Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More

Chalermsook, Parinya; Laekhanukit, Bundit; Nanongkai, Danupon

doi:10.1137/1.9781611973105.112

Cited by 38 publications

(51 citation statements)

References 40 publications

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“…This is indicated by the fact that we have no heuristics or approximation algorithms to produce realizers of partial orders that have reasonable size. A hardness result for approximations was first obtained by Hegde and Jain [12] and recently strengthened by Chalermsook et al [4]. They show that unless NP = ZPP there exists no polynomial algorithm to approximate the dimension of a partial order with a factor of O(n 1−ε ) for any ε > 0, where n is the number of elements of the input order.…”

Section: Introductionmentioning

confidence: 95%

The Complexity of the Partial Order Dimension Problem: Closing the Gap

Felsner¹,

Mustaţă²,

Pergel³

2017

SIAM J. Discrete Math.

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The dimension of a partial order P is the minimum number of linear orders whose intersection is P . There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [25] showed that for k ≥ 3 to test if a partial order has dimension ≤ k is NP-complete. The height of a partial order P is the maximum size of a chain in P . Yannakakis also showed that for k ≥ 4 to test if a partial order of height 2 has dimension ≤ k is NP-complete. The complexity of deciding whether an order of height 2 has dimension 3 was left open. This question became one of the best known open problems in dimension theory for partial orders. We show that the problem is NP-complete.Technically, we show that the decision problem (3DH2) for dimension is equivalent to deciding for the existence of bipartite triangle containment representations (BTCon). This problem then allows a reduction from a class of planar satisfiability problems (P-3-CON-3-SAT(4)) which is known to be NP-hard.Mathematics Subject Classifications (2010) 06A07, 68Q25, 05C62,

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Section: Introductionmentioning

confidence: 95%

The Complexity of the Partial Order Dimension Problem: Closing the Gap

Felsner¹,

Mustaţă²,

Pergel³

2017

SIAM J. Discrete Math.

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“…Since, a maximal induced matching in any bi partite graph with B nodes can be recovered upon optimally solving (6) for the corresponding instance (which in turn can be obtained in linear time for the given graph) , we have established a reduction to (6) from the problem of finding a maximal induced matching in a general bi-partite graph with B nodes. The claimed hardness result in the theorem follows upon invoking the recently discovered hardness result in [13] for the latter problem.…”

Section: Optimal Two Time-scale Policymentioning

confidence: 79%

“…Then, note that the user association problem in (9) can be re-formulated as max{f(9)}, f E ;?;. - (13) which is the problem of maximizing a non-negative non decreasing submodular set function subject to a matroid con straint. 0 As a consequence of Proposition 1 we can invoke the famous result in [15 ], which shows that a simple greedy algorithm yields a constant factor approximation for the prob lem of maximizing a non-negative non-decreasing submodular set function subject to one matroid constraint.…”

Section: Optimal Two Time-scale Policymentioning

confidence: 99%

A two time scale approach for coordinated multi-point transmission and reception over practical backhaul

Prasad

Arslan

Rangarajan

2014

2014 Sixth International Conference on Communication Systems and Networks (COMSNETS)

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We consider the problem of maximizing a given utility over heterogeneous wireless networks (HetNets) that have a practical (non ideal) backhaul. Such a back haul is characterized by high latency and possibly limited capacity, which necessitates the use of semi-static schemes for resource management. The management in these semi-static schemes is done at two time scale s over a coordination unit within the HetNet, that comprises of a set of transmission points (TPs) and a set of users. In this paper we consider one such promising semi-static resource management scheme in which the group of beam vectors that each TP in the coordination unit should employ and the set of users that it should serve is decided at a coarse time scale in a centralized manner, while the fine time scale per-slot scheduling is done in a distributed fashion with each TP using its assigned beam group (precoder) to serve its associated set of users. We propose an optimal two time scale policy for this scheme that guarantees a long-term optimality with respect to the given utility.However, this policy involves repeatedly solving a joint precoder set assignment and user association problem, which we prove is strongly NP-hard. Consequently, we develop an approximation algorithm for the latter optimization problem and also prove that the two time scale policy that employs this approximation algorithm also yields an approximate long-term optimality with respect to the given utility. Finally, we also incorporate another practically relevant constraint which demands that the per-slot scheduling policy of each TP remain unaltered.

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“…This is a result due to Chalermsook et al [9] using the hardness reduction of Adiga, Bhowmick and Chandran [2]. There is, however, an approximation algorithm with factor o(n) for general graphs; see Adiga et al [1].…”

Section: Theoremmentioning

confidence: 94%

Structural Parameterizations for Boxicity

et al. 2015

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The boxicity of a graph G is the least integer d such that G has an intersection model of axis-aligned d-dimensional boxes. Boxicity, the problem of deciding whether a given graph G has boxicity at most d, is NP-complete for every fixed d ≥ 2. We show that Boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al. (2010), that Boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that Boxicity admits an additive 1-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of that Boxicity remains NP-complete even on graphs of constant treewidth.

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Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More

Abstract: Graph product is a fundamental tool with rich applications in both graph theory and theoretical computer science.

Cited by 38 publications

References 40 publications

The Complexity of the Partial Order Dimension Problem: Closing the Gap

The Complexity of the Partial Order Dimension Problem: Closing the Gap

A two time scale approach for coordinated multi-point transmission and reception over practical backhaul

Structural Parameterizations for Boxicity

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