3,312 research outputs found
Computational Topology and the Unique Games Conjecture
Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial\u27s 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O\u27Donnell, FOCS \u2704; SICOMP \u2707) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade.
Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization.
Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future
Multicast Network Design Game on a Ring
In this paper we study quality measures of different solution concepts for
the multicast network design game on a ring topology. We recall from the
literature a lower bound of 4/3 and prove a matching upper bound for the price
of stability, which is the ratio of the social costs of a best Nash equilibrium
and of a general optimum. Therefore, we answer an open question posed by
Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs
of a potential optimizer and of an optimum, provide a construction of a lower
bound, and give a computer-assisted argument that it reaches for any
precision. We then turn our attention to players arriving one by one and
playing myopically their best response. We provide matching lower and upper
bounds of 2 for the myopic sequential price of anarchy (achieved for a
worst-case order of the arrival of the players). We then initiate the study of
myopic sequential price of stability and for the multicast game on the ring we
construct a lower bound of 4/3, and provide an upper bound of 26/19. To the
end, we conjecture and argue that the right answer is 4/3.Comment: 12 pages, 4 figure
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
- …