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Mateusz Skomra

Publications and source records attributed to Mateusz Skomra.

17 recordsLinked to original sources
arXiv2026

Duality attainment and strict feasibility of the generalized moment problem and its relaxations

The generalized moment problem (GMP) is an infinite dimensional linear problem over the cone of finite nonnegative Borel measures. When a GMP instance involves finitely many polynomial moment constraints, moment/sum-of-squares hierarchies provide a sequence of bounds converging to the optimal value. We consider GMP instances with measures supported over a compact basic semialgebraic set $X$. We study the case when $X$ has nonempty interior, and the case when $X$ is the vanishing set of prescribed polynomials forming a Gröbner basis of the ideal they generate, which we assume is real radical. Under a relative interior assumption, we show attainment of the infinite dimensional dual problem, and attainment of each associated finite dimensional sum-of-squares strengthening. For the latter we present two disjoint proofs. The first is obtained by adapting results regarding the closedness of quadratic modules, and the second builds on Csiszár's work on exponential density constructions to find a strictly feasible measure. Finally, we discuss the special case where $X$ is the product of spheres, and applications of our results to GMP instances arising from tensor optimization and quantum information theory.

math.OC↗
arXiv2025

Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres

We study the polynomial optimization problem of minimizing a multihomogeneous polynomial over the product of spheres. This polynomial optimization problem models the tensor optimization problem of finding the best rank one approximation of an arbitrary tensor. We show that the moment-SOS hierarchy has finite convergence in this case, for a generic multihomogeneous objective function. To show finite convergence of the hierarchy, we use a result of Huang et al. [SIAM J. Optim. 34(4) (2024), pp 3399-3428], which relies on local optimality conditions. To prove that the local optimality conditions hold generically, we use techniques from differential geometry and Morse theory. This work generalizes the main result of Huang [Optim. Lett. 17(5) (2023), pp 1263-1270], which shows finite convergence for the case of a homogeneous polynomial over a single sphere.

math.OC↗
arXiv2025

Reducing Stochastic Games to Semidefinite Program Feasibility

We present a polynomial-time reduction from max-plus-average constraints to the feasibility problem for semidefinite programs. This shows that Condon's simple stochastic games, stochastic mean payoff games, and in particular mean payoff games and parity games can all be reduced to semidefinite programming.

math.OC↗
arXiv2025

Games on Graphs: From Logic and Automata to Algorithms

The objective of this book is to give a comprehensive presentation of the research field concerned with infinite duration games on graphs. Historically, these game models appeared in the study of automata and logic, and they later became important for program verification and synthesis. They have many more applications, in particular some of the models investigated in this book were introduced and studied in neighbouring research communities such as optimisation, reinforcement learning, model theory, and set theory.

cs.GT↗
arXiv2025

Cycle Patterns and Mean Payoff Games

We introduce the concept of a \emph{cycle pattern} for directed graphs as functions from the set of cycles to the set $\{-,0,+\}$. The key example for such a pattern is derived from a weight function, giving rise to the sign of the total weight of the edges for each cycle. Hence, cycle patterns describe a fundamental structure of a weighted digraph, and they arise naturally in games on graphs, in particular parity games, mean payoff games, and energy games. Our contribution is threefold: we analyze the structure and derive hardness results for the realization of cycle patterns by weight functions. Then we use them to show hardness of solving games given the limited information of a cycle pattern. Finally, we identify a novel geometric hardness measure for solving mean payoff games (MPG) using the framework of linear decision trees, and use cycle patterns to derive lower bounds with respect to this measure, for large classes of algorithms for MPGs.

cs.GT↗
arXiv2024

Universal Complexity Bounds Based on Value Iteration for Stochastic Mean Payoff Games and Entropy Games

We develop value iteration-based algorithms to solve in a unified manner different classes of combinatorial zero-sum games with mean-payoff type rewards. These algorithms rely on an oracle, evaluating the dynamic programming operator up to a given precision. We show that the number of calls to the oracle needed to determine exact optimal (positional) strategies is, up to a factor polynomial in the dimension, of order R/sep, where the "separation" sep is defined as the minimal difference between distinct values arising from strategies, and R is a metric estimate, involving the norm of approximate sub and super-eigenvectors of the dynamic programming operator. We illustrate this method by two applications. The first one is a new proof, leading to improved complexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino, showing that turn-based mean-payoff games with a fixed number of random positions can be solved in pseudo-polynomial time. The second one concerns entropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. The rank of an entropy game is defined as the maximal rank among all the ambiguity matrices determined by strategies of the two players. We show that entropy games with a fixed rank, in their original formulation, can be solved in polynomial time, and that an extension of entropy games incorporating weights can be solved in pseudo-polynomial time under the same fixed rank condition.

cs.GT↗
arXiv2024

Signed tropical halfspaces and convexity

We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and Végh (ITCS 2020). We study two notions of convexity for signed tropical numbers called 'TO-convexity' (formerly 'signed tropical convexity') and the novel notion 'TC-convexity'. We derive several separation results for TO-convexity and TC-convexity. A key ingredient is a thorough understanding of TC-hemispaces - those TC-convex sets whose complement is also TC-convex. Furthermore, we use new insights in the interplay between convexity over Puiseux series and its signed valuation. Remarkably, TC-convexity can be seen as a natural convexity notion for representing oriented matroids as it arises from a generalization of the composition operation of vectors in an oriented matroid. We make this explicit by giving representations of linear spaces over the real tropical hyperfield in terms of TC-convexity.

math.CO↗
arXiv2024

Smoothed analysis of deterministic discounted and mean-payoff games

We devise a policy-iteration algorithm for deterministic two-player discounted and mean-payoff games, that runs in polynomial time with high probability, on any input where each payoff is chosen independently from a sufficiently random distribution. This includes the case where an arbitrary set of payoffs has been perturbed by a Gaussian, showing for the first time that deterministic two-player games can be solved efficiently, in the sense of smoothed analysis. More generally, we devise a condition number for deterministic discounted and mean-payoff games, and show that our algorithm runs in time polynomial in this condition number. Our result confirms a previous conjecture of Boros et al., which was claimed as a theorem and later retracted. It stands in contrast with a recent counter-example by Christ and Yannakakis, showing that Howard's policy-iteration algorithm does not run in smoothed polynomial time on stochastic single-player mean-payoff games. Our approach is inspired by the analysis of random optimal assignment instances by Frieze and Sorkin, and the analysis of bias-induced policies for mean-payoff games by Akian, Gaubert and Hochart.

cs.GT↗
arXiv2021

Optimal bounds for bit-sizes of stationary distributions in finite Markov chains

An irreducible stochastic matrix with rational entries has a stationary distribution given by a vector of rational numbers. We give an upper bound on the lowest common denominator of the entries of this vector. Bounds of this kind are used to study the complexity of algorithms for solving stochastic mean payoff games. They are usually derived using the Hadamard inequality, but this leads to suboptimal results. We replace the Hadamard inequality with the Markov chain tree formula in order to obtain optimal bounds. We also adapt our approach to obtain bounds on the absorption probabilities of finite Markov chains and on the gains and bias vectors of Markov chains with rewards.

math.CO↗
arXiv2021

Derandomization and absolute reconstruction for sums of powers of linear forms

We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be written as a sum of cubes of linearly independent linear forms with complex coefficients. Compared to previous algorithms for the same problem, the two main novel features of this algorithm are: (i) It is an algebraic algorithm, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial. In particular, it does not make any appeal to polynomial factorization. (ii) For an input polynomial with rational coefficients, the algorithm runs in polynomial time when implemented in the bit model of computation. The algorithm relies on methods from linear and multilinear algebra (symmetric tensor decomposition by simultaneous diagonalization). We also give a version of our algorithm for decomposition over the field of real numbers. In this case, the algorithm performs arithmetic operations and comparisons on the input coefficients. Finally we give several related derandomization results on black box polynomial identity testing, the minimization of the number of variables in a polynomial, the computation of Lie algebras and factorization into products of linear forms.

cs.CC↗
arXiv2020

Tropical spectrahedra

We introduce tropical spectrahedra, defined as the images by the nonarchimedean valuation of spectrahedra over the field of real Puiseux series. We provide an explicit polyhedral characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. One of the key ingredients is Denef-Pas quantifier elimination result over valued fields. We obtain from this that the nonarchimedean valuation maps semialgebraic sets to semilinear sets that are closed. We also prove that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities which define it.

math.AG↗
arXiv2019

Intersection multiplicity of a sparse curve and a low-degree curve

Let $F(x, y) \in \mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) \in \mathbb{C}[x,y]$ be a polynomial with $t$ monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x,y) = G(x,y) = 0$. Our main result is that the multiplicity of any isolated solution $(a,b) \in \mathbb{C}^2$ with nonzero coordinates is no greater than $\frac{5}{2}d^2t^2$. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of $F$ and $G$, and we briefly review some connections between sparse polynomials and algebraic complexity theory.

math.AG↗
arXiv2018

Individual Security and Network Design with Malicious Nodes

Networks are beneficial to those being connected but can also be used as carriers of contagious hostile attacks. These attacks are often facilitated by exploiting corrupt network users. To protect against the attacks, users can resort to costly defense. The decentralized nature of such protection is known to be inefficient but the inefficiencies can be mitigated by a careful network design. Is network design still effective when not all users can be trusted? We propose a model of network design and defense with byzantine nodes to address this question. We study the optimal defended networks in the case of centralized defense and, for the case of decentralized defense, we show that the inefficiencies due to decentralization can be fully mitigated, despite the presence of the byzantine nodes.

cs.GT↗
arXiv2018

Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming

Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by means of stochastic game algorithms. In this paper, we show that the mean payoff of these games can be interpreted as a condition number for the corresponding nonarchimedean feasibility problems. This number measures how close a feasible instance is from being infeasible, and vice versa. We show that it coincides with the maximal radius of a ball in Hilbert's projective metric, that is included in the feasible set. The geometric interpretation of the condition number relies in particular on a duality theorem for tropical semidefinite feasibility programs. Then, we bound the complexity of the feasibility problem in terms of the condition number. We finally give explicit bounds for this condition number, in terms of the characteristics of the stochastic game. As a consequence, we show that the simplest algorithm to decide whether a stochastic mean payoff game is winning, namely value iteration, has a pseudopolynomial complexity when the number of random positions is fixed.

math.OC↗
arXiv2018

Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility problems given by generic matrices. Our approach is based on tropical geometry. It relies on tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish a correspondence between generic tropical spectrahedra and zero-sum stochastic games with perfect information. The latter have been well studied in algorithmic game theory. This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.

math.OC↗
arXiv2018

The tropical analogue of the Helton-Nie conjecture is true

Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods.

math.AG↗