arXiv subjects

István Miklós

Publications and source records attributed to István Miklós.

At least 19 recordsLinked to original sources

arXiv2025

On the Complexity of Bipartite Degree Realizability

We study the \emph{Bipartite Degree Realization} (BDR) problem: given a graphic degree sequence $D$, decide whether it admits a realization as a bipartite graph. While bipartite realizability for a fixed vertex partition can be decided in polynomial time via the Gale--Ryser theorem, the computational complexity of BDR without a prescribed partition remains unresolved. We address this question through a parameterized analysis. For constants $0 \le c_1 \le c_2 \le 1$, we define $\mathrm{BDR}_{c_1,c_2}$ as the restriction of BDR to degree sequences of length $n$ whose degrees lie in the interval $[c_1 n, c_2 n]$. Our main result shows that $\mathrm{BDR}_{c_1,c_2}$ is solvable in polynomial time whenever $0 \le c_1 \le c_2 \le \frac{\sqrt{c_1(c_1+4)}-c_1}{2}$, as well as for all $c_1 > \tfrac12$. The proof relies on a reduction to extremal \emph{least balanced degree sequences} and a detailed verification of the critical Gale--Ryser inequalities, combined with a bounded subset-sum formulation. We further show that, assuming the NP-completeness of unrestricted BDR, the problem $\mathrm{BDR}_{c_1,c_2}$ remains NP-complete for all $0 < c_2 < \frac{1}{2}$ and $c_1 < 1 - c_2 - \sqrt{1-2c_2}$. % This establishes a sharp conditional boundary between tractable and intractable parameter regimes. Our results clarify the algorithmic landscape of bipartite degree realization and contribute to the broader study of potentially bipartite graphic degree sequences.

math.CO↗

arXiv2025

A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem

We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphic degree sequence problem, $\mathrm{sparse}\text{-}t\text{-}\mathrm{uni}\text{-}\mathrm{HDS}_{α',α}$. For any fixed $t \ge 3$, given parameters $0 \le α' \le α< t-1$, the input consists of degree sequences $D$ of length $n$ with degrees between $n^{α'}$ and $6n^α$. We show that the problem is NP-complete whenever $α' \le \frac{t(α- 1) + 1}{t - 1}$, and solvable in linear time when $α' > \frac{t(α- 1) + 1}{t - 1}$. This establishes a sharp boundary between polynomial-time solvable and NP-complete instances, thereby characterizing the computational complexity across all degree exponent regimes. The result extends the earlier NP-completeness of dense hypergraphicality to a unified framework covering both sparse and dense regimes, revealing that even extremely sparse instances (with maximum degree $o(n)$ but $Ω(n^{\frac{t-1}{t}})$) remain NP-complete. On the other hand, the $t$-uniform hypergraphicality solvable in linear time when the maximum degree is $o(n^{\frac{t-1}{t}})$. This dichotomy provides a comprehensive classification of the complexity landscape for hypergraphic degree sequences.

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arXiv2024

A dichotomy theorem on the complexity of 3-uniform hypergraphic degree sequence graphicality

We present a dichotomy theorem on the parameterized complexity of the 3-uniform hypergraphicality problem. Given $0<c_1\le c_2 < 1$, the parameterized 3-uniform Hypergraphic Degree Sequence problem, $3uni-HDS_{c_1,c_2}$, considers degree sequences $D$ of length $n$ such that all degrees are between $c_1 {n-1 \choose 2}$ and $c_2 {n-1\choose 2}$ and it asks if there is a 3-uniform hypergraph with degree sequence $D$. We prove that for any $0<c_2< 1$, there exists a unique, polynomial-time computable $c_1^*$ with the following properties. For any $ c_1\in (c_1^*,c_2]$, $3uni-HDS_{c_1,c_2}$ can be solved in linear time. In fact, for any $c_1\in (c_1^*,c_2]$ there exists an easy-to-compute $n_0$ such that any degree sequence $D$ of length $n\ge n_0$ and all degrees between $c_1 {n-1\choose 2}$ and $c_2 {n-1\choose 2}$ has a 3-uniform hypergraph realization if and only if the sum of the degrees can be divided by $3$. Further, $n_0$ grows polynomially with the inverse of $c_1-c_1^*$. On the other hand, we prove that for all $c_1<c_1^*$, $3uni-HDS_{c_1,c_2}$ is NP-complete. Finally, we briefly consider an extension of the hypergraphicality problem to arbitrary $t$-uniformity. We show that the interval where degree sequences (satisfying divisibility conditions) always have $t$-uniform hypergraph realizations must become increasingly narrow, with interval width tending to $0$ as $t \rightarrow \infty$.

math.CO↗

arXiv2024

Fully graphic degree sequences and P-stable degree sequences

The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for $P$-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of $P$-stable degree sequence families. At first we will show that every known infinite $P$-stable degree sequence set, described by inequalities of the parameters $n, c_1, c_2, Σ$ (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if $Σ$ does not occur in the determining inequality, then the notions of $P$-stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about $P$-stability, describing the maximal $P$-stable sequence set by $n, c_1, c_2$. Furthermore we conjecture that similar equivalences occur in cases if $Σ$ also part of the defining inequality.

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arXiv2023

Superpolynomial period lengths of the winning positions in the subtraction game

Given a finite set of positive integers, $A$, and starting with a heap of $n$ chips, Alice and Bob alternate turns and on each turn a player chooses $x\in A$ with $x$ smaller or equal than the current number of chips and subtract $x$ chips from the heap. The game terminates when the current number of chips becomes smaller than $\min\{A\}$ and no moves are possible. The player who makes the last move is the winner. We can define $w^A(n)$ to be $1$ if Alice has a winning strategy with a starting heap of $n$ chips and $0$ if Bob has a winning strategy. By the Pigeonhole Principle, $w^A(n)$ becomes periodic, and it is easy to see that the period length is at most an exponential function of $\max\{A\}$. The typical period length is a linear function of $\max\{A\}$, and it is a long time open question if exponential period length is possible. We consider a slight modification of this game by introducing an initial seed $S$ that tells for the few initial numbers of chips whether the current or the opposite player is the winner. In this paper we show that the initial seed cannot change the period length of $w^A(n)$ if the size of $A$ is $1$ or $2$, but it can change the period length with $|A| \geq 3$. Further, we exhibit a class of sets $A$ of size $3$ and corresponding initial seeds such that the period length becomes a superpolynomial function of $\max\{A\}$.

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arXiv2022

Approximate Sampling of Graphs with Near-$P$-stable Degree Intervals

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller-Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well-studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently Amanatidis and Kleer published a beautiful paper (arXiv:2110.09068), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper we extend substantially their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centred at P-stable degree sequences.

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arXiv2021

Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

Let $D = d_1, d_2, \ldots, d_n$ and $F = f_1, f_2,\ldots, f_n$ be two sequences of positive integers. We consider the following decision problems: is there a $i)$ multigraph, $ii)$ loopless multigraph, $iii)$ simple graph, $iv)$ connected simple graph, $v)$ tree, $vi)$ caterpillar $G = (V,E)$ such that for all $k$, $d(v_k) = d_k$ and $\sum_{w\in \mathcal{N}(v_k)} d(w) = f_k$ ($d(v)$ is the degree of $v$ and $\mathcal{N}(v)$ is the set of neighbors of $v$). Here we show that all these decision problems can be solved in polynomial time if $\max_{k} d_k$ is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.

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arXiv2021

The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $γ>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erdős-Rényi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.

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arXiv2020

Half-graphs, other non-stable degree sequences, and the switch Markov chain

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called $P$-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not $P$-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.

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arXiv2019

Edge Disjoint Caterpillar Realizations

In this paper, we consider the edge disjoint caterpillar realizations of tree degree sequences. We give the necessary and sufficient conditions when two tree degree sequences have edge disjoint caterpillar realizations. We conjecture that an arbitrary number of tree degree sequences have edge disjoint realizations if every vertex is a leaf in at most one tree. We prove that the conjecture is true if the number of tree degree sequences is at most $4$. We also prove that the conjecture is true if $n \ge \max\{22k-11, 396\}$, where $n$ is the number of vertices and $k$ is the number of tree degree sequences.

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arXiv2018

Efficiently sampling the realizations of irregular, but linearly bounded bipartite and directed degree sequences

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. Several results were proved on rapidly mixing Markov chains on regular simple, on regular directed, on half-regular directed and on half-regular bipartite degree sequences. In this paper, the main result is the following: Let $U$ and $V$ be disjoint finite sets, and let $0 < c_1 \le c_2 < |U|$ and $0 < d_1 \le d_2 < |V|$ be integers. Furthermore, assume that the bipartite degree sequence on $U \cup V$ satisfies $c_1 \le d(v) \le c_2\ : \ \forall v\in V$ and $d_1 \le d(u) \le d_2 \ : \ \forall u\in U$. Finally assume that $(c_2-c_1 -1)(d_2 -d_1 -1) < 1 + \max \{c_1(|V| -d_2), d_1(|V|- c_2) \}$. Then the swap Markov chain on this bipartite degree sequence is rapidly mixing. The technique applies on directed degree sequences as well, with very similar parameter values. These results are germane to the recent results of Greenhill and Sfragara about fast mixing MCMC processes on simple and directed degree sequences, where the maximum degrees are $O(\sqrt{\# \mbox{ of edges}})$. The results are somewhat comparable on directed degree sequences: while the GS results are better applicable for degree sequences developed under some scale-free random process, our new results are better fitted to degree sequences developed under the Erdős -- Rényi model. For example our results cover all regular degree sequences, the GS model is not applicable when the average degree is $ > n/16.$

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arXiv2017

Packing tree degree sequences

We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge packing problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. We also consider the following variant of the degree matrix problem: given two degree sequences $D_1$ and $D_2$ such that $D_2$ is a tree degree sequence, decide if there exists edge-disjoint realizations of $D_1$ and $D_2$ where the realization of $D_2$ is not necessarily a tree. We show that this problem is already $\NP$-complete. Counting, or just estimating the number of distinct realizations of degree sequences is challenging in general. We show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

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arXiv2017

Graph realizations constrained by skeleton graphs

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper we apply the new {\em skeleton graph} model to describe the same information as the PAM model. Our model is more convenient for handling problems with low number of partition classes or with special topological restrictions among the classes. We investigate two particular cases in detail: (i) when there are only two vertex classes and (ii) when the skeleton graph contains at most one cycle.

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arXiv2017

The computational complexity of calculating partition functions of optimal medians with Hamming distance

In this paper, we show that calculating the partition function of optimal medians of binary strings with Hamming distance is \#P-complete for several weight functions. The case when the weight function is the factorial function has application in bioinformatics. In that case, the partition function counts the most parsimonious evolutionary scenarios on a star tree under several models in bioinformatics. The results are extended to binary trees and we show that it is also \#P-complete to calculate the most parsimonious evolutionary scenarios on an arbitrary binary tree under the substitution model of biological sequences and under the Single Cut-or-Join model for genome rearrangements.

cs.CC↗

arXiv2016

New classes of degree sequences with fast mixing swap Markov chain sampling

In network modeling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a joint degree matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence and conjectured that it mixes rapidly (in poly-time) for arbitrary degree sequences. While the conjecture is still open, it was proven for special degree sequences, in particular, for those of undirected and directed regular simple graphs, of half-regular bipartite graphs, and of graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite splitted degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

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arXiv2016

Not all simple looking degree sequence problems are easy

Degree sequence (DS) problems are around for at least hundred twenty years, and with the advent of network science, more and more complicated, structured DS problems were invented. Interestingly enough all those problems so far are computationally easy. It is clear, however, that we will find soon computationally hard DS problems. In this paper we want to find such hard DS problems with relatively simple definition. For a vertex $v$ in the simple graph $G$ denote $d_i(v)$ the number of vertices at distance exactly $i$ from $v$. Then $d_1(v)$ is the usual degree of vertex $v.$ The vector $\mathbf{d}^2(G)=( (d_1(v_1), d_2(v_1)), \ldots,$ $(d_1(v_n), d_2(v_n))$ is the {\bf second order degree sequence} of the graph $G$. In this note we show that the problem to decide whether a sequence of natural numbers $((i_1,j_1),\ldots (i_n,j_n))$ is a second order degree sequence of a simple undirected graph $G$ is strongly NP-complete. Then we will discuss some further NP-complete DS problems.

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arXiv2015

Exact sampling of graphs with prescribed degree correlations

Many real-world networks exhibit correlations between the node degrees. For instance, in social networks nodes tend to connect to nodes of similar degree. Conversely, in biological and technological networks, high-degree nodes tend to be linked with low-degree nodes. Degree correlations also affect the dynamics of processes supported by a network structure, such as the spread of opinions or epidemics. The proper modelling of these systems, i.e., without uncontrolled biases, requires the sampling of networks with a specified set of constraints. We present a solution to the sampling problem when the constraints imposed are the degree correlations. In particular, we develop an efficient and exact method to construct and sample graphs with a specified joint-degree matrix, which is a matrix providing the number of edges between all the sets of nodes of a given degree, for all degrees, thus completely specifying all pairwise degree correlations, and additionally, the degree sequence itself. Our algorithm always produces independent samples without backtracking. The complexity of the graph construction algorithm is O(NM) where N is the number of nodes and M is the number of edges.

cs.DM↗

arXiv2014

A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two degree groups are placed as uniformly as possible. We prove that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes $n$. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

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