We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form $\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of $f$. The constraint function can take both positive and negative values, allowing for cancellations. The dichotomy extends easily to rational valued functions of the same type. In addition, we discover a new phenomenon: there is a set $\mathcal{F}$ with the property that for every $f \in \mathcal{F}$ the problem $\operatorname{Holant}\left(f\mid =_3 \right)$ is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by $\left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right]$ to FKT together with an independent global argument.
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We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We also present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. The main application of our analysis is the description of the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massive, free QFT in two spacetime dimensions, thus complementing recent results in higher space dimensions. In particular, we have the formula for the local entropy of a one dimensional Klein-Gordon wave packet and Araki's vacuum relative entropy of a coherent state on a double cone von Neumann algebra. Besides, we derive the type III_1 factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.
We study the problem of checking the existence of a step-by-step transformation of $d$-regular induced subgraphs in a graph, where $d \ge 0$ and each step in the transformation must follow a fixed reconfiguration rule. Our problem for $d=0$ is equivalent to \textsc{Independent Set Reconfiguration}, which is one of the most well-studied reconfiguration problems. In this paper, we systematically investigate the complexity of the problem, in particular, on chordal graphs and bipartite graphs. Our results give interesting contrasts to known ones for \textsc{Independent Set Reconfiguration}.
We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory $\mathrm{APC}_2$ of [Je\v{r}\'abek 2009]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory $T^2_2$, this shows that $\mathrm{APC}_2$ does not prove every $\forall \Sigma^b_1$ sentence which is provable in bounded arithmetic. This answers the question posed in [Buss, Ko{\l}odziejczyk, Thapen 2014] and represents some progress in the programme of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the "fixing lemma" from [Pudl\'ak, Thapen 2017], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a $\textrm{P}^{\textrm{NP}}$ oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.
We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.
We study the problem of checking the existence of a step-by-step transformation of $d$-regular induced subgraphs in a graph, where $d \ge 0$ and each step in the transformation must follow a fixed reconfiguration rule. Our problem for $d=0$ is equivalent to \textsc{Independent Set Reconfiguration}, which is one of the most well-studied reconfiguration problems. In this paper, we systematically investigate the complexity of the problem, in particular, on chordal graphs and bipartite graphs. Our results give interesting contrasts to known ones for \textsc{Independent Set Reconfiguration}.
Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-coloured graph $H$ is called "rainbow" if no two vertices of $H$ have the same colour. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-coloured $\mathcal{F}$-free graph $G$ having $\chi(G)\geq \ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gy\`arf\`as and S\`ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. We show that $\ell(s,K_{2,r})\leq (r-1)(s-1)s/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2+s}{2}$. This significantly improves Gy\`arf\`as and S\`ark\"ozy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some new results on an oriented variant of the Gy\`arf\`as-Sumner conjecture. Let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph having chromatic number at least $(r-1)(s-1)(s-2)+s$ and every bikernel-perfect oriented graph with girth $g\geq 5$ having chromatic number at least $2s^{1+\frac{4}{g-4}}$ contains every oriented tree on at most $s$ vertices as an induced subgraph.
A common approach to tackle a combinatorial optimization problem is to first solve a continuous relaxation and then round the obtained fractional solution. For the latter, the framework of contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak, and Zenklusen, is a general and successful tool. A CR scheme takes a fractional point $x$ in a relaxation polytope, rounds each coordinate $x_i$ independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the independence constraints. Intuitively, a CR scheme is $c$-balanced if every element $i$ is selected with probability at least $c \cdot x_i$. It is known that general matroids admit a $(1-1/e)$-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple and explicit monotone CR scheme with a balancedness of $1 - \binom{n}{k}\:\left(1-\frac{k}{n}\right)^{n+1-k}\:\left(\frac{k}{n}\right)^k$, and show that this is optimal. As $n$ grows, this expression converges from above to $1 - e^{-k}k^k/k!$. While this asymptotic bound can be obtained by combining previously known results, these require defining an exponential-sized linear program, as well as using random sampling and the ellipsoid algorithm. Our procedure, on the other hand, has the advantage of being simple and explicit. Moreover, this scheme generalizes into an optimal CR scheme for partition matroids.
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function $f$, there is an algorithm $\mathfrak{A}$ that takes as input a CMSO sentence $\varphi$, a positive integer $t$, and a connected graph $G$ of maximum degree at most $\Delta$, and determines, in time $f(|\varphi|,t)\cdot 2^{O(\Delta \cdot t)}\cdot |G|^{O(t)}$, whether $G$ has a supergraph $G'$ of treewidth at most $t$ such that $G'\models \varphi$. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time $f(d)\cdot 2^{O(\Delta \cdot d)}\cdot |G|^{O(d)}$, whether a connected graph of maximum degree $\Delta$ has a planar supergraph of diameter at most $d$. Additionally, we show that for each fixed $k$, the problem of determining whether $G$ has an $k$-outerplanar supergraph of diameter at most $d$ is strongly uniformly fixed parameter tractable with respect to the parameter $d$. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter $\mathbf{p}$. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.
For integers $d \geq 2$ and $k \geq d+1$, a $k$-hole in a set $S$ of points in general position in $\mathbb{R}^d$ is a $k$-tuple of points from $S$ in convex position such that the interior of their convex hull does not contain any point from $S$. For a convex body $K \subseteq \mathbb{R}^d$ of unit $d$-dimensional volume, we study the expected number $EH^K_{d,k}(n)$ of $k$-holes in a set of $n$ points drawn uniformly and independently at random from $K$. We prove an asymptotically tight lower bound on $EH^K_{d,k}(n)$ by showing that, for all fixed integers $d \geq 2$ and $k\geq d+1$, the number $EH_{d,k}^K(n)$ is at least $\Omega(n^d)$. For some small holes, we even determine the leading constant $\lim_{n \to \infty}n^{-d}EH^K_{d,k}(n)$ exactly. We improve the currently best known lower bound on $\lim_{n \to \infty}n^{-d}EH^K_{d,d+1}(n)$ by Reitzner and Temesvari (2019). In the plane, we show that the constant $\lim_{n \to \infty}n^{-2}EH^K_{2,k}(n)$ is independent of $K$ for every fixed $k \geq 3$ and we compute it exactly for $k=4$, improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche (2015) and by the authors (2020).
A matroid $\mathcal{M}$ on a set $E$ of elements has the $\alpha$-partition property, for some $\alpha>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $\alpha$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $\alpha$-partition property for any constant $\alpha>0$ (independent of $d$). Furthermore, we refute a recent conjecture of B\'erczi, Schwarcz, and Yamaguchi by showing the same matroid is $2^d/d$-colorable but cannot be reduced to an $\alpha 2^d/d$-colorable partition matroid for any $\alpha$ that is sublinear in $d$.
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