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37
The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond
 In SODA ’08: Proceedings of the nineteenth annual ACMSIAM symposium on Discrete algorithms
, 2008
"... We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. W ..."
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Cited by 34 (4 self)
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We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACEcomplete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).
Reconfiguration of list edgecolorings in a graph
 PROC. OF WADS 2009, IN: LNCS
, 2009
"... We study the problem of reconfiguring one list edgecoloring of a graph into another list edgecoloring by changing one edge color at a time, while at all times maintaining a list edgecoloring, given a list of allowed colors for each edge. First we show that this problem is PSPACEcomplete, even ..."
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Cited by 20 (6 self)
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We study the problem of reconfiguring one list edgecoloring of a graph into another list edgecoloring by changing one edge color at a time, while at all times maintaining a list edgecoloring, given a list of allowed colors for each edge. First we show that this problem is PSPACEcomplete, even for planar graphs of maximum degree 3 and just six colors. Then we consider the problem restricted to trees. We show that any list edgecoloring can be transformed into any other under the sufficient condition that the number of allowed colors for each edge is strictly larger than the degrees of both its endpoints. This sufficient condition is best possible in some sense. Our proof yields a polynomialtime algorithm that finds a transformation between two given list edgecolorings of a tree with n vertices using O(n 2) recolor steps. This worstcase bound is tight: we give an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n 2) recolor steps.
On the Complexity of Reconfiguration Problems
"... Abstract. Reconfiguration problems arise when we wish to find a stepbystep transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NPcomplete problems are PSPACEcomplete, w ..."
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Abstract. Reconfiguration problems arise when we wish to find a stepbystep transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NPcomplete problems are PSPACEcomplete, while some are also NPhard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time. 1
The complexity of rerouting shortest paths
 In Proc. of Mathematical Foundations of Computer Science
, 2012
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On the parameterized complexity of reconfiguration problems
, 2013
"... We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration version of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps that can be applied to transform S into ..."
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We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration version of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps that can be applied to transform S into T such that each step results in a feasible solution to Q. For most of the results in this paper, S and T are subsets of vertices of a given graph and a reconfiguration step adds or deletes a vertex. Our study is motivated by recent results establishing that for most NPhard problems, the classical complexity of reconfiguration is PSPACEcomplete. We address the question for several important graph properties under two natural parameterizations: k, the size of the solutions, and `, the length of the sequence of steps. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixedparameter algorithms for the reconfiguration versions of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set, all parameterized by k. In contrast, we show that reconfiguring Unbounded Hitting Set is W [2]hard when parameterized by k+`. We also demonstrate the W [1]hardness of the reconfiguration versions of a large class of maximization problems parameterized by k + `, and of their corresponding deletion problems parameterized by `; in doing so, we show that there exist problems in FPT when parameterized by k, but whose reconfiguration versions are W [1]hard when parameterized by k + `.
Recoloring bounded treewidth graphs
 In Proceedings of the 7th LatinAmerican Algorithms, Graphs, and Optimization Symposium
, 2013
"... Let k be an integer. Two vertex kcolorings of a graph are adjacent if they differ on exactly one vertex. A graph is kmixing if any proper kcoloring can be transformed into any other through a sequence of adjacent proper kcolorings. Any graph is (tw + 2)mixing, where tw is the treewidth of the g ..."
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Let k be an integer. Two vertex kcolorings of a graph are adjacent if they differ on exactly one vertex. A graph is kmixing if any proper kcoloring can be transformed into any other through a sequence of adjacent proper kcolorings. Any graph is (tw + 2)mixing, where tw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two (tw + 2)colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is kmixing if k is at least the maximum degree plus two. We improve Jerrum’s bound using the grundy number, which is the worst number of colors in a greedy coloring.
On the Boolean connectivity problem for Horn relations
 IN PROC. 10 TH INTL. CONFERENCE ON THEORY AND APPLICATIONS OF SATISFIABILITY TESTING (SAT), 2007
, 2007
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Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs
, 2012
"... A kcolouring of a graph G = (V, E) is a mapping c: V {1, 2,..., k} such that c(u) = c(v) whenever uv is an edge. The reconfiguration graph of the kcolourings of G contains as its vertex set the kcolourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex ..."
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Cited by 7 (3 self)
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A kcolouring of a graph G = (V, E) is a mapping c: V {1, 2,..., k} such that c(u) = c(v) whenever uv is an edge. The reconfiguration graph of the kcolourings of G contains as its vertex set the kcolourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of kcolourable graphs, which we call kcolourdense graphs. We show that for each kcolourdense graph G, the reconfiguration graph of the ℓcolourings of G is connected and has diameter O(V  2), for all ℓ ≥ k + 1. We show that this graph class contains the kcolourable chordal graphs and that it contains all chordal bipartite graphs when k = 2. Moreover, we prove that for each k ≥ 2 there is a kcolourable chordal graph G whose reconfiguration graph of the (k + 1)colourings has diameter Θ(V  2).
Rerouting shortest paths in planar graphs
"... A rerouting sequence is a sequence of shortest stpaths such that consecutive paths differ in one vertex. We study the Shortest Path Rerouting Problem, which asks, given two shortest stpaths P and Q in a graph G, whether a rerouting sequence exists from P to Q. This problem is PSPACEhard in genera ..."
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Cited by 6 (2 self)
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A rerouting sequence is a sequence of shortest stpaths such that consecutive paths differ in one vertex. We study the Shortest Path Rerouting Problem, which asks, given two shortest stpaths P and Q in a graph G, whether a rerouting sequence exists from P to Q. This problem is PSPACEhard in general, but we show that it can be solved in polynomial time if G is planar. To this end, we introduce a dynamic programming method for reconfiguration problems.