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Given a graph $G$ and two graph homomorphisms $\alpha$ and $\beta$ from $G$ to a fixed graph $H$, the problem $H$-Recoloring asks whether there is a transformation from $\alpha$ to $\beta$ that changes the image of a single vertex at each step and keeps a graph homomorphism throughout. The complexity of the problem depends among other things on the presence of loops on the vertices. We provide a simple reduction that, using a known algorithmic result for $H$-Recoloring for square-free irreflexive graphs $H$, yields a polynomial-time algorithm for $H$-Recoloring for square-free reflexive graphs $H$. This generalizes all known algorithmic results for $H$-Recoloring for reflexive graphs $H$. Furthermore, the construction allows us to recover some of the known hardness results. Finally, we provide a partial inverse of the construction for bipartite instances., Comment: 12 pages, 2 figures

The problem \textsc{Token Jumping} asks whether, given a graph $G$ and two independent sets of \emph{tokens} $I$ and $J$ of $G$, we can transform $I$ into $J$ by changing the position of a single token in each step and having an independent set of tokens throughout. We show that there is a polynomial-time algorithm that, given an instance of \textsc{Token Jumping}, computes an equivalent instance of size $O(g^2 + gk + k^2)$, where $g$ is the genus of the input graph and $k$ is the size of the independent sets., Comment: 11 pages, 1 figure

Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev's theorem, showing that the problem is PSPACE-complete unless $H$ is a path or a subdivision of the claw. We then show that under the token sliding rule, the problem admits a polynomial-time algorithm if the input graph is fork-free., Comment: 19 pages, 4 figures

The orientation theorem of Nash-Williams states that an undirected graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. Recently, Ito et al. showed that any orientation of an undirected $2k$-edge-connected graph can be transformed into a $k$-arc-connected orientation by reorienting one arc at a time without decreasing the arc-connectivity at any step, thus providing an algorithmic proof of Nash-Williams' theorem. We generalize their result to hypergraphs and therefore provide an algorithmic proof of the characterization of hypergraphs with a $k$-hyperarc-connected orientation originally given by Frank et al. We prove that any orientation of an undirected $(k,k)$-partition-connected hypergraph can be transformed into a $k$-hyperarc-connected orientation by reorienting one hyperarc at a time without decreasing the hyperarc-connectivity in any step. Furthermore, we provide a simple combinatorial algorithm for computing such a transformation in polynomial time., Comment: 18 pages, 3 figures, 3 algorithms

For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G to H throughout. We extend an algorithm of Wrochna for H-Recoloring where H is a square-free loopless undirected graph to the more general setting of directed graphs. We obtain a polynomial-time algorithm for H-Recoloring in this setting whenever H is a loopless digraph that does not contain a 4-cycle of algebraic girth zero and whenever H is a reflexive digraph that contains neither a 3-cycle of algebraic girth 1 nor a 4-cycle of algebraic girth zero.

An approximation algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying a $(1 - f(\epsilon))$-fraction of the constraints on any $(1-\epsilon)$-satisfiable instance, where the loss function $f$ is such that $f(\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. Moreover, the runtime of a robust algorithm should not depend in any way on $\epsilon$. In this paper, we present such an algorithm for Min-Unique-Games on complete graphs with $q$ labels. Specifically, the loss function is $f(\epsilon) = (\epsilon + c_{\epsilon} \epsilon^2)$, where $c_{\epsilon}$ is a constant depending on $\epsilon$ such that $\lim_{\epsilon \rightarrow 0} c_{\epsilon} = 16$. The runtime of our algorithm is $O(qn^3)$ (with no dependence on $\epsilon$) and can run in time $O(qn^2)$ using a randomized implementation with a slightly larger constant $c_{\epsilon}$. Our algorithm is combinatorial and uses voting to find an assignment. It can furthermore be used to provide a PTAS for Min-Unique-Games on complete graphs, recovering a result of Karpinski and Schudy with a simpler algorithm and proof. We also prove NP-hardness for Min-Unique-Games on complete graphs and (using a randomized reduction) even in the case where the constraints form a cyclic permutation, which is also known as Min-Linear-Equations-mod-$q$ on complete graphs.

Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures \PSPACE\xspace and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains \PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar \textsc{and/or}\xspace graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of \textsc{and}\xspace or \textsc{or}\xspace vertices of an \textsc{and/or}\xspace constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of \textsc{and}\xspace vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing \PSPACE.

The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform fault model. Such a model assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ may fail. While this approach yields problems with clean combinatorial structure and good algorithms, it often fails to capture the true nature of the scenario set coming from applications. One natural refinement of the uniform model is obtained by partitioning the set of resources into vulnerable and safe resources. The scenario set contains every subset of at most $k$ faulty resources. This work studies the Fault-Tolerant Path (FTP) problem, the counterpart of the Shortest Path problem in this fault model and the Fault-Tolerant Flow problem (FTF), the counterpart of the $\ell$-disjoint Shortest $s$-$t$ Path problem. We present complexity results alongside exact and approximation algorithms for both models. We emphasize the vast increase in the complexity of the problem with respect to the uniform analogue, the Edge-Disjoint Paths problem., Comment: arXiv admin note: substantial text overlap with arXiv:1301.6299

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (TAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study \emph{Flexible Graph Connectivity} (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges. While this non-uniform nature of the problem makes it appealing from the modeling perspective, it also renders most existing algorithmic tools for dealing with network design problems unfit for approximating FGC. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are very close to the best known bounds for many special cases, such as 2-ECSS and TAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem., Comment: Improved presentation of the paper

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as $P_4$-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

Meta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a meta-heuristic for a large class of optimization problems over discrete domains based on the particle swarm optimization (PSO) paradigm. We provide a comprehensive formal analysis of the performance of this algorithm on certain "easy" reference problems in a black-box setting, namely the sorting problem and the problem OneMax. In our analysis we use a Markov model of the proposed algorithm to obtain upper and lower bounds on its expected optimization time. Our bounds are essentially tight with respect to the Markov model. We show that for a suitable choice of algorithm parameters the expected optimization time is comparable to that of known algorithms and, furthermore, for other parameter regimes, the algorithm behaves less greedy and more explorative, which can be desirable in practice in order to escape local optima. Our analysis provides a precise insight on the tradeoff between optimization time and exploration. To obtain our results we introduce the notion of indistinguishability of states of a Markov chain and provide bounds on the solution of a recurrence equation with non-constant coefficients by integration.

Imagine that unlabelled tokens are placed on the edges of a graph, such that no two tokens are placed on incident edges. A token can jump to another edge if the edges having tokens remain independent. We study the problem of determining the distance between two token configurations (resp., the corresponding matchings), which is given by the length of a shortest transformation. We give a polynomial-time algorithm for the case that at least one of the two configurations is not inclusion-wise maximal and show that otherwise, the problem admits no polynomial-time sublogarithmic-factor approximation unless P = NP. Furthermore, we show that the distance of two configurations in bipartite graphs is fixed-parameter tractable parameterized by the size $d$ of the symmetric difference of the source and target configurations, and obtain a $d^\varepsilon$-factor approximation algorithm for every $\varepsilon > 0$ if additionally the configurations correspond to maximum matchings. Our two main technical tools are the Edmonds-Gallai decomposition and a close relation to the Directed Steiner Tree problem. Using the former, we also characterize those graphs whose corresponding configuration graphs are connected. Finally, we show that deciding if the distance between two configurations is equal to a given number $\ell$ is complete for the class $D^P$, and deciding if the diameter of the graph of configurations is equal to $\ell$ is $D^P$-hard., Comment: 31 pages, 3 figures

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We provide efficient exact and approximation algorithms. In particular, for the unit-cost problem, we provide a $\log_2 n$-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we settle the approximability of the problem and show a close relation to the Directed Steiner Forest Problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. Our methods rely on a close relationship to the classical strong connectivity augmentation problem and directed Steiner problems., Comment: 32 pages, 5 figures

The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a degree-constrained subgraph instance, can we transform one solution into the other by adding and removing individual edges, such that each intermediate subgraph satisfies the degree constraints and contains at least a certain minimum number of edges? This problem is a generalization of the matching reconfiguration problem, which is known to be in P. We show that even in the more general setting the reconfiguration problem is in P., Comment: Full version of the paper published at Mathematical Foundations of Computer Science (MFCS) 2015

We investigate the connectedness of clash-free timetables with respect to the Kempe-exchange operation. This investigation is related to the connectedness of the search space of timetabling problem instances, which is a desirable property, for example for two-step algorithms using the Kempe-exchange during the optimization step. The theoretical framework for our investigations is based on the study of reconfiguration graphs, which model the search space of timetabling problems. We contribute to this framework by including timeslot availability requirements in the analysis and we derive improved conditions for the connectedness of clash-free timetables in this setting. We apply the theoretical insights to establish the connectedness of clash-free timetables for a number of benchmark instances.

We propose a decomposition of the max-min fair curriculum-based course timetabling (MMF-CB-CTT) problem. The decomposition models the room assignment subproblem as a generalized lexicographic bottleneck optimization problem (LBOP). We show that the generalized LBOP can be solved efficiently if the corresponding sum optimization problem can be solved efficiently. As a consequence, the room assignment subproblem of the MMF-CB-CTT problem can be solved efficiently. We use this insight to improve a previously proposed heuristic algorithm for the MMF-CB-CTT problem. Our experimental results indicate that using the new decomposition improves the performance of the algorithm on most of the 21 ITC2007 test instances with respect to the quality of the best solution found. Furthermore, we introduce a measure of the quality of a solution to a max-min fair optimization problem. This measure helps to overcome some limitations imposed by the qualitative nature of max-min fairness and aids the statistical evaluation of the performance of randomized algorithms for such problems. We use this measure to show that using the new decomposition the algorithm outperforms the original one on most instances with respect to the average solution quality., Comment: revised version (fixed problems in the notation and general improvements); original paper: 16 pages, accepted for publication at the Multidisciplinary International Scheduling Conference 2013 (MISTA 2013)

We consider the problem of creating fair course timetables in the setting of a university. Our motivation is to improve the overall satisfaction of individuals concerned (students, teachers, etc.) by providing a fair timetable to them. The central idea is that undesirable arrangements in the course timetable, i.e., violations of soft constraints, should be distributed in a fair way among the individuals. We propose two formulations for the fair course timetabling problem that are based on max-min fairness and Jain's fairness index, respectively. Furthermore, we present and experimentally evaluate an optimization algorithm based on simulated annealing for solving max-min fair course timetabling problems. The new contribution is concerned with measuring the energy difference between two timetables, i.e., how much worse a timetable is compared to another timetable with respect to max-min fairness. We introduce three different energy difference measures and evaluate their impact on the overall algorithm performance. The second proposed problem formulation focuses on the tradeoff between fairness and the total amount of soft constraint violations. Our experimental evaluation shows that the known best solutions to the ITC2007 curriculum-based course timetabling instances are quite fair with respect to Jain's fairness index. However, the experiments also show that the fairness can be improved further for only a rather small increase in the total amount of soft constraint violations., Comment: appeared in PATAT 2012, pp. 114-130