Depth, balancing, and limits of the Elo model

-Much work has been devoted to the computational complexity of games. However, they are not necessarily relevant for estimating the complexity in human terms. Therefore, human-centered measures have been proposed, e.g. the depth. This paper discusses the depth of various games, extends it to a continuous measure. We provide new depth results and present tool (given-first-move, pie rule, size extension) for increasing it. We also use these measures for analyzing games and opening moves in Y, NoGo, Killall Go, and the effect of pie rules

Complexity of Token Swapping and its Variants

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

Complexity of token swapping and its variants

AbstractIn the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)no(k/logk) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases

Complexity of token swapping and its variants

AbstractIn the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)no(k/logk) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases

Lower Bounds for Existential Pebble Games and k-Consistency Tests

The existential k-pebble game characterizes the expressive power of the existential-positive k-variable fragment of first-order logic on finite structures. The winner of the existential k-pebble game on two given finite structures can be determined in time O(n2k) by dynamic programming on the graph of game configurations. We show that there is no O(n(k-3)/12)-time algorithm that decides which player can win the existential k-pebble game on two given structures. This lower bound is unconditional and does not rely on any complexity-theoretic assumptions. Establishing strong k-consistency is a well-known heuristic for solving the constraint satisfaction problem (CSP). By the game characterization of Kolaitis and Vardi our result implies that there is no O(n(k-3)/12)-time algorithm that decides if strong k-consistency can be established for a given CSP-instance

A parametric analysis of the state-explosion problem in model checking

AbstractIn model checking, the state-explosion problem occurs when one checks a nonflat system, i.e., a system implicitly described as a synchronized product of elementary subsystems. In this paper, we investigate the complexity of a wide variety of model-checking problems for nonflat systems under the light of parameterized complexity, taking the number of synchronized components as a parameter. We provide precise complexity measures (in the parameterized sense) for most of the problems we investigate, and evidence that the results are robust

Catena: A Memory-Consuming Password-Scrambling Framework

It is a common wisdom that servers should store the one-way hash of their clients’ passwords, rather than storing the password in the clear. In this paper we introduce a set of functional properties a key-derivation function (password scrambler) should have. Unfortunately, none of the existing algorithms satisfies our requirements and therefore, we introduce a novel and provably secure password scrambling framework (PSF) called Catena. Furthermore, we introduce two instantiations of Catena based on a memory-consuming one-way functions. Thus, Catena excellently thwarts massively parallel attacks on cheap memory-constrained hardware, such as recent graphical processing units (GPUs). Additionally, we show that Catena is also a good key-derivation function, since – in the random oracle model – it is indistinguishable from a random function. Furthermore, the memory-access pattern of both instantiations is password-independent and therefore, Catena provides resistance against cache-timing attacks. Moreover, Catena is the first PSF which naturally supports (1) client-independent updates (the server can increase the security parameters and update the password hash without user interaction or knowing the password), (2) an optional server relief protocol (saving the server’s resources at the cost of the client), and (3) a variant Catena-KG for secure key derivation (to securely generate many cryptographic keys of arbitrary lengths such that compromising some keys does not help to break others). We denote a password scrambler as a PSF with a certain instantiation