Shapley and Shubik characterization of the core of the assignment game. A key difference is the following technical novelty: whereas their characterization follows from total unimodularity, ours follows from total dual integralityComment: 16 pages. arXiv admin note: text overlap with arXiv:2209.0490

General Non-structure Theory

The theme of the first two sections, is to prepare the framework of how from a "complicated" family of index models I in K_1 we build many and/or complicated structures in a class K_2. The index models are characteristically linear orders, trees with kappa+1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graph, for this we phrase relevant complicatedness properties (called bigness). We say when M in K_2 is represented in I in K_1. We give sufficient conditions when {M_I:I\in K^1_\lambda} is complicated where for each I in K^1_lambda we build M_I in K^2 (usually in K^2_lambda) represented in it and reflecting to some degree its structure (e.g. for I a linear order we can build a model of an unstable first order class reflecting the order). If we understand enough we can even build e.g. rigid members of K^2_lambda. Note that we mention "stable", "superstable", but in a self contained way, using an equivalent definition which is useful here and explicitly given. We also frame the use of generalizations of Ramsey and Erdos-Rado theorems to get models in which any I from the relevant K_1 is reflected. We give in some detail how this may apply to the class of separable reduced Abelian p-group and how we get relevant models for ordered graphs (via forcing). In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality lambda>aleph_0 we can attach a model M_I in K_lambda in which the linear order can be embedded so that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^lambda non-isomorphic cases