arXiv subjects

Kilian Risse

Publications and source records attributed to Kilian Risse.

9 recordsLinked to original sources

arXiv2026

On Cutting Cakes and Crossing Curves

We consider the classic envy-free cake-cutting problem where the goal is to cut and allocate a divisible resource among a set of agents in a way that avoids any envy between them. When the agents' valuation functions are continuous and nonnegative, an envy-free solution is guaranteed to exist where each agent is allocated a contiguous piece of the resource. Such a solution can be efficiently computed using the standard cut-and-choose algorithm for two agents, but the problem is known to be hard when there are at least four agents. The setting with three agents has remained open. We show that the problem remains intractable for three agents. We obtain this result by uncovering a novel connection between cake-cutting and a computational problem corresponding to the Jordan curve theorem, introduced by Adler, Daskalakis, and Demaine (2016). As our main technical contribution, we provide the first lower bounds for the Jordan curve problem in the form of a query lower bound as well as hardness for the class UEOPL, a subclass of PPAD containing notoriously challenging problems such as Simple Stochastic Games and the P-matrix Linear Complementarity Problem.

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arXiv2026

Superpolynomial Length Lower Bounds for Tree-Like Semantic Proof Systems with Bounded Line Size

We prove superpolynomial length lower bounds for the semantic tree-like Frege refutation system with bounded line size. Concretely, for any function $n^{2-\varepsilon} \leq s(n) \leq 2^{n^{1-\varepsilon}}$ we exhibit an explicit family $\mathcal{A}$ of $n$-variate CNF formulas $A$, each of size $|A| \le s(n)^{1+\varepsilon}$, such that if $A$ is chosen uniformly from $\mathcal{A}$, then asymptotically almost surely any tree-like Frege refutation of $A$ in line-size $s(n)$ is of length super-polynomial in $|A|$. Our lower bounds apply also to tree-like degree-$d$ threshold systems, for $d \approx \log\bigl(s(n)\bigr)$, that is, for $d$ up to $n^{1-\varepsilon}$. More generally, our lower bounds apply to the semantic version of these systems and to any semantic tree-like proof system where the number of distinct lines is bounded by $\exp\bigl(s(n)\bigr)$.

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arXiv2025

On bounded depth proofs for Tseitin formulas on the grid; revisited

We study Frege proofs using depth-$d$ Boolean formulas for the Tseitin contradiction on $n \times n$ grids. We prove that if each line in the proof is of size $M$ then the number of lines is exponential in $n/(\log M)^{O(d)}$. This strengthens a recent result of Pitassi et al. [PRT22]. The key technical step is a multi-switching lemma extending the switching lemma of Håstad [Hås20] for a space of restrictions related to the Tseitin contradiction. The strengthened lemma also allows us to improve the lower bound for standard proof size of bounded depth Frege refutations from exponential in $\tilde Ω(n^{1/59d})$ to exponential in $\tilde Ω(n^{1/d})$. This strengthens the bounds given in the preliminary version of this paper [HR22].

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arXiv2025

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

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arXiv2025

Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

We prove that polynomial calculus (and hence also Nullstellensatz) over any field requires linear degree to refute that sparse random regular graphs, as well as sparse Erdős-Rényi random graphs, are $3$-colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on proof size.

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arXiv2024

Supercritical Tradeoffs for Monotone Circuits

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: We introduce a new family of unsatisfiable 3-CNF formulas (called bracket formulas) that admit resolution refutations of quasipolynomial size while any refutation of polynomial depth requires exponential size.

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arXiv2024

Clique Is Hard on Average for Sherali-Adams with Bounded Coefficients

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{Ω(d)}$ to rule out the existence of an $n^{Θ(1)}$-clique in Erdős-Rényi random graphs whose maximum clique is of size $d\leq 2\log n$. This lower bound is tight up to the multiplicative constant in the exponent. We obtain this result by introducing a technique inspired by pseudo-calibration which may be of independent interest. The technique involves defining a measure on monomials that precisely captures the contribution of a monomial to a refutation. This measure intuitively captures progress and should have further applications in proof complexity.

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arXiv2023

Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $Ω(s^{1-ε})$ to prove that $f$ does not have circuits of size $s$ (for any $s > \mathrm{poly}(n)$). As a corollary we obtain that there are no low degree SoS proofs of the statement NP $\not \subseteq $ P/poly. We also show that for any $0 < α< 1$ there are Boolean functions with circuit complexity larger than $2^{n^α}$ but SoS requires size $2^{2^{Ω(n^α)}}$ to prove this. In addition we prove analogous results on the minimum \emph{monotone} circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system $Q$ has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, $Q$ is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for $Q$.

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arXiv2022

Perfect Matching in Random Graphs is as Hard as Tseitin

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $Ω(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires $n^δ$ rounds to refute these formulas for some $δ> 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

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