Exact Weight Subgraphs and the k-Sum Conjecture

@inproceedings{Abboud2013ExactWS,
  title={Exact Weight Subgraphs and the k-Sum Conjecture},
  author={Amir Abboud and Kevin Lewi},
  booktitle={ICALP},
  year={2013}
}
We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-sum problem. In particular, we show that under the k-sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O… 
Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model
TLDR
It is formally proved that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.
Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal
TLDR
This paper demonstrates the existence of maximally hard pattern graphs H that require time n tw( H )+1 − o (1) and proves the following asymptotic statement for large treewidth t: under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory.
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
TLDR
This paper considers partially dynamic versions of three classic problems in graph theory, and shows that no algorithm with amortized update time for incremental or decremental maximum flow in directed and weighted sparse graphs or undirected and weighted graphs.
Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems
TLDR
It is proved that sufficient progress would imply a breakthrough on one of five major open problems in the theory of algorithms, including dynamic versions of bipartite perfect matching, bipartites maximum weight matching, single source reachability, single sources shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems.
Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
TLDR
Novel reductions from 3-SUM, APSP, and CNF-SAT are designed, and interesting consequences of this very plausible conjecture are derived, including tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs and new conditional lower bound for the Single-Source-Max-Flow problem.
On the parameterized complexity of k-SUM
TLDR
These results effectively resolve the parameterized complexity of k-SUM, initially posed in 1992 by Downey and Fellows in their seminal paper on parameterized intractability.
On the parameterized complexity of approximating dominating set
TLDR
To prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis the authors rely on, generalizing the ideas from a recent breakthrough work of Abboud et al.
On the Parameterized Complexity of Approximating Dominating Set
TLDR
The non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under Gap-ETH and to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis the authors rely on.
Budgeted Dominating Sets in Uncertain Graphs
TLDR
It is shown that the PBDS problem is NP-complete even when restricted to uncertain trees of diameter at most four, and its NP-hardness is proved by a reduction from the well-known k-SUM problem, presenting a close relationship between the two problems.
Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms
  • Amir Abboud, Søren Dahlgaard
  • Computer Science, Mathematics
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
TLDR
A new framework which is inspired by techniques from the literatures on distance labelling schemes and on parameterized complexity is introduced, showing that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized O(n1/2-ε) time, unless the classical all-pairs-shortest-paths problem can be solved in truly subcubic time.
...
...

References

SHOWING 1-10 OF 25 REFERENCES
Finding, minimizing, and counting weighted subgraphs
TLDR
These algorithms rely on fast algorithms for computing the permanent of a k x n matrix, over rings and semirings and give a new (algorithmic) application of multiparty communication complexity.
Faster algorithms for finding and counting subgraphs
Towards polynomial lower bounds for dynamic problems
TLDR
This work describes a carefully-chosen dynamic version of set disjointness (the "multiphase problem"), and conjecture that it requires n^Omega(1) time per operation, and forms the first nonalgebraic reduction from 3SUM, which allows3SUM-hardness results for combinatorial problems.
Lower bounds for linear satisfiability problems
We prove an Ω(ndr/2e) lower bound for the following problem: For some fixed linear equation in r variables, given n real numbers, do any r of them satisfy the equation? Our lower bound holds in a
Finding and counting small induced subgraphs efficiently
TLDR
Two algorithms for listing all simplicial vertices of a graph running in time O (n α ) and O (e 2α/(α+1) )= O ( e 1.41) are given, and it is shown that counting the number of K 4 's in a graph can be done in timeO (e ( α+1)/2 ) .
Counting and detecting small subgraphs via equations and matrix multiplication
We present a general technique for detecting and counting small subgraphs. It consists in forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed
Fixed-Parameter Tractability and Completeness I: Basic Results
TLDR
This paper establishes the main results of a completeness program which addresses the apparent fixed-parameter intractability of many parameterized problems and gives a compendium of currently known hardness results.
Fixed-Parameter Tractability and Completeness II: On Completeness for W[1]
On the possibility of faster SAT algorithms
We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNF-SAT) to several natural algorithmic problems. We show that attaining any of the following bounds
Finding a minimum circuit in a graph
TLDR
Finding minimum circuits in graphs and digraphs is discussed and an algorithm to find an almost minimum circuit is presented and an alternative method is to reduce the problem of finding a minimum circuit to that of finding an auxiliary graph.
...
...