Strong Colorings of Hypergraphs

Agnarsson, Geir; Halldórsson, Magnús M.

doi:10.1007/978-3-540-31833-0_21

Cited by 26 publications

(25 citation statements)

References 36 publications

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“…Because hypergraph L-strong k-colorability is NP-complete [1], Thm. 6.4 shows that minimizing probing delay subject to a linkload restriction is NP-complete as well.…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

“…THEOREM 6.4. Polynomial-time reductions exist between the following two problems: (1) deciding whether a hypergraph is Lstrongly k-colorable; and (2) deciding, for a given probing-designproblem input, whether all the probing paths in P can be probed within k timesteps without imposing a load of more than L on any link in any timestep.…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

“…Reduction from (1) to (2): Conversely, consider a hypergraph H = (V H , E H ) and fixed values of k and L; we wish to decide whether H is L-strongly k colorable. Fix an orderingê 1 , .…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

“…Reduction from (2) to (1): Consider a probing problem consisting of a graph G = (V, E) and the set P of probing paths. Construct a hypergraph H as follows: The vertices of H correspond (bijectively) to the elements of P. For each edge e ∈ G, add as an edge in H the set of those vertices in H that correspond to the paths in P that include the edge e. Viewing the colors of vertices in H as assignments of elements of P to timesteps, the colorings of H with i colors in which j is the maximum number of vertices in any one edge with a single color bijectively correspond to the assignments of elements of P to i timesteps in which the maximum load on any edge in any timestep is j.…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

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The design space of probing algorithms for network-performance measurement

Jaggard

Kopparty

Ramachandran

et al. 2013

SIGMETRICS Perform. Eval. Rev.

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We present a framework for the design and analysis of probing methods to monitor network performance, an important technique for collecting measurements in tasks such as fault detection. We use this framework to study the interaction among numerous, possibly conflicting, optimization goals in the design of a probing algorithm. We present a rigorous definition of a probing-algorithm design problem that can apply broadly to network-measurement scenarios. We also present several metrics relevant to the analysis of probing algorithms, including probing frequency and network coverage, communication and computational overhead, and the amount of algorithm state required. We show inherent tradeoffs among optimization goals and give hardness results for achieving some combinations of optimization goals. We also consider the possibility of developing approximation algorithms for achieving some of the goals and describe a randomized approach as an alternative, evaluating it using our framework. Our work aids future development of low-overhead probing techniques and introduces principles from IP-based networking to theoretically grounded approaches for concurrent path-selection problems.

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“…Because hypergraph L-strong k-colorability is NP-complete [1], Thm. 6.4 shows that minimizing probing delay subject to a linkload restriction is NP-complete as well.…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

“…Reduction from (1) to (2): Conversely, consider a hypergraph H = (V H , E H ) and fixed values of k and L; we wish to decide whether H is L-strongly k colorable. Fix an orderingê 1 , .…”

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

Section: Definition 63 (L-strong K-colorable Hypergraph) a Hypergramentioning

confidence: 99%

See 2 more Smart Citations

The design space of probing algorithms for network-performance measurement

Jaggard

Kopparty

Ramachandran

et al. 2013

SIGMETRICS Perform. Eval. Rev.

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“…H is called γ -uniform if every hyperedge is incident to γ vertices, i.e., |e n | = γ for all e n ∈ E. H is called strongly γ -colorable [43] if there exists a partition of V into γ sets, denoted by…”

Section: Appendix 1: Proof Of Theoremmentioning

confidence: 99%

Performance characterization and transmission schemes for instantly decodable network coding in wireless broadcast

Sadeghi

Aboutorab

2015

EURASIP J. Adv. Signal Process.

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We consider broadcasting a block of packets to multiple wireless receivers under random packet erasures using instantly decodable network coding (IDNC). The sender first broadcasts each packet uncoded once, then generates coded packets according to receivers' feedback about their missing packets. We focus on strict IDNC (S-IDNC), where each coded packet includes at most one missing packet of every receiver. But, we will also study its relation with generalized IDNC (G-IDNC), where this condition is relaxed. We characterize two fundamental performance limits of S-IDNC: (1) the number of transmissions to complete the broadcast, which measures throughput and (2) average packet decoding delay, which measures how fast each packet is decoded at each receiver on average. We derive a closed-form expression for the expected minimum number of transmissions in terms of the number of packets and receivers and the erasure probability. We prove that it is NP-hard to minimize the average packet decoding delay of S-IDNC. We also prove that the graph models of S-and G-IDNC share the same chromatic number. Next, we design efficient S-IDNC transmission schemes and coding algorithms with full/intermittent receiver feedback. We present simulation results to corroborate the developed theory and compare our schemes with existing ones.

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