4 research outputs found
Distributed Exact Shortest Paths in Sublinear Time
The distributed single-source shortest paths problem is one of the most
fundamental and central problems in the message-passing distributed computing.
Classical Bellman-Ford algorithm solves it in time, where is the
number of vertices in the input graph . Peleg and Rubinovich (FOCS'99)
showed a lower bound of for this problem, where
is the hop-diameter of .
Whether or not this problem can be solved in time when is
relatively small is a major notorious open question. Despite intensive research
\cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the
approximate variant of this problem, no progress was reported for the original
problem.
In this paper we answer this question in the affirmative. We devise an
algorithm that requires time, for , and time, for larger . This
running time is sublinear in in almost the entire range of parameters,
specifically, for . For the all-pairs shortest paths
problem, our algorithm requires time, regardless of
the value of .
We also devise the first algorithm with non-trivial complexity guarantees for
computing exact shortest paths in the multipass semi-streaming model of
computation.
From the technical viewpoint, our algorithm computes a hopset of a
skeleton graph of without first computing itself. We then conduct
a Bellman-Ford exploration in , while computing the required edges
of on the fly. As a result, our algorithm computes exactly those edges of
that it really needs, rather than computing approximately the entire
New Tools and Connections for Exponential-Time Approximation
In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of
1.
r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r)))
time,
2.
r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r)))
time,
3.
(2−1/r)
for minimum vertex cover in O∗(exp(n/rΩ(r)))
time, and
4.
(k−1/r)
for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr)))
time.
(Throughout, O~
and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp(n1−o(1)/r1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O∗(2n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016)
New Tools and Connections for Exponential-Time Approximation
In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of
1.
r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r)))
time,
2.
r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r)))
time,
3.
(2−1/r)
for minimum vertex cover in O∗(exp(n/rΩ(r)))
time, and
4.
(k−1/r)
for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr)))
time.
(Throughout, O~
and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. i