On the diameter of Kneser graphs

Valencia-Pabon, Mario; Vera, Juan C.

doi:10.1016/j.disc.2005.10.001

Cited by 46 publications

(28 citation statements)

References 3 publications

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“…This is done via a reduction from local search on a class of graphs known as the Odd graphs, for which we prove an exponential lower bound. In combination with results due to Dinh and Russell [13] and Valencia-Pabon and Vera [30], this yields an analogous exponential lower bound for randomized algorithms. Dobzinski et al [14] also use a local search reduction to prove a lower bound on the number of value queries required to find a certain type of equilibrium in a simultaneous second price auction, for bidders with XOS (i.e., fractionally subadditive) valuations.…”

Section: Exponential Query Complexity Lower Boundsupporting

confidence: 67%

“…To complete the lower bound, we proved an exponential lower bound on the number of queries required to find a local maximum on K(2k+1, k). We used results from Dinh and Russell [13] and Valencia-Pabon and Vera [30] to obtain an exponential lower bound for randomized algorithms as well. Our EFX lower bounds hold even for two players with identical submodular valuations.…”

Section: Discussionmentioning

confidence: 99%

“…Our reduction from Local Search to EFX Allocation also yields an exponential lower bound for randomized algorithms for free, thanks to results due to Dinh and Russell [13] and Valencia-Pabon and Vera [30]. Let R[LS(G)] be the minimum number of queries required to solve Local Search on G by a randomized algorithm: the algorithm should output a local maximum with probability at least 2/3 (say) over its internal coin flips.…”

Section: Randomized Query Complexitymentioning

confidence: 97%

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Almost Envy-Freeness with General Valuations

Plaut¹,

Roughgarden²

2020

SIAM J. Discrete Math.

157

250

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The goal of fair division is to distribute resources among competing players in a "fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.removed; this is a thought experiment used in the definition of envy-freeness up to one good. An EF1 allocation always exists, and can be computed in polynomial time [20]. 1 Caragiannis et al. [10] proposed another fairness criterion, one which is strictly stronger than EF1, but strictly weaker than full envy-freeness. An allocation is envy-free up to any good (EFX) if for any i, j where player i envies player j, removing any good from j's allocation would eliminate i's envy. Do EFX allocations always exist? This paper takes the first steps toward answering this question.

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Section: Exponential Query Complexity Lower Boundsupporting

confidence: 67%

Section: Discussionmentioning

confidence: 99%

Section: Randomized Query Complexitymentioning

confidence: 97%

See 1 more Smart Citation

Almost Envy-Freeness with General Valuations

Plaut¹,

Roughgarden²

2020

SIAM J. Discrete Math.

157

250

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“…Note that G is a triangle-free graph. From general results proved in Valencia-Pabon and Vera [6], we know that the diameter of G is k and the following holds.…”

Section: Kneser Graphmentioning

confidence: 82%

Coloring the square of the Kneser graph KG(2k+1,k) and the Schrijver graph
Chen¹,
Lih²,
Wu³
2009
Discrete Applied Mathematics

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a b s t r a c tThe Kneser graph KG(n, k) is the graph whose vertex set consists of all k-subsets of an n-set, and two vertices are adjacent if and only if they are disjoint. The Schrijver graph SG(n, k) is the subgraph of KG(n, k) induced by all vertices that are 2-stable subsets. The square G 2 of a graph G is defined on the vertex set of G such that distinct vertices within distance two in G are joined by an edge. The span λ(G) of G is the smallest integer m such that an L(2, 1)-labeling of G can be constructed using labels belonging to the set {0, 1, . . . , m}.The following results are established. (1) χ (KG 2 (2k + 1, k)) 3k + 2 for k 3 and χ (KG 2 (9, 4)) 12;(2) χ (SG 2 (2k + 2, k)) = λ(SG(2k + 2, k)) = 2k + 2 for k 4, χ (SG 2 (8, 3)) = 8, λ(SG(8, 3)) = 9, χ (SG 2 (6, 2)) = 9, and λ(SG(6, 2)) = 8.

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“…If n and k are positive integers with n ≥ k, then the Kneser graph K(n, k) has as vertices all the k-element subsets of the set [n], vertices being adjacent if the corresponding sets are disjoint. For more on Kneser graph see [2,3,15,19].…”

Section: Kneser Graphsmentioning

confidence: 99%

The general position problem on Kneser graphs and on some graph operations

Ghorbani¹,

Klavžar²,

Maimani³

et al. 2021

Discuss. Math. Graph Theory

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A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number (gp-number) gp(G) of G. The gp-number is determined for some families of Kneser graphs, in particular for K(n, 2) and K(n, 3). A sharp lower bound on the gp-number is proved for Cartesian products of graphs. The gp-number is also determined for joins of graphs, coronas over graphs, and line graphs of complete graphs.

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On the diameter of Kneser graphs

Cited by 46 publications

References 3 publications

Almost Envy-Freeness with General Valuations

Almost Envy-Freeness with General Valuations

Coloring the square of the Kneser graph KG(2k+1,k) and the Schrijver graph
Chen¹,
Lih²,
Wu³
2009
Discrete Applied Mathematics

The general position problem on Kneser graphs and on some graph operations

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On the diameter of Kneser graphs

Cited by 46 publications

References 3 publications

Almost Envy-Freeness with General Valuations

Almost Envy-Freeness with General Valuations

Coloring the square of the Kneser graph KG(2k+1,k) and the Schrijver graph Chen1, Lih2, Wu3 2009Discrete Applied Mathematics

The general position problem on Kneser graphs and on some graph operations

Contact Info

Product

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Coloring the square of the Kneser graph KG(2k+1,k) and the Schrijver graph
Chen¹,
Lih²,
Wu³
2009
Discrete Applied Mathematics