18 research outputs found
A linear optimization technique for graph pebbling
Graph pebbling is a network model for studying whether or not a given supply
of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling
move across an edge of a graph takes two pebbles from one endpoint and places
one pebble at the other endpoint; the other pebble is lost in transit as a
toll. It has been shown that deciding whether a supply can meet a demand on a
graph is NP-complete. The pebbling number of a graph is the smallest t such
that every supply of t pebbles can satisfy every demand of one pebble. Deciding
if the pebbling number is at most k is \Pi_2^P-complete. In this paper we
develop a tool, called the Weight Function Lemma, for computing upper bounds
and sometimes exact values for pebbling numbers with the assistance of linear
optimization. With this tool we are able to calculate the pebbling numbers of
much larger graphs than in previous algorithms, and much more quickly as well.
We also obtain results for many families of graphs, in many cases by hand, with
much simpler and remarkably shorter proofs than given in previously existing
arguments (certificates typically of size at most the number of vertices times
the maximum degree), especially for highly symmetric graphs. Here we apply the
Weight Function Lemma to several specific graphs, including the Petersen,
Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a
number of infinite families of graphs, such as trees, cycles, graph powers of
cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly
answers a question of Pachter, et al., by computing the pebbling exponent of
cycles to within an asymptotically small range. It is conceivable that this
method yields an approximation algorithm for graph pebbling
A linear optimization technique for graph pebbling
Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is NP 2 -complete. In this paper we develop a tool, called theWeight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply theWeight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling
Zero-sum problems for abelian p-groups and covers of the integers by residue classes
Zero-sum problems for abelian groups and covers of the integers by residue
classes, are two different active topics initiated by P. Erdos more than 40
years ago and investigated by many researchers separately since then. In an
earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60],
the author claimed some surprising connections among these seemingly unrelated
fascinating areas. In this paper we establish further connections between
zero-sum problems for abelian p-groups and covers of the integers. For example,
we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If
{a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly
2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there
exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in
I}c_s=0. Our main theorem in this paper unifies many results in the two realms
and also implies an extension of the Alon-Friedland-Kalai result on regular
subgraphs
IST Austria Thesis
A search problem lies in the complexity class FNP if a solution to the given instance of the problem can be verified efficiently. The complexity class TFNP consists of all search problems in FNP that are total in the sense that a solution is guaranteed to exist. TFNP contains a host of interesting problems from fields such as algorithmic game theory, computational topology, number theory and combinatorics. Since TFNP is a semantic class, it is unlikely to have a complete problem. Instead, one studies its syntactic subclasses which are defined based on the combinatorial principle used to argue totality. Of particular interest is the subclass PPAD, which contains important problems
like computing Nash equilibrium for bimatrix games and computational counterparts of several fixed-point theorems as complete. In the thesis, we undertake the study of averagecase hardness of TFNP, and in particular its subclass PPAD.
Almost nothing was known about average-case hardness of PPAD before a series of recent results showed how to achieve it using a cryptographic primitive called program obfuscation.
However, it is currently not known how to construct program obfuscation from standard cryptographic assumptions. Therefore, it is desirable to relax the assumption under which average-case hardness of PPAD can be shown. In the thesis we take a step in this direction. First, we show that assuming the (average-case) hardness of a numbertheoretic
problem related to factoring of integers, which we call Iterated-Squaring, PPAD is hard-on-average in the random-oracle model. Then we strengthen this result to show that the average-case hardness of PPAD reduces to the (adaptive) soundness of the Fiat-Shamir Transform, a well-known technique used to compile a public-coin interactive protocol into a non-interactive one. As a corollary, we obtain average-case hardness for PPAD in the random-oracle model assuming the worst-case hardness of #SAT. Moreover, the above results can all be strengthened to obtain average-case hardness for the class CLS ⊆ PPAD.
Our main technical contribution is constructing incrementally-verifiable procedures for computing Iterated-Squaring and #SAT. By incrementally-verifiable, we mean that every intermediate state of the computation includes a proof of its correctness, and the proof can be updated and verified in polynomial time. Previous constructions of such procedures relied on strong, non-standard assumptions. Instead, we introduce a technique called recursive proof-merging to obtain the same from weaker assumptions
Unique End of Potential Line
This paper studies the complexity of problems in PPAD PLS that have
unique solutions. Three well-known examples of such problems are the problem of
finding a fixpoint of a contraction map, finding the unique sink of a Unique
Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem
(P-LCP). Each of these are promise-problems, and when the promise holds, they
always possess unique solutions.
We define the complexity class UEOPL to capture problems of this type. We
first define a class that we call EOPL, which consists of all problems that can
be reduced to End-of-Potential-Line. This problem merges the canonical
PPAD-complete problem End-of-Line, with the canonical PLS-complete problem
Sink-of-Dag, and so EOPL captures problems that can be solved by a
line-following algorithm that also simultaneously decreases a potential
function.
Promise-UEOPL is a promise-subclass of EOPL in which the line in the
End-of-Potential-Line instance is guaranteed to be unique via a promise. We
turn this into a non-promise class UEOPL, by adding an extra solution type to
EOPL that captures any pair of points that are provably on two different lines.
We show that UEOPL EOPL CLS, and that all of our
motivating problems are contained in UEOPL: specifically USO, P-LCP, and
finding a fixpoint of a Piecewise-Linear Contraction under an -norm all
lie in UEOPL. Our results also imply that parity games, mean-payoff games,
discounted games, and simple-stochastic games lie in UEOPL.
All of our containment results are proved via a reduction to a problem that
we call One-Permutation Discrete Contraction (OPDC). This problem is motivated
by a discretized version of contraction, but it is also closely related to the
USO problem. We show that OPDC lies in UEOPL, and we are also able to show that
OPDC is UEOPL-complete.Comment: This paper substantially revises and extends the work described in
our previous preprint "End of Potential Line'' (arXiv:1804.03450). The
abstract has been shortened to meet the arXiv character limi
Unique End of Potential Line
This paper studies the complexity of problems in PPAD PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions. We define the complexity class UEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to End-of-Potential-Line. This problem merges the canonical PPAD-complete problem End-of-Line, with the canonical PLS-complete problem Sink-of-Dag, and so EOPL captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function. Promise-UEOPL is a promise-subclass of EOPL in which the line in the End-of-Potential-Line instance is guaranteed to be unique via a promise. We turn this into a non-promise class UEOPL, by adding an extra solution type to EOPL that captures any pair of points that are provably on two different lines. We show that UEOPL EOPL CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an -norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL. All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete