arXiv subjects

Serge Gaspers

Publications and source records attributed to Serge Gaspers.

At least 19 recordsLinked to original sources

arXiv2026

NP-Hardness and Additive Inapproximability of Deterministic Communication Complexity

Given the full truth table of a Boolean function whose input is split between two players, we study the complexity of determining the minimum worst-case number of bits they must communicate to compute it. This is a basic meta-complexity question, posed by Yao in 1979. We prove NP-hardness of approximation within every fixed additive error for deterministic communication complexity, under deterministic polynomial-time reductions. The result holds both in the standard model and in Yao's original model, where the players alternate after every bit and only the receiver of the last bit must determine the answer. In particular, it resolves the exact computation question under Yao's original conventions. Earlier versions of this same paper established exact NP-hardness for the standard model, independently and concurrently with Hirahara, Ilango, and Loff. In both models, a polynomial-time estimator with any fixed additive error would yield a polynomial-time $3$-SAT algorithm, deterministic or bounded-error randomized according to the estimator. For the standard model, we also give quantitative implications for errors that grow with the number $T$ of table entries. A bounded-error randomized polynomial-time estimator with additive error $c\log T/\log\log T$, for any fixed $0\le c<1/8$, yields a bounded-error randomized $3$-SAT algorithm on $N$ variables and $O(N)$ clauses running in time $2^{O(N^α)}$ for every fixed $8c<α<1$. A deterministic polynomial-time estimator with additive error $c_*\log\log T$, for an effective absolute constant $c_*>0$, yields a deterministic algorithm running in time $2^{O((\log N)^2)}$ on such formulas.

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arXiv2026

Path Decompositions of Sparse Graphs via Gaussian Sweeps and Faster Exponential Algorithms

Using the bisection bound of Monien and Preis, Fomin and Høie proved in 2006 that every graph of maximum degree 3 on $n$ vertices has pathwidth at most $n/6+o(n)$. We prove that the pathwidth, bisection width and cutwidth of these graphs are at most $c_3 n+o(n)$, where $c_3=\frac{3}{4π}\arccos\frac56\approx0.1398$. For bisection width this improves the bound $n/6+o(n)$ of Monien and Preis (2006) and extends to all such graphs the bound that Lichev and Mitsche (2023) obtained for random cubic graphs. For maximum degree 4, we prove that bisection width and cutwidth are at most $n/3+o(n)$, improving the bound $2n/5+o(n)$ of Monien and Preis for 4-regular graphs, and that pathwidth is at most $0.2458n$, improving the bound $n/3$ of Fomin, Gaspers, Saurabh and Stepanov (2009). We show that every graph with $m$ edges has pathwidth at most $7m/48+o(n)$, improving the bound $13m/75$ of Scott and Sorkin (2007), and we give further bounds of the form $\sum_{d\ge3}a_d n_d+o(n)$ with $a_d<1$, where $n_d$ is the number of vertices of degree $d$. The path decompositions, and linear orderings of the stated cutwidth, can be computed in polynomial time. Instead of deriving pathwidth bounds from bisection width, we give each vertex a Gaussian label, a weighted sum of independent standard normal variables over a ball around it, and sweep a threshold through the labels; the bags are frontiers, the vertices above the threshold with a neighbor below it. We bound the probability that a vertex lies in a frontier by comparison with an explicit Gaussian model of a vertex and its neighbors. Faster exact algorithms follow: for feedback vertex set and the traveling salesperson problem on graphs of bounded maximum or average degree, and on general graphs for minimum weight dominating set (by branching until the instance is sparse), Max 2-CSP and counting subgraphs with $k$ edges (by the bound in $m$).

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arXiv2026

Monotone Local Search with Sample-Dependent Running Times

Monotone Local Search turns an algorithm that extends a given partial solution into an algorithm that finds a solution from scratch: sample a few elements, hope that they lie in an unknown solution, and complete them. Its known analyses bound every call of the extension algorithm by a worst-case time that depends only on the number of sampled elements, and Esmer et al. (SODA 2024) showed that no algorithm that uses the extension algorithm as a black box with this running time beats the original form by more than a polynomial factor. We let this running time depend on the actual sample. If the extension algorithm decides in time $O^*(2^{g(I,X,k)})$, for a polynomial-time computable $g$, whether a sample $X$ can be completed to a solution with at most $k$ elements, our main theorem gives a randomized algorithm with one-sided error in time $O^*(2^{ψn})$ whenever, for some solution $S$ of size $k$ and some $t$, the exponential moment $\sum_{X\subseteq S,|X|=t}2^{-g(I,X,k)}$ is at least $2^{-ψn}\binom{n}{t}$. A corollary, multivariate Monotone Local Search, generalizes the theorems of Fomin et al. (STOC 2016, J. ACM 2019) and of Gaspers and Lee (ICALP 2017). For Max Leaf Spanning Tree and, equivalently, Connected Dominating Set, the previous best polynomial-space bound is $O^*(1.8619^n)$. Completing a sample is a Steiner tree problem, solved in time exponential in the number of terminals, which varies strongly with the sample. We obtain a randomized polynomial-space algorithm in time $O(1.6991^n)$ and a deterministic one in time $(\sqrt5-\frac12)^n 2^{o(n)}=O(1.7361^n)$. For connectivity problems in which the extension algorithm's running time doubles with every component of the sample, an analysis through a partition function gives randomized polynomial-space algorithms in time $O^*(1.8^n)$ for Connected Feedback Vertex Set and $O(1.6444^n)$ for Connected Vertex Cover.

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arXiv2026

Faster Exponential-Time Approximate Counting via Bounded Self-Reductions

We give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general $n$-vertex graphs, our independent-set counter runs in $O^{\ast}(1.1869^{n})$ time, improving the previous $O^{\ast}(1.2041^{n})$ general-graph bound. For $n$-variable \#\textsc{2-SAT}, we obtain an $O^{\ast}(1.2373^{n})$-time approximation algorithm, narrowly below Wahlstr{ö}m's currently cited $O^{\ast}(1.2377^{n})$ variable-parameter exact bound. The new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with $f(x)$ positive leaves and recursion-compatible upper bound $b(x)$, an enumerate-or-sample estimator gives an $(\varepsilon,δ)$-approximation in \[ O^{\ast}\!\left(\sqrt{b(x)}\,\varepsilon^{-2}\log \tfrac1δ\right) \] time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays \[ O^{\ast}\!\left(\sqrt{\sum_i b_i(x_i)}\,\varepsilon^{-2}\log \tfrac1δ\right), \] rather than estimating the cores separately at cost $\sum_i \sqrt{b_i(x_i)}$. The same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.

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arXiv2025

A Faster Randomized Algorithm for Vertex Cover: An Automated Approach

This work introduces two techniques for the design and analysis of branching algorithms, illustrated through the case study of the Vertex Cover problem. First, we present a method for automatically generating branching rules through a systematic case analysis of local structures. Second, we develop a new technique for analyzing randomized branching algorithms using the Measure & Conquer method, offering greater flexibility in formulating branching rules. By combining these innovations with additional techniques, we obtain the fastest known randomized algorithms in different parameters for the Vertex Cover problem on graphs with bounded degree (up to 6) and on general graphs. For example, our algorithm solves Vertex Cover on subcubic graphs in $O^*(1.07625^n)$ time and $O^*(1.13132^k)$ time, respectively. For graphs with maximum degree 4, we achieve running times of $O^*(1.13735^n)$ and $O^*(1.21103^k)$, while for general graphs we achieve $O^*(1.25281^k)$.

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arXiv2024

A Piecewise Approach for the Analysis of Exact Algorithms

To analyze the worst-case running time of branching algorithms, the majority of work in exponential time algorithms focuses on designing complicated branching rules over developing better analysis methods for simple algorithms. In the mid-$2000$s, Fomin et al. [2005] introduced measure & conquer, an advanced general analysis method, sparking widespread adoption for obtaining tighter worst-case running time upper bounds for many fundamental NP-complete problems. Yet, much potential in this direction remains untapped, as most subsequent work applied it without further advancement. Motivated by this, we present piecewise analysis, a new general method that analyzes the running time of branching algorithms. Our approach is to define a similarity ratio that divides instances into groups and then analyze the running time within each group separately. The similarity ratio is a scale between two parameters of an instance I. Instead of relying on a single measure and a single analysis for the whole instance space, our method allows to take advantage of different intrinsic properties of instances with different similarity ratios. To showcase its potential, we reanalyze two $17$-year-old algorithms from Fomin et al. [2007] that solve $4$-Coloring and #$3$-Coloring respectively. The original analysis in their paper gave running times of $O(1.7272^n)$ and $O(1.6262^n)$ respectively for these algorithms, our analysis improves these running times to $O(1.7207^n)$ and $O(1.6225^n)$.

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arXiv2023

Quantum Algorithms for Graph Coloring and other Partitioning, Covering, and Packing Problems

Let U be a universe on n elements, let k be a positive integer, and let F be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from F, covering U with k sets from F, and packing k non-intersecting sets from F into U. Classically, these problems can be solved via inclusion-exclusion in O*(2^n) time [BjorklundHK09]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140^n) [ShimizuM22] and for Set Cover with a small number of sets with running time O(1.7274^n |F|^O(1)) [AmbainisBIKPV19]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a subset X of U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in O*(c^{|X|}) time with c<2. Our bounded-error quantum algorithm runs in O*((2+c)^(n/2)) for Set Partition, Set Cover, and Set Packing. When c<=1.147899, our algorithm is slightly faster than O*((2+c)^(n/2)); when c approaches 1, it matches the running time of [AmbainisBIKPV19] for Set Cover when |F| is subexponential in n. For Graph Coloring, we further improve the running time to O(1.7956^n) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number, we obtain a O((2-ε)^n) running time for some ε>0.

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arXiv2023

Cluster Editing with Vertex Splitting

Cluster Editing, also known as Correlation Clustering, is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to $k$ edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Other strong motivations come from biological network analysis and from language networks. Trying to cluster words with similar usage in the latter can be confounded by homonyms, that is, words with multiple meanings like "bat." In this paper, we introduce a new variant of Cluster Editing whereby a vertex can be split into two or more vertices. First used in the context of graph drawing, this operation allows a vertex $v$ to be replaced by two vertices whose combined neighborhood is the neighborhood of $v$ (and thus $v$ can belong to more than one cluster). We call the new problem Cluster Editing with Vertex Splitting and we initiate the study of it. We show that it is NP-complete and fixed-parameter tractable when parameterized by the total number $k$ of allowed vertex-splitting and edge-editing operations. In particular, we obtain an $O(2^{9k log k} + n + m)$-time algorithm and a $6k$-vertex kernel.

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arXiv2021

Making the Most of Parallel Composition in Differential Privacy

We show that the `optimal' use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that `overlap' on the data domain, a quantity we call the \emph{maximum overlap} of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of $f$-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for $f$-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to $10^{20000}$), and that its application can reduce the noise added to query answers by up to 60\%.

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arXiv2020

From Matching with Diversity Constraints to Matching with Regional Quotas

In the past few years, several new matching models have been proposed and studied that take into account complex distributional constraints. Relevant lines of work include (1) school choice with diversity constraints where students have (possibly overlapping) types and (2) hospital-doctor matching where various regional quotas are imposed. In this paper, we present a polynomial-time reduction to transform an instance of (1) to an instance of (2) and we show how the feasibility and stability of corresponding matchings are preserved under the reduction. Our reduction provides a formal connection between two important strands of work on matching with distributional constraints. We then apply the reduction in two ways. Firstly, we show that it is NP-complete to check whether a feasible and stable outcome for (1) exists. Due to our reduction, these NP-completeness results carry over to setting (2). In view of this, we help unify some of the results that have been presented in the literature. Secondly, if we have positive results for (2), then we have corresponding results for (1). One key conclusion of our results is that further developments on axiomatic and algorithmic aspects of hospital-doctor matching with regional quotas will result in corresponding results for school choice with diversity constraints.

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arXiv2019

Minimizing and Computing the Inverse Geodesic Length on Trees

For any fixed measure $H$ that maps graphs to real numbers, the MinH problem is defined as follows: given a graph $G$, an integer $k$, and a target $τ$, is there a set $S$ of $k$ vertices that can be deleted, so that $H(G - S)$ is at most $τ$? In this paper, we consider the MinH problem on trees. We call $H$ "balanced on trees" if, whenever $G$ is a tree, there is an optimal choice of $S$ such that the components of $G-S$ have sizes bounded by a polynomial in $n/k$. We show that MinH on trees is FPT for parameter $n/k$, and furthermore, can be solved in subexponential time, and polynomial space, if $H$ is additive, balanced on trees, and computable in polynomial time. A measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: $IGL(G)=\sum_{\{u,v\} \subseteq V} \frac{1}{d_G(u,v)}$. While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in $2^{o(k+n+m)}$ time, even on bipartite graphs with $n$ vertices and $m$ edges, the complexity status of the problem remains open on trees. We show that IGL is balanced on trees, to give a $2^{O((n\log n)^{5/6})}$ time, polynomial space algorithm. The distance distribution of $G$ is the sequence $\{a_i\}$ describing the number of vertex pairs distance $i$ apart in $G$: $a_i=|\{\{u, v\}: d_G(u, v)=i\}|$. We show that the distance distribution of a tree can be computed in $O(n\log^2 n)$ time by reduction to polynomial multiplication. We extend our result to graphs with small treewidth by showing that the first $p$ values of the distance distribution can be computed in $2^{O(tw(G))} n^{1+\varepsilon} \sqrt{p}$ time, and the entire distance distribution can be computed in $2^{O(tw(G))} n^{1+\varepsilon}$ time, when the diameter of $G$ is $O(n^{\varepsilon'})$ for every $\varepsilon'>0$.

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arXiv2019

Enumeration of Preferred Extensions in Almost Oriented Digraphs

In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all preferred extensions can be enumerated in $O^*(3^{n/3})$ time and there are directed graphs with $Ω(3^{n/3})$ preferred extensions. We give faster enumeration algorithms for directed graphs with at most $0.8004\cdot n$ vertices occurring in $2$-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time $O(1.2321^n)$, and we show that there are oriented graphs with $Ω(3^{n/6}) > Ω(1.2009^n)$ preferred extensions. A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in $2$-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017).

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arXiv2019

Linearly $χ$-Bounding $(P_6,C_4)$-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on $t$ vertices and the cycle on $s$ vertices, respectively. In this paper we show that for any $(P_6,C_4)$-free graph $G$ it holds that $χ(G)\le \frac{3}{2}ω(G)$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. %Our bound is attained by $C_5$ and the Petersen graph. Our bound is attained by several graphs, for instance, the five-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all $4$-critical $(P_6,C_4)$-free graphs other than $K_4$ (see \cite{HH17}). The new result unifies previously known results on the existence of linear $χ$-binding functions for several graph classes. Our proof is based on a novel structure theorem on $(P_6,C_4)$-free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time $3/2$-approximation algorithm for coloring $(P_6,C_4)$-free graphs. Our algorithm computes a coloring with $\frac{3}{2}ω(G)$ colors for any $(P_6,C_4)$-free graph $G$ in $O(n^2m)$ time.

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arXiv2018

$(2P_2,K_4)$-Free Graphs are 4-Colorable

In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for $(4P_1,C_4)$-free graphs. Lozin and Malyshev \cite{LM17} conjecture that the answer is yes. As an application of our main result, we provide the first positive evidence to the conjecture by giving a 2-approximation algorithm for coloring $(4P_1,C_4)$-free graphs.

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arXiv2018

Colouring Square-Free Graphs without Long Induced Paths

The complexity of {\sc Colouring} is fully understood for $H$-free graphs, but there are still major complexity gaps if two induced subgraphs $H_1$ and $H_2$ are forbidden. Let $H_1$ be the $s$-vertex cycle $C_s$ and $H_2$ be the $t$-vertex path $P_t$. We show that {\sc Colouring} is polynomial-time solvable for $s=4$ and $t\leq 6$, strengthening several known results. Our main approach is to initiate a study into the boundedness of the clique-width of atoms (graphs with no clique cutset) of a hereditary graph class. We first show that the classifications of boundedness of clique-width of $H$-free graphs and $H$-free atoms coincide. We then show that this is not the case if two graphs are forbidden: we prove that $(C_4,P_6)$-free atoms have clique-width at most~18. Our key proof ingredients are a divide-and-conquer approach for bounding the clique-width of a subclass of $C_4$-free graphs and the construction of a new bound on the clique-width for (general) graphs in terms of the clique-width of recursively defined subgraphs induced by homogeneous pairs and triples of sets. As a complementary result we prove that {\sc Colouring} is \NP-complete for $s=4$ and $t\geq 9$, which is the first hardness result on {\sc Colouring} for $(C_4,P_t)$-free graphs. Combining our new results with known results leads to an almost complete dichotomy for \cn restricted to $(C_s,P_t)$-free graphs.

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arXiv2017

Separate, Measure and Conquer: Faster Algorithms for Max 2-CSP and Counting Dominating Sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. An instance of the problem Max (r,2)-CSP, or simply Max 2-CSP, is parametrized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G), and the number of constraints m (edges in G). When G is cubic, and omitting sub-exponential terms here for clarity, we give an algorithm running in time r^((1/5)n) = r^((2/15)m); the previous best was r^((1/4)n) = r^((1/6)m). By known results, this improvement for the cubic case results in an algorithm running in time r^((9/50)m) for general instances; the previous best was r^((19/100)m). We show that the analysis of the earlier algorithm was tight: our improvement is in the algorithm, not just the analysis. The new algorithm, like the old, extends to Polynomial and Ring CSP. We also give faster algorithms for #Dominating Set, counting the dominating sets of every cardinality 0,...,n for a graph G of order n. For cubic graphs, our algorithm runs in time 3^((1/6)n); the previous best was 2^((1/2)n). For general graphs, we give an unrelated algorithm running in time 1.5183^n; the previous best was 1.5673^n. The previous best algorithms for these problems all used local transformations and were analyzed by the "Measure and Conquer" method. Our new algorithms capitalize on the existence of small balanced separators for cubic graphs - a non-local property - and the ability to tailor the local algorithms always to "pivot" on a vertex in the separator. The new algorithms perform much as the old ones until the separator is empty, at which point they gain because the remaining vertices are split into two independent problem instances that can be solved recursively. It is likely that such algorithms can be effective for other problems too, and we present their design and analysis in a general framework.

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arXiv2017

Barrier Coverage with Non-uniform Lengths to Minimize Aggregate Movements

Given a line segment $I=[0,L]$, the so-called barrier, and a set of $n$ sensors with varying ranges positioned on the line containing $I$, the barrier coverage problem is to move the sensors so that they cover $I$, while minimising the total movement. In the case when all the sensors have the same radius the problem can be solved in $O(n \log n)$ time (Andrews and Wang, Algorithmica 2017). If the sensors have different radii the problem is known to be NP-hard to approximate within a constant factor (Czyzowicz et al., ADHOC-NOW 2009). We strengthen this result and prove that no polynomial time $ρ^{1-\varepsilon}$-approximation algorithm exists unless $P=NP$, where $ρ$ is the ratio between the largest radius and the smallest radius. Even when we restrict the number of sensors that are allowed to move by a parameter $k$, the problem turns out to be W[1]-hard. On the positive side we show that a $((2+\varepsilon)ρ+2/\varepsilon)$-approximation can be computed in $O(n^3/\varepsilon^2)$ time and we prove fixed-parameter tractability when parameterized by the total movement assuming all numbers in the input are integers.

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arXiv2017

The Parameterized Complexity of Positional Games

We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows' influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterized by formula size. We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise. Short Maker-Maker is AW[*]-complete, whereas Short Maker-Breaker is W[1]-complete and Short Enforcer-Avoider co-W[1]-complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the W-hierarchy when the winning configurations only depend on which vertices one player has been able to pick, but AW[*]-completeness when the winning condition depends on which vertices both players have picked. However, some positional games where the board and the winning configurations are highly structured are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when parameterized by the number of moves.

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