arXiv subjects

Mikhail Vyalyi

Publications and source records attributed to Mikhail Vyalyi.

7 recordsLinked to original sources

arXiv2025

Two-person Positive Shortest Path Games Have Nash Equilibria in Pure Stationary Strategies

We prove that every finite two-person shortest path game, where the local cost of every move is positive for each player, has a Nash equilibrium (NE) in pure stationary strategies, which can be computed in polynomial time. We also extend the existence result to infinite graphs with finite out-degrees. Moreover, our proof gives that a terminal NE (in which the play is a path from the initial position to a terminal) exists provided at least one of the two players can guarantee reaching a terminal. If none of the players can do it, in other words, if each of the two players has a strategy that separates all terminals from the initial position $s$, then, obviously, a cyclic NE exists, although its cost is infinite for both players, since we restrict ourselves to positive games. We conjecture that a terminal NE exists too, provided there exists a directed path from $s$ to a terminal. However, this is open. We extend our result to short paths interdiction games, where at each vertex, we allow one player to block some of the arcs and the other player to choose one of the non-blocked arcs. Assuming that blocking sets are chosen from an independence system given by an oracle, we give an algorithm for computing a NE in time $O(|E|(\log|V|+τ))$, where $V$ is the set of vertices, $E$ is the set of arcs, and $τ$ is the maximum time taken by the oracle on any input.

cs.DM↗

arXiv2025

Avoidability beyond paths

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order $k$ also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which appear to be related to the recent work by Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175].

math.CO↗

arXiv2025

Growing Trees and Amoebas' Replications

An amoeba is a tree together with instructions how to iteratively grow trees by adding paths of a fixed length $\ell$. This paper analyses such a growth process. An amoeba is mortal if all versions of the process are finite, and it is immortal if they are all infinite. We obtain some necessary and some sufficient conditions for mortality. In particular, for growing caterpillars in the case $\ell=1$ we characterize mortal amoebas. We discuss variations of the mortality concept, conjecture that some of them are equivalent, and support this conjecture for $\ell\in\{1,2\}$.

math.CO↗

arXiv2022

Automata Equipped with Auxiliary Data Structures and Regular Realizability Problems

We consider general computational models: one-way and two-way finite automata, and logarithmic space Turing machines, all equipped with an auxiliary data structure (ADS). The definition of an ADS is based on the language of protocols of work with the ADS. We describe the connection of automata-based models with ``Balloon automata'' that are another general formalization of automata equipped with an ADS presented by Hopcroft and Ullman in 1967. This definition establishes the connection between the non-emptiness problem for one-way automata with ADS, languages recognizable by nondeterministic log-space Turing machines equipped with the same ADS, and a regular realizability problem (NRR) for the language of ADS' protocols. The NRR problem is to verify whether the regular language on the input has a non-empty intersection with the language of protocols. The computational complexity of these problems (and languages) is the same up to log-space reductions.

cs.FL↗

arXiv2021

Shifting paths to avoidable ones

An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced path of the same length, and proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths are shifts of each other if their union is an induced path with $k+1$ vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.

math.CO↗

arXiv2019

An application of communication complexity, Kolmogorov complexity and extremal combinatorics to parity games

So-called separation automata are in the core of several recently invented quasi-polynomial time algorithms for parity games. An explicit $q$-state separation automaton implies an algorithm for parity games with running time polynomial in $q$. It is open whether a polynomial-state separation automaton exists. A positive answer will lead to a polynomial-time algorithm for parity games, while a negative answer will at least demonstrate impossibility to construct such an algorithm using separation approach. In this work we prove exponential lower bound for a restricted class of separation automata. Our technique combines communication complexity and Kolmogorov complexity. One of our technical contributions belongs to extremal combinatorics. Namely, we prove a new upper bound on the product of sizes of two families of sets with small pairwise intersection.

cs.FL↗

arXiv2019

Re-pairing brackets

Consider the following one-player game. Take a well-formed sequence of opening and closing brackets. As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, and the complexity of a strategy is measured by its width: the maximum number of nonempty segments of symbols (separated by blank space) seen during the play. For various initial sequences, we prove upper and lower bounds on the minimum width sufficient for re-pairing. (In particular, the sequence associated with the complete binary tree of height $n$ admits a strategy of width sub-exponential in $\log n$.) Our two key contributions are (1) lower bounds on the width and (2) their application in automata theory: quasi-polynomial lower bounds on the translation from one-counter automata to Parikh-equivalent nondeterministic finite automata. The latter result answers a question by Atig et al. (2016).

cs.FL↗