arXiv subjects

Tillmann Miltzow

Publications and source records attributed to Tillmann Miltzow.

At least 19 recordsLinked to original sources

arXiv2026

Beyond Bits: An Introduction to Computation over the Reals

We introduce a lightweight and accessible approach to computation over the real numbers, with the aim of clarifying both the underlying concepts and their relevance in modern research. The material is intended for a broad audience, including instructors who wish to incorporate real computation into algorithms courses, their students, and PhD students encountering the subject for the first time. Rather than striving for completeness, we focus on a carefully selected set of results that can be presented and proved in a classroom setting. This allows us to highlight core techniques and recurring ideas while maintaining an approachable exposition. In some places, the presentation is intentionally informal, prioritizing intuition and practical understanding over full technical precision. We position our exposition relative to existing literature, including Matousek's lecture notes on ER-completeness and the recent compendium of ER-complete problems by Schaefer, Cardinal, and Miltzow. While these works provide deep and comprehensive perspectives, our goal is to offer an accessible entry point with proofs and examples suitable for teaching. Our approach follows modern formulations of real computation that emphasize binary input, real-valued witnesses, and restricted use of constants, aligning more closely with contemporary complexity theory, while acknowledging the foundational contributions of the Blum--Shub--Smale model.

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arXiv2026

Sometimes Two Irrational Guards are Needed

In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.

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arXiv2025

Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete

Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly $k$ bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as an intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines among the few remaining natural cases where the recognition complexity remained open. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete.

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arXiv2025

Devil's Games and $\text{Q}\mathbb{R}$: Continuous Games complete for the First-Order Theory of the Reals

We introduce the complexity class Quantified Reals ($\text{Q}\mathbb{R}$). Let FOTR be the set of true sentences in the first-order theory of the reals. A language $L$ is in $\text{Q}\mathbb{R}$, if there is a polynomial time reduction from $L$ to FOTR. This seems the first time this complexity class is studied. We show that $\text{Q}\mathbb{R}$ can also be defined using real Turing machines. It is known that deciding FOTR requires at least exponential time unconditionally [Berman, 1980]. We focus on devil's games with two defining properties: (1) Players (human and devil) alternate turns and (2) each turn has a continuum of options. First, we show that FOTRINV is $\text{Q}\mathbb{R}$-complete. FOTRINV has only inversion and addition constraints and all variables are in a compact interval. FOTRINV is a stepping stone for further reductions. Second, we show that the Packing Game is $\text{Q}\mathbb{R}$-complete. In the Packing Game we are given a container and two sets of pieces. One set of pieces for the human and one set for the devil. The human and the devil alternate by placing a piece into the container. Both rotations and translations are allowed. The first player that cannot place a piece loses. Third, we show that the Planar Extension Game is $\text{Q}\mathbb{R}$-complete. We are given a partially drawn plane graph and the human and the devil alternate by placing vertices and the corresponding edges in a straight-line manner. The vertices and edges to be placed are prescribed before hand. The first player that cannot place a vertex loses. Finally, we show that the Order Type Game is $\text{Q}\mathbb{R}$-complete. We are given an order-type together with a linear order. The human and the devil alternate in placing a point in the Euclidean plane following the linear order. The first player that cannot place a point correctly loses.

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arXiv2025

Oracle Separations for RPH

While theoretical computer science primarily works with discrete models of computation, like the Turing machine and the wordRAM, there are many scenarios in which introducing real computation models is more adequate. We want to compare real models of computation with discrete models of computation. We do this by means of oracle separation results. We define the notion of a real Turing machine as an extension of the (binary) Turing machine by adding a real tape. Using those machines, we define and study the real polynomial hierarchy RPH. We are interested in RPH as the first level of the hierarchy corresponds to the well-known complexity class ER. It is known that $NP \subseteq ER \subseteq PSPACE$ and furthermore $PH \subseteq RPH \subseteq PSPACE$. We are interested to know if any of those inclusions are tight. In the absence of unconditional separations of complexity classes, we turn to oracle separation. We develop a technique that allows us to transform oracle separation results from the binary world to the real world. As applications, we show there are oracles such that: - $RPH^O$ proper subset of $PSPACE^O$, - $Σ_{k+1}^O$ not contained in $Σ_kR^O$, for all $k\geq 0$, - $Σ_kR^O$ proper subset of $Σ_{k+1}R^O$, for all $k\geq 0$, - $BQP^O$ not contained in $RPH^O$. Our results hint that ER is strictly contained in PSPACE and that there is a separation between the different levels of the real polynomial hierarchy. We also bound the power of real computations by showing that NP-hard problems are unlikely to be solvable using polynomial time on a realRAM. Furthermore, our oracle separations hint that polynomial-time quantum computing cannot be simulated on an efficient real Turing machine.

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arXiv2025

On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation

We investigate machine models similar to Turing machines that are augmented with the operations of a first-order structure $\mathcal{R}$, and we show that under weak conditions on $\mathcal{R}$, the complexity class $\Sigma_k \mathcal{R}$ may be characterized in four equivalent ways: (1) by polynomial-time algorithms implemented on $\mathcal{R}$-machines together with witness strings, (2) by the $\Sigma_k\mathcal{R}$-complete problem $\Sigma_k\text{SAT}(\mathcal{R})$, (3) by the $k$th existential fragment of second-order metafinite logic over $\mathcal{R}$ via descriptive complexity, and (4) via oracles. By characterizing $\Sigma_k\mathcal{R}$ in these four ways, we extend previous work and embed it in one coherent framework. In addition, we derive similar results for $\exists_k \mathcal{R}$, the constant-free Boolean part of $\Sigma_k\mathcal{R}$, by showing that $\exists_k\mathcal{R}$ may be characterized in four analogous ways. Some conditions on $\mathcal{R}$ must be assumed in order to achieve the above quaternity because there are infinite-vocabulary structures for which $\text{NP}(\mathcal{R}) = \Sigma_1 \mathcal{R}$ does not have a complete problem. Surprisingly, even in these cases, we show that $\text{NP}(\mathcal{R})$ does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces.

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arXiv2024

The Existential Theory of the Reals as a Complexity Class: A Compendium

We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mnëv and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which $\exists \mathbb{R}$ rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. $\exists \mathbb{R}$ has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for $\exists \mathbb{R}$, as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.

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arXiv2024

Representing Matroids over the Reals is $\exists \mathbb R$-complete

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $φ\colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $φ(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.

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arXiv2024

Geometric Thickness of Multigraphs is $\exists \mathbb{R}$-complete

We say that a (multi)graph $G = (V,E)$ has geometric thickness $t$ if there exists a straight-line drawing $φ: V \rightarrow \mathbb{R}^2$ and a $t$-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The \textsc{Geometric Thickness} problem asks whether a given multigraph has geometric thickness at most $t$. This problem was shown to be NP-hard for $t=2$ [Durocher, Gethner, and Mondal, CG 2016]. In this paper, we settle the computational complexity of \textsc{Geometric Thickness} by showing that it is $\exists \mathbb{R}$-complete already for thickness $30$. Moreover, our reduction shows that the problem is $\exists \mathbb{R}$-complete for $4392$-planar graphs, where a graph is $k$-planar if it admits a topological drawing with at most $k$ crossings per edge. In the course of our paper, we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of $31$ edge-disjoint graphs and pseudo-segment stretchability with chromatic number $30$ are $\exists \mathbb{R}$-complete.

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arXiv2024

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $δ$ or a diminished guard whose vision is by an angle of $δ$ "blocked" by reflex vertices. A polygon $P$ has vision stability $δ$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

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arXiv2024

Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

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arXiv2024

On Classifying Continuous Constraint Satisfaction Problems

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : Φ(x_1, \ldots, x_n)$, where $Φ$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

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arXiv2024

Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\exists\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\mathsf{NP}=\exists\mathbb{R}$.

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arXiv2024

Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

We devise a data structure that can answer shortest path queries for two query points in a polygonal domain $P$ on $n$ vertices. For any $\varepsilon > 0$, the space complexity of the data structure is $O(n^{10+\varepsilon })$ and queries can be answered in $O(\log n)$ time. Alternatively, we can achieve a space complexity of $O(n^{9+\varepsilon })$ by relaxing the query time to $O(\log^2 n)$. This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They present a data structure with $O(n^{11})$ space complexity and $O(\log n)$ query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with $O(n^{9+\varepsilon}/\hspace{1pt} \ell^{4 + O(\varepsilon )})$ space complexity and $O(\ell \log^2 n )$ query time, for any integer $1 \leq \ell \leq n$. Furthermore, we present improved data structures with $O(\log n)$ query time for the special case where we restrict one (or both) of the query points to lie on the boundary of $P$. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes $O(n^{6+\varepsilon})$. When both query points are on the boundary, the space complexity is decreased further to $O(n^{4+\varepsilon })$, thereby improving an earlier result of Bae and Okamoto.

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arXiv2023

The Complexity of Recognizing Geometric Hypergraphs

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is associated with a point $p_v\in \mathbb{R}^d$ and each hyperedge $e\in E$ is associated with a connected set $s_e\subset \mathbb{R}^d$ such that $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$ for all $e\in E$. We say that a given hypergraph $H$ is representable by some (infinite) family $F$ of sets in $\mathbb{R}^d$, if there exist $P\subset \mathbb{R}^d$ and $S \subseteq F$ such that $(P,S)$ is a geometric representation of $H$. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is $\exists\mathbb{R}$-hard for halfspaces in $\mathbb{R}^d$. We study the families of translates of balls and ellipsoids in $\mathbb{R}^d$, as well as of other convex sets, and show that their RECOGNITION problems are also $\exists\mathbb{R}$-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

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arXiv2023

Topological Art in Simple Galleries

Let $P$ be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in $P$. We say two points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is contained in $P$. We denote by $V(P)$ the family of all minimum guard placements. The Hausdorff distance makes $V(P)$ a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy equivalent to $S$. Furthermore, for various concrete topological spaces $T$, we describe instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to $T$.

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arXiv2023

Token Swapping on Trees

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.

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arXiv2022

Avoider-Enforcer Game is NP-hard

In an Avoider-Enforcer game, we are given a hypergraph. Avoider and Enforcer alternate in claiming an unclaimed vertex, until all the vertices of the hypergraph are claimed. Enforcer wins if Avoider claims all vertices of an edge; Avoider wins otherwise. We show that it is NP-hard to decide if Avoider has a winning strategy.

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