292 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Infinitary cut-elimination via finite approximations
We investigate non-wellfounded proof systems based on parsimonious logic, a
weaker variant of linear logic where the exponential modality ! is interpreted
as a constructor for streams over finite data. Logical consistency is
maintained at a global level by adapting a standard progressing criterion. We
present an infinitary version of cut-elimination based on finite
approximations, and we prove that, in presence of the progressing criterion, it
returns well-defined non-wellfounded proofs at its limit. Furthermore, we show
that cut-elimination preserves the progressive criterion and various regularity
conditions internalizing degrees of proof-theoretical uniformity. Finally, we
provide a denotational semantics for our systems based on the relational model
Independent Sets in Elimination Graphs with a Submodular Objective
Maximum weight independent set (MWIS) admits a -approximation in
inductively -independent graphs and a -approximation in
-perfectly orientable graphs. These are a a parameterized class of graphs
that generalize -degenerate graphs, chordal graphs, and intersection graphs
of various geometric shapes such as intervals, pseudo-disks, and several
others. We consider a generalization of MWIS to a submodular objective. Given a
graph and a non-negative submodular function , the goal is to approximately solve where is the set of independent sets of . We obtain an
-approximation for this problem in the two mentioned graph
classes. The first approach is via the multilinear relaxation framework and a
simple contention resolution scheme, and this results in a randomized algorithm
with approximation ratio at least . This approach also yields
parallel (or low-adaptivity) approximations. Motivated by the goal of designing
efficient and deterministic algorithms, we describe two other algorithms for
inductively -independent graphs that are inspired by work on streaming
algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In
addition to being simpler and faster, these algorithms, in the monotone
submodular case, yield the first deterministic constant factor approximations
for various special cases that have been previously considered such as
intersection graphs of intervals, disks and pseudo-disks.Comment: Extended abstract to appear in Proceedings of APPROX 2023. v2
corrects technical typos in few place
On the Succinctness of Good-for-MDPs Automata
Good-for-MDPs and good-for-games automata are two recent classes of
nondeterministic automata that reside between general nondeterministic and
deterministic automata. Deterministic automata are good-for-games, and
good-for-games automata are good-for-MDPs, but not vice versa. One of the
question this raises is how these classes relate in terms of succinctness.
Good-for-games automata are known to be exponentially more succinct than
deterministic automata, but the gap between good-for-MDPs and good-for-games
automata as well as the gap between ordinary nondeterministic automata and
those that are good-for-MDPs have been open. We establish that these gaps are
exponential, and sharpen this result by showing that the latter gap remains
exponential when restricting the nondeterministic automata to separating safety
or unambiguous reachability automata.Comment: 18 page
The Worst-Case Complexity of Symmetric Strategy Improvement
Symmetric strategy improvement is an algorithm introduced by Schewe et al.
(ICALP 2015) that can be used to solve two-player games on directed graphs such
as parity games and mean payoff games. In contrast to the usual well-known
strategy improvement algorithm, it iterates over strategies of both players
simultaneously. The symmetric version solves the known worst-case examples for
strategy improvement quickly, however its worst-case complexity remained open.
We present a class of worst-case examples for symmetric strategy improvement
on which this symmetric version also takes exponentially many steps.
Remarkably, our examples exhibit this behaviour for any choice of improvement
rule, which is in contrast to classical strategy improvement where hard
instances are usually hand-crafted for a specific improvement rule. We present
a generalized version of symmetric strategy iteration depending less rigidly on
the interplay of the strategies of both players. However, it turns out it has
the same shortcomings
Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics
The "short cycle removal" technique was recently introduced by Abboud,
Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of
approximation. Its main technical result is that listing all triangles in an
-regular graph is -hard under the 3-SUM conjecture even
when the number of short cycles is small; namely, when the number of -cycles
is for .
Abboud et al. achieve by applying structure vs. randomness
arguments on graphs. In this paper, we take a step back and apply conceptually
similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve
the best possible and the following lower bounds under the 3-SUM
conjecture:
* Approximate distance oracles: The seminal Thorup-Zwick distance oracles
achieve stretch after preprocessing a graph in
time. For the same stretch, and assuming the query time is Abboud et
al. proved an lower bound on the
preprocessing time; we improve it to which is only a
factor 2 away from the upper bound. We also obtain tight bounds for stretch
and and higher lower bounds for dynamic shortest paths.
* Listing 4-cycles: Abboud et al. proved the first super-linear lower bound
for listing all 4-cycles in a graph, ruling out time
algorithms where is the number of 4-cycles. We settle the complexity of
this basic problem by showing that the
upper bound is tight up to factors.
Our results exploit a rich tool set from additive combinatorics, most notably
the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key
ingredient that may be of independent interest is a subquadratic algorithm for
3-SUM if one of the sets has small doubling.Comment: Abstract shortened to fit arXiv requirement
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Asymptotically Tight Bounds on the Time Complexity of Broadcast and Its Variants in Dynamic Networks
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