175 research outputs found
Inapproximability of Rainbow Colouring
No full textA rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one path in which no two edges are coloured the same. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Chakraborty, Fischer, Matsliah and Yuster have shown that it is NP-hard to compute the rainbow connection number of graphs [J. Comb. Optim., 2011]. Basavaraju, Chandran, Rajendraprasad and Ramaswamy have reported an (r+3)-factor approximation algorithm to rainbow colour any graph of radius r [Graphs and Combinatorics, 2012]. In this article, we use a result of Guruswami, Håstad and Sudan on the NP-hardness of colouring a 2-colourable 4-uniform hypergraph using constantly many
colours [SIAM J. Comput., 2002] to show that for every positive integer k, it is NP-hard to distinguish between graphs with rainbow connection number 2k+2 and 4k+2. This, in turn, implies that there cannot exist a polynomial time algorithm to rainbow colour graphs with less than twice the optimum number of colours, unless P=NP.
The authors have earlier shown that the rainbow connection number problem remains NP-hard even when restricted to the class of chordal graphs, though in this case a 4-factor approximation algorithm is available [COCOON, 2012]. In this article, we improve upon the 4-factor approximation algorithm to design a linear-time algorithm that can rainbow colour a chordal graph G using at most 3/2 times the minimum number of colours if G is bridgeless and at most 5/2 times the minimum number of colours otherwise. Finally we show that the rainbow connection number of bridgeless chordal graphs cannot be polynomial-time approximated to a factor less than 5/4, unless P=NP
Cubicity, Degeneracy, and Crossing Number
No full textA k-box B=(R_1,R_2,...,R_k), where each R_i is a closed interval on the real line, is defined to be the Cartesian product R_1 X R_2 X ... X R_k. If each R_i is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes.
It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan. Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/0607092, 2006.] that, for a graph G with maximum degree \Delta, cub(G) <= \lceil 4(\Delta +1) ln n\rceil. In this paper we show that, for a k-degenerate graph G, cub(G) <= (k+2) \lceil 2e log n \rceil. Since k is at most \Delta and can be much lower, this clearly is a stronger result. We also give an efficient deterministic algorithm that runs in O(n^2k) time to output a 8k(\lceil 2.42 log n\rceil + 1) dimensional cube representation for G.
The crossing number of a graph G, denoted as CR(G), is the minimum number of crossing pairs of edges, over all drawings of G in the plane. An important consequence of the above result is that if the crossing number of a graph G is t, then box(G) is O(t^{1/4}{\lceil log t\rceil}^{3/4}) . This bound is tight upto a factor of O((log t)^{3/4}).
Let (P,\leq) be a partially ordered set and let G_{P} denote its underlying comparability graph. Let dim(P) denote the poset dimension of P. Another interesting consequence of our result is to show that dim(P) \leq 2(k+2) \lceil 2e \log n \rceil, where k denotes the degeneracy of G_{P}. Also, we get a deterministic algorithm that runs in O(n^2k) time to construct a 16k(\lceil 2.42 log n\rceil + 1) sized realizer for P. As far as we know, though very good upper bounds exist for poset dimension in terms of maximum degree of its underlying comparability graph, no upper bounds in terms of the degeneracy of the underlying comparability graph is seen in the literature
Combinatorial Lower Bounds for 3-Query LDCs
No full textA code is called a q-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index i and a received word w close to an encoding of a message x, outputs x_i by querying only at most q coordinates of w. Understanding the tradeoffs between the dimension, length and query complexity of LDCs is a fascinating and unresolved research challenge. In particular, for 3-query binary LDC’s of dimension k and length n, the best known bounds are: 2^{k^o(1)} ≥ n ≥ Ω ̃(k²).
In this work, we take a second look at binary 3-query LDCs. We investigate a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query LDCs. We prove an upper bound on the number of edges in these hypergraphs, reproducing the known lower bound of Ω ̃(k²) for the length of strong 3-query LDCs. In contrast to previous work, our techniques are purely combinatorial and do not rely on a direct reduction to 2-query LDCs, opening up a potentially different approach to analyzing 3-query LDCs
Spanning Tree Congestion and Computation of Generalized Györi-Lovász Partition
No full textWe study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges.
For any general connected graph with n vertices and m edges, we show that its STC is at most O(sqrt{mn}), which is asymptotically optimal since we also demonstrate graphs with STC at least Omega(sqrt{mn}). We present a polynomial-time algorithm which computes a spanning tree with congestion O(sqrt{mn}* log n). We also present another algorithm for computing a spanning tree with congestion O(sqrt{mn}); this algorithm runs in sub-exponential time when m = omega(n log^2 n).
For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time O^*(4^n). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest.
We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Theta(n) with high probability
Boxicity of line graphs
No full textThe boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R(k). In this paper we show that for a line graph G of a multigraph, box(G) <= 2 Delta (G)(inverted right perpendicularlog(2) log(2) Delta(G)inverted left perpendicular + 3) + 1, where Delta(G) denotes the maximum degree of G. Since G is a line graph, Delta(G) <= 2(chi (G) - 1), where chi (G) denotes the chromatic number of G, and therefore, box(G) = 0(chi (G) log(2) log(2) (chi (G))). For the d-dimensional hypercube Q(d), we prove that box(Q(d)) >= 1/2 (inverted right perpendicularlog(2) log(2) dinverted left perpendicular + 1). The question of finding a nontrivial lower bound for box(Q(d)) was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795-5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once). (C) 2011 Elsevier B.V. All rights reserved
Krenn-Gu Conjecture for Sparse Graphs
No full textGreenberger–Horne–Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the GHZ graphs. On such GHZ graphs, a graph parameter called dimension can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than 4 vertices is at most 2. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. Moreover, this would save huge computational resources used for finding experiments which lead to higher dimensional GHZ states. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results.
In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be 4-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them
Boxicity of Line Graphs
No full textThe boxicity of a graph H, denoted by View the MathML source, is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in View the MathML source. In this paper we show that for a line graph G of a multigraph, View the MathML source, where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, View the MathML source. For the d-dimensional hypercube Qd, we prove that View the MathML source. The question of finding a nontrivial lower bound for View the MathML source was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800].
The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once)
Boxicity of line graphs
No full textThe boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in Rk. In this paper we show that for a line graph G of a multigraph, box(G)≤2Δ(G)(log2log2Δ(G)⌉+3)+1, where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)-1), where χ(G) denotes the chromatic number of G, and therefore, box(G)=O(χ(G)log2log2(χ(G))). For the d-dimensional hypercube Qd, we prove that box( Qd)<12(log2log2d⌉+1). The question of finding a nontrivial lower bound for box(Qd) was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 57955800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once)
Two Results on LPT: A Near-Linear Time Algorithm and Parcel Delivery Using Drones
No full textThe focus of this paper is to increase our understanding of the Longest Processing Time First (LPT) heuristic. LPT is a classical heuristic for the fundamental problem of uniform machine scheduling. For different machine speeds, LPT was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). Since then, extensive work has been done to improve the approximation factor of the LPT heuristic. However, all known implementations of the LPT heuristic take O(mn) time, where m is the number of machines and n is the number of jobs. In this work, we come up with the first near-linear time implementation for LPT. Specifically, the running time is O((n+m)(log²m + log n)). Somewhat surprisingly, the result is obtained by mapping the problem to dynamic maintenance of lower envelope of lines, which has been well studied in the computational geometry community.
Our second contribution is to analyze the performance of LPT for the Drones Warehouse Problem (DWP), which is a natural generalization of the uniform machine scheduling problem motivated by drone-based parcel delivery from a warehouse. In this problem, a warehouse has multiple drones and wants to deliver parcels to several customers. Each drone picks a parcel from the warehouse, delivers it, and returns to the warehouse (where it can also get charged). The speeds and battery lives of the drones could be different, and due to the limited battery life, each drone has a bounded range in which it can deliver parcels. The goal is to assign parcels to the drones so that the time taken to deliver all the parcels is minimized. We prove that the natural approach of solving this problem via the LPT heuristic has an approximation factor of ϕ, where ϕ ≈ 1.62 is the golden ratio
On the Parameterized Complexity of Biclique Cover and Partition
No full textGiven a bipartite graph G, we consider the decision problem called BicliqueCover for a fixed positive integer parameter k where we are asked whether the edges of G can be covered with at most k complete bipartite subgraphs (a.k.a. bicliques). In the BicliquePartition problem, we have the additional constraint that each edge should appear in exactly one of the k bicliques. These problems are both known to be NP-complete but fixed parameter tractable. However, the known FPT algorithms have a running time that is doubly exponential in k, and the best known kernel for both problems is exponential in k. We build on this kernel and improve the running time for BicliquePartition to O*(2^{2k^2+k*log(k)+k}) by exploiting a linear algebraic view on this problem. On the other hand, we show that no such improvement is possible for BicliqueCover unless the Exponential Time Hypothesis (ETH) is false by proving a doubly exponential lower bound on the running time. We achieve this by giving a reduction from 3SAT on n variables to an instance of BicliqueCover with k=O(log(n)). As a further consequence of this reduction, we show that there is no subexponential kernel for BicliqueCover unless P=NP. Finally, we point out the significance of the exponential kernel mentioned above for the design of polynomial-time approximation algorithms for the optimization versions of both problems. That is, we show that it is possible to obtain approximation factors of n/log(n) for both problems, whereas the previous best approximation factor was n/sqrt(log(n))
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