# NYCAC 4 - Abstracts

Friday, November 18, 2011,   New York City
Room 4102

## The New York Colloquium on Algorithms and Complexity

#### Eric Allender

Kolmogorov Complexity, Circuits, and the Strength of Logical Theories of Arithmetic
Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to R_K (the set of Kolmogorov-random strings) that lies between BPP and PSPACE. In this talk, we investigate improving this upper bound from PSPACE to PSPACE intersect P/poly. (P/poly is the class of problems with polynomial-size circuits.)
More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP \subseteq C \subseteq PSPACE intersect P/poly.
We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.
Joint work with George Davie, Luke Friedman, Sam Hopkins, and Iddo Tzameret.
slides

#### Martin Fürer

Approximating k-Set Covers
(joint work with Huiwen Yu)
The k-set cover problem asks to cover a set U with a minimal number of subsets of U from a given collection C of such subsets, all of which have size at most k. The Greedy algorithm obtains an approximation ratio of H(k) ≥ ln k + 1, where H(k) is the k-th harmonic number. Trevisan 2001 has shown a hardness bound of ln k - O(log log k) based on Feige's 1998 hardness result for the general set cover problem.
We conjecture that the best ratio achievable in polynomial time is H(k) - O(1). Indeed, there have been many approximation algorithms improving the O(1) term. Much motivation comes from the trivial observation that for k=2, the problem is in P, a gain of 1/2 in the approximation ratio from H(2). To retain this benefit of 1/2 for larger k, one has to avoid the occurrence of unnecessary singleton sets. Here it is assumed that the collection C is closed under subsets. Thus it is sufficient to consider disjoint covers.
For k ≥ 4, packing based algorithms are better than Greedy. It turns out to be quite challenging to determine the performance of sophisticated packing algorithms that avoid the occurrence of singleton sets.

#### Bill Gasarch

Using Ramsey's Theorem to Prove Programs Terminate
Given a program we want to know that no matter what crazy thing the user does, it will terminate. And we want to prove this. This looks like HALT but actually it is much harder.
We will look at some programs where you can actually PROVE that no matter what the user does they will terminate. We will do this in two ways:
a) Traditional Method due to Floyd: Find some parameter that is decreasing and such that when it hits bottom the program must have terminated. This usually involves reasoning about single steps (which is easy) but may involve a complicated ordering (Note that I said `decrease' but it may be in some complicated ordering)
b) Ramsey Method due to Lee, Jones, Ben-Amram, Cook, Podelski, Rybalchenko (and probably others). This involved reasoning about several steps (which is harder), applyig Ramsey's Theorem (which I think is COOL!), and using rather simple orderings.
We will discuss these two approaches and some open problems that arise.
slides

#### George Karakostas

Using reputation and misinformation to influence users in selfish routing
Can we use deception for the manipulation of the selfish users in a network towards a desireable equilibrium. In this talk, we will examine reputation as an instigator of beneficial user behavior in selfish routing and when the network users rely on the network operator for information on the network parameters. Instead of the use of tolls or artificial delays, the network operator takes advantage of the users' insufficient data, in order to manipulate them through the information he provides. The issue that arises then is what can be the operator's gain without compromising (by too much) the trust users put on the information provided, i.e., by maintaining a reputation for (at least some) trustworthiness. We will see an example of modeling such a system as a repeated game of incomplete information in the case of single-commodity general networks with affine latency functions. This allows us to apply known folk-like theorems to get bounds on the price of anarchy that are better (if that is possible at all) than the well-known $4/3$ bound without any data fudging.
slides

#### Michael Lampis

The Landscape of Structural Graph Parameters
In traditional computational complexity we measure algorithm running times as functions of one variable, the size of the input. Though in this setting our goal is usually to design polynomial-time algorithms, most interesting graph problems are unfortunately believed to require exponential time to solve exactly. Parameterized complexity theory refines this by introducing a second variable, called the parameter, which is supposed to quantify the “hardness” of each specific instance. The goal now becomes to confine the combinatorial explosion to the parameter, by designing an algorithm that runs in time polynomial in the size of the input, though inevitably exponential in the parameter. This will allow us to tackle instances where the parameter value is much more modest than the input size, which will happen often if the parameter is chosen well.
Of course, this setting is rather general and there are countless ways in which one may attempt to measure the hardness of specific instances, depending on the problem. The parameterized graph algorithms literature has been dominated for a long time by a very successful notion called treewidth, which measures graph complexity by quantifying a graph's similarity to a tree. However, more recently the study of alternative graph parameters and widths has become more popular. In this (self-contained) talk we will attempt to explore the algorithmic properties of treewidth, its related graph width parameters, as well as other graph invariants which may serve as measurements of a graph's complexity such as Vertex Cover, or Max-Leaf number. We will focus especially on general results which prove tractability for whole classes of problems characterized by expressibility in certain logics, results often referred to as "algorithmic meta-theorems".
slides

#### Aris Tentes

Hardness Preserving Constructions of Pseudorandom Functions
We show a hardness-preserving construction of a PRF from any length doubling PRG which improves upon known constructions whenever we can put a non-trivial upper bound $q$ on the number of queries to the PRF. Our construction requires only $O(\log q)$ invocations to the underlying PRG with each query. In comparision, the number of invocations by the best previous hardness-preserving construction (GGM using Levin's trick) is logarithmic in the \emph{hardness} of the PRG.
For example, starting from an exponentially secure PRG $\set{0,1}^n\mapsto \set{0,1}^{2n}$, we get a PRF which is exponentially secure if queried at most $q=\exp(\sqrt n)$ times and where each invocation of the PRF requires $\Theta(\sqrt n)$ queries to the underlying PRG. This is much less than the $\Theta(n)$ required by known constructions.

#### Christos Tzamos

Winner-Imposing Strategyproof Mechanisms for Multiple Facility Location Games
We study Facility Location games, where a number of facilities are placed in a metric space based on locations reported by strategic agents. A mechanism maps the agents’ locations to a set of facilities. The agents seek to minimize their connection cost, namely the distance of their true location to the nearest facility, and may misreport their location. We are interested in mechanisms that are strategyproof, i.e., ensure that no agent can bene?t from misreporting her location, do not resort to monetary transfers, and approximate the optimal social cost. We focus on the closely related problems of k-Facility Location and Facility Location with a uniform facility opening cost, and mostly study winner-imposing mechanisms, which allocate facilities to the agents and require that each agent allocated a facility should connect to it. We show that the winner-imposing version of the Proportional Mechanism (Lu et al., EC ’10) is stategyproof and 4kapproximate for the k-Facility Location game. For the Facility Location game, we show that the winner-imposing version of the randomized algorithm of (Meyerson, FOCS ’01), which has an approximation ratio of 8, is strategyproof. Furthermore, we present a deterministic non-imposing group strategyproof O(log n)- approximate mechanism for the Facility Location game on the line

#### Vassilis Zikas

Player-Centric Byzantine Agreement
Most of the existing feasibility results for Byzantine Agreement (BA) are of an all-or-nothing fashion: in Broadcast they address the question whether or not there exists a protocol which allows any player to broadcast his input. Similarly, in Consensus the question is whether or not consensus can be reached which respects pre-agreement on the inputs of all correct players. In this work, we introduce the natural notion of player-centric BA which is a class of BA primitives, denoted as PCBA={PCBA_C}_C, parametrized by subsets C of the player set P. For each primitive PCBA_C the validity is defined on the input(s) of the players in C. Broadcast (with sender p) and Consensus are special (extreme) cases of PCB primitives for C={p} and C=P, respectively.
We study feasibility of PCBA in the presence of a general (aka non-threshold) mixed (active/passive) adversary, and give a complete characterization for perfect, statistical, and computational security. Our results expose an asymmetry of Broadcast which has, so far, been neglected in the literature: there exist non-trivial adversaries which can be tolerated for Broadcast with sender some player p_i but not for every player p_j being the sender. Finally, we extend the definition of PCBA by adding fail corruption to the adversary's capabilities, and give exact feasibility bounds for computationally secure PCBA_P (aka Consensus) in this setting. This answers an open problem from ASIACRYPT~2008 concerning feasibility of computationally secure multi-party computation in this model.