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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 21 (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).
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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Cited by 3 (1 self)
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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
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 of list edgecolorings in a graph, in
 Proc. of WADS 2009, in: LNCS
"... Abstract. 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 PSPACEcomplet ..."
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Cited by 2 (1 self)
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Abstract. 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. 1
Approximability of the Subset Sum Reconfiguration Problem
"... Abstract. The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this pa ..."
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Cited by 2 (2 self)
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Abstract. The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NPhard, and is PSPACEcomplete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in the reconfiguration. We show that this maximization problem admits a polynomialtime approximation scheme (PTAS), while the problem is APXhard if we are given a conflict graph. 1
Finding paths between . . .
, 2007
"... Suppose we are given a graph G together with two proper vertex kcolourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper kcolouring of G? This decision problem is trivial for k = ..."
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Suppose we are given a graph G together with two proper vertex kcolourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper kcolouring of G? This decision problem is trivial for k = 2, and decidable in polynomial time for k = 3. Here we prove it is PSPACEcomplete for all k ≥ 4. In particular, we prove that the problem remains PSPACEcomplete for bipartite graphs, as well as for: (i) planar graphs and 4 ≤ k ≤ 6, and (ii) bipartite planar graphs and k = 4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour α to β. We also exhibit, for every k ≥ 4, a class of graphs {GN,k: N ∈ N ∗}, together with two kcolourings for each GN,k, such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2 N), whereas the size of GN is O(N 2). This is in stark contrast to the k = 3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4 ≤ k ≤ 6 planar graphs and (ii) for k = 4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.
DOI: 10.1007/s108780129490y Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs ∗†
, 2013
"... 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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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).
Journal of Combinatorial Optimization manuscript No. (will be inserted by the editor) Approximability of the Subset Sum Reconfiguration Problem
"... Abstract The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this pap ..."
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Abstract The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NPhard, and is PSPACEcomplete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomialtime approximation scheme (PTAS), while the problem is APXhard if we are given a conflict graph. Keywords approximation algorithm · PTAS · reachability on solution space · subset sum 1
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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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.