23 research outputs found
The Complexity of Finding Dense Subgraphs in Graphs with Large Cliques
The GapDensest-k-Subgraph(d) problem (GapDkS(d)) is defined as follows: given a graph G and parameters k,d, distinguish between the case that G contains a k-clique, and the case that every k-subgraph of G has density at most d.
GapDkS(d) is a natural relaxation of the standard Clique problem, which is known to be NP-complete. For d very close to 1, the GapDkS(d) problem is equivalent to the Clique problem, and when d is very close to 0 the GapDkS(d) problem can easily be solved in polynomial time. However, despite much work on both the algorithmic and hardness front, the exact k and d parameter values for which GapDkS(d) can be solved in polynomial time are still unknown. In particular, the best polynomial-time algorithms can solve GapDkS(d) when d is an inverse polynomial in the number of vertices n, but there have been no NP-hardness results beyond the trivial result.
This thesis attempts to understand the GapDkS(d) problem better by studying the case when k is restricted to be linear in n (where n is the number of vertices in G). In particular, we survey the GapDkS(d) algorithms and hardness results that can be best applied to this restriction in an attempt to determine the threshold for when the problem becomes NP-hard. With some modifications to the algorithms and proofs, we produce algorithms and hardness results for the GapDkS(d) problem with k linear in n.
In addition, we study the connection between GapDkS(d) and MaxClique, and show that despite having strong hardness results for MaxClique, reductions from MaxClique do not give strong hardness bounds for GapDkS(d)
Graphs without a partition into two proportionally dense subgraphs
A proportionally dense subgraph (PDS) is an induced subgraph of a graph such that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a 2-PDS partition, with and without additional constraint of connectivity of the subgraphs. We present two infinite classes of graphs: one with graphs without a 2-PDS partition, and another with graphs that only admit a disconnected 2-PDS partition. These results answer some questions proposed by Bazgan et al. (2018)
Where Graph Topology Matters: The Robust Subgraph Problem
Robustness is a critical measure of the resilience of large networked
systems, such as transportation and communication networks. Most prior works
focus on the global robustness of a given graph at large, e.g., by measuring
its overall vulnerability to external attacks or random failures. In this
paper, we turn attention to local robustness and pose a novel problem in the
lines of subgraph mining: given a large graph, how can we find its most robust
local subgraph (RLS)?
We define a robust subgraph as a subset of nodes with high communicability
among them, and formulate the RLS-PROBLEM of finding a subgraph of given size
with maximum robustness in the host graph. Our formulation is related to the
recently proposed general framework for the densest subgraph problem, however
differs from it substantially in that besides the number of edges in the
subgraph, robustness also concerns with the placement of edges, i.e., the
subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two
heuristic algorithms based on top-down and bottom-up search strategies.
Further, we present modifications of our algorithms to handle three practical
variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs
demonstrate that we find subgraphs with larger robustness than the densest
subgraphs even at lower densities, suggesting that the existing approaches are
not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only
Inapproximability of Maximum Biclique Problems, Minimum -Cut and Densest At-Least--Subgraph from the Small Set Expansion Hypothesis
The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly
states that it is NP-hard to distinguish between a graph with a small subset of
vertices whose edge expansion is almost zero and one in which all small subsets
of vertices have expansion almost one. In this work, we prove inapproximability
results for the following graph problems based on this hypothesis:
- Maximum Edge Biclique (MEB): given a bipartite graph , find a complete
bipartite subgraph of with maximum number of edges.
- Maximum Balanced Biclique (MBB): given a bipartite graph , find a
balanced complete bipartite subgraph of with maximum number of vertices.
- Minimum -Cut: given a weighted graph , find a set of edges with
minimum total weight whose removal partitions into connected
components.
- Densest At-Least--Subgraph (DALS): given a weighted graph , find a
set of at least vertices such that the induced subgraph on has
maximum density (the ratio between the total weight of edges and the number of
vertices).
We show that, assuming SSEH and NP BPP, no polynomial time
algorithm gives -approximation for MEB or MBB for every
constant . Moreover, assuming SSEH, we show that it is NP-hard
to approximate Minimum -Cut and DALS to within factor
of the optimum for every constant .
The ratios in our results are essentially tight since trivial algorithms give
-approximation to both MEB and MBB and efficient -approximation
algorithms are known for Minimum -Cut [SV95] and DALS [And07, KS09].
Our first result is proved by combining a technique developed by Raghavendra
et al. [RST12] to avoid locality of gadget reductions with a generalization of
Bansal and Khot's long code test [BK09] whereas our second result is shown via
elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a
different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced
Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page