scholarly journals Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

2021 ◽  
pp. 23-38
Author(s):  
Václav Blažej ◽  
Pratibha Choudhary ◽  
Dušan Knop ◽  
Jan Matyáš Křišt’an ◽  
Ondřej Suchý ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Fault Tolerant ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Constant Factor ◽  
Weighted Graphs ◽  
Feedback Vertex Set ◽  
Fixed Integer ◽  
Vertex Set

AbstractConsider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

scholarly journals Convexity of graph-restricted games induced by minimum partitions

2019 ◽  
Vol 53 (3) ◽  
pp. 841-866 ◽  
Author(s):  
Alexandre Skoda
Keyword(s):  
Necessary Conditions ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Connected Components ◽  
Weighted Graphs ◽  
Classical Definition ◽  
Restricted Game ◽  
Definition Of ◽  
Adjacent Cycles ◽  
Restricted Games

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition 𝒫min is i nduced by the deletion of the minimum weight edges. We provide five necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with 𝒫min. Then, we establish that these conditions are also sufficient for a weaker condition, called ℱ-convexity, obtained by restriction of convexity to connected subsets. Moreover, we prove that inheritance of convexity for Myerson restricted game associated with a given graph G is equivalent to inheritance of ℱ-convexity for the 𝒫min-restricted game associated with a particular weighted graph G′ built from G by adding a dominating vertex, and with only two different edge-weights. Then, we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G′.

scholarly journals 2-Approximating Feedback Vertex Set in Tournaments

2021 ◽  
Vol 17 (2) ◽  
pp. 1-14
Author(s):  
Daniel Lokshtanov ◽  
Pranabendu Misra ◽  
Joydeep Mukherjee ◽  
Fahad Panolan ◽  
Geevarghese Philip ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Weight Function ◽  
Directed Graph ◽  
Polynomial Time ◽  
Optimal Solutions ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Unique Games Conjecture ◽  
Unique Games

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

Reduced space-time and time costs Ising dislocation codes and arbitrary ancillas

2015 ◽  
Vol 15 (11&12) ◽  
pp. 962-986
Author(s):  
Matthew B. Hastings ◽  
A. Geller
Keyword(s):  
Fault Tolerant ◽  
Minimum Weight ◽  
Finite Size ◽  
Constant Factor ◽  
Measurement Methods ◽  
Finite Size Scaling ◽  
Amortized Analysis ◽  
Parallel Performance ◽  
Logical Qubits ◽  
Reduced Space

We propose two distinct methods of improving quantum computing protocols based on surface codes. First, we analyze the use of dislocations instead of holes to produce logical qubits, potentially reducing spacetime volume required. Dislocations\cite{dis2,dis} induce defects which, in many respects, behave like Majorana quasi-particles. We construct circuits to implement these codes and present fault-tolerant measurement methods for these and other defects which may reduce spatial overhead. One advantage of these codes is that Hadamard gates take exactly $0$ time to implement. We numerically study the performance of these codes using a minimum weight and a greedy decoder using finite-size scaling. Second, we consider state injection of arbitrary ancillas to produce arbitrary rotations. This avoids the logarithmic (in precision) overhead in online cost required if $T$ gates are used to synthesize arbitrary rotations. While this has been considered before\cite{ancilla}, we consider also the parallel performance of this protocol. Arbitrary ancilla injection leads to a probabilistic protocol in which there is a constant chance of success on each round; we use an amortized analysis to show that even in a parallel setting this leads to only a constant factor slowdown as opposed to the logarithmic slowdown that might be expected naively.

scholarly journals A Constant Factor Approximation Algorithm for Fault-Tolerant k -Median

2016 ◽  
Vol 12 (3) ◽  
pp. 1-19 ◽  
Author(s):  
Mohammadtaghi Hajiaghayi ◽  
Wei Hu ◽  
Jian Li ◽  
Shi Li ◽  
Barna Saha
Keyword(s):  
Approximation Algorithm ◽  
Fault Tolerant ◽  
Constant Factor

scholarly journals Stabilizing Weighted Graphs

2020 ◽  
Vol 45 (4) ◽  
pp. 1318-1341
Author(s):  
Zhuan Khye Koh ◽  
Laura Sanità
Keyword(s):  
Approximation Algorithm ◽  
Weighted Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Constant Factor ◽  
Maximum Weight ◽  
Minimum Cardinality ◽  
Matching Games ◽  
Maximum Weight Matching ◽  
Network Bargaining

An edge-weighted graph [Formula: see text] is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in network bargaining games and cooperative matching games, because they characterize instances that admit stable outcomes. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P = NP. In this setting, we develop an O(Δ)-approximation algorithm for the problem, where Δ is the maximum degree of a node in G.

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