Elimination graphs

TL;DR: It turns out that if the authors take P as a graph with maximum independent set size no greater than , then this definition gives a natural generalization of both chordal graphs and (

Abstract: In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = (V, E) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v1, …, vn such that the property P holds on the induced subgraph G[N(vi)∩ Vi], where N(vi) is the neighborhood of vi and Vi = {vi, …, vn}. It turns out that if we take P as a graph with maximum independent set size no greater than k, then this definition gives a natural generalization of both chordal graphs and (k + 1)-claw-free graphs. We refer to such graphs as inductive k-independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3-independent graph; furthermore, any planar graph is an inductive 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for inductive k-independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.

TL;DR: In this article, the authors considered the problem of scheduling arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests, and gave an algorithm that achieves an approximation ratio of O(log n c log log Δ), where n is the number of links and Δ is the ratio between the longest and the shortest link length.

Abstract: We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints.We give an algorithm that attains an approximation ratio of O(log n c log log Δ), where n is the number of links and Δ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on Δ is unavoidable, showing that any reasonably behaving oblivious power assignment results in a Ω(log log Δ)-approximation.These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore the utility of graph models in capturing wireless interference.

TL;DR: A novel LP formulation for combinatorial auctions with conflict graph using a non-standard graph parameter, the so-called inductive independence number, is suggested to bypass the well-known lower bound of Ω(n1-ε) on the approximability of independent set in general graphs with n nodes (bidders).

Abstract: We study combinatorial auctions for the secondary spectrum market. In this market, short-term licenses shall be given to wireless nodes for communication in their local neighborhood. In contrast to the primary market, channels can be assigned to multiple bidders, provided that the corresponding devices are well separated such that the interference is sufficiently low. Interference conflicts are described in terms of a conflict graph in which the nodes represent the bidders and the edges represent conflicts such that the feasible allocations for a channel correspond to the independent sets in the conflict graph.In this paper, we suggest a novel LP formulation for combinatorial auctions with conflict graph using a non-standard graph parameter, the so-called inductive independence number. Taking into account this parameter enables us to bypass the well-known lower bound of Ω(n1-e) on the approximability of independent set in general graphs with n nodes (bidders). We achieve significantly better approximation results by showing that interference constraints for wireless networks yield conflict graphs with bounded inductive independence number.Our framework covers various established models of wireless communication, e.g., the protocol or the physical model. For the protocol model, we achieve an O(√k)-approximation, where k is the number of available channels. For the more realistic physical model, we achieve an O(√k log2n) approximation based on edge-weighted conflict graphs. Combining our approach with the LP-based framework of Lavi and Swamy, we obtain incentive compatible mechanisms for general bidders with arbitrary valuations on bundles of channels specified in terms of demand oracles.

TL;DR: A polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs is presented and is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph.

Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + e)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.

TL;DR: This work investigates online algorithms for maximum (weight) independent set on graph classes with bounded inductive independence number ρ like interval and disk graphs with applications to, e.g., task scheduling, spectrum allocation and admission control.

Abstract: We investigate online algorithms for maximum (weight) independent set on graph classes with bounded inductive independence number ρ like interval and disk graphs with applications to, e.g., task scheduling, spectrum allocation and admission control. In the online setting, nodes of an unknown graph arrive one by one over time. An online algorithm has to decide whether an arriving node should be included into the independent set.

TL;DR: The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant.

Abstract: We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.

TL;DR: The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.

Abstract: L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

TL;DR: A graph G is a set of elements called vertices and a finite set of edges called edges such that each edge meets exactly two vertices, called the end-points of the edge as mentioned in this paper.

Abstract: A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.

TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.

Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

TL;DR: A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.

Abstract: This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decompos- ing a planar graph into subgraphs of a form we call k-outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least k/(k + 1)optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = (c log log nl or k = (c log nl, where n is the number of nodes and c is some constant, we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominat- ing set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time.

TL;DR: Linear-algebra rank is the solution to an especially tractable optimization problem which are linear programs relative to certain derived polyhedra.

Abstract: Linear-algebra rank is the solution to an especially tractable optimization problem This tractability is viewed abstractly, and extended to certain more general optimization problems which are linear programs relative to certain derived polyhedra