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Reports tagged with width:
TR05-066 | 4th June 2005
Jakob Nordström

Narrow Proofs May Be Spacious: Separating Space and Width in Resolution

Revisions: 2 , Comments: 1

The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of memory cells used if the proof is only allowed to resolve on clauses kept in memory. Both of these measures have previously ... more >>>

TR07-114 | 28th September 2007
Jakob Nordström

A Simplified Way of Proving Trade-off Results for Resolution

We present a greatly simplified proof of the length-space
trade-off result for resolution in Hertel and Pitassi (2007), and
also prove a couple of other theorems in the same vein. We point
out two important ingredients needed for our proofs to work, and
discuss possible conclusions to be drawn regarding ... more >>>

TR08-048 | 8th April 2008
Meena Mahajan, B. V. Raghavendra Rao

Arithmetic circuits, syntactic multilinearity, and the limitations of skew formulae

Functions in arithmetic NC1 are known to have equivalent constant
width polynomial degree circuits, but the converse containment is
unknown. In a partial answer to this question, we show that syntactic
multilinear circuits of constant width and polynomial degree can be
depth-reduced, though the resulting circuits need not be ... more >>>

TR09-003 | 6th January 2009
Alex Hertel, Alasdair Urquhart

Comments on ECCC Report TR06-133: The Resolution Width Problem is EXPTIME-Complete

We discovered a serious error in one of our previous submissions to ECCC and wish to make sure that this mistake is publicly known.

The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (Exists,k)-pebble game, ... more >>>

TR10-085 | 20th May 2010
Eli Ben-Sasson, Jan Johannsen

Lower bounds for width-restricted clause learning on small width formulas

It has been observed empirically that clause learning does not significantly improve the performance of a SAT solver when restricted
to learning clauses of small width only. This experience is supported by lower bound theorems. It is shown that lower bounds on the runtime of width-restricted clause learning follow from ... more >>>

TR14-081 | 13th June 2014
Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools ... more >>>

TR14-093 | 22nd July 2014
Dmitry Itsykson, Mikhail Slabodkin, Dmitry Sokolov

Resolution complexity of perfect mathcing principles for sparse graphs

The resolution complexity of the perfect matching principle was studied by Razborov [Raz04], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph $G_n$ such that the resolution complexity of the perfect matching principle for $G_n$ is $2^{\Omega(n)}$, where $n$ is ... more >>>

TR14-118 | 9th September 2014
Albert Atserias, Massimo Lauria, Jakob Nordström

Narrow Proofs May Be Maximally Long

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. ... more >>>

TR16-010 | 28th January 2016
Alexander Razborov

On the Width of Semi-Algebraic Proofs and Algorithms

In this paper we initiate the study of width in semi-algebraic proof systems
and various cut-based procedures in integer programming. We focus on two
important systems: Gomory-Chv\'atal cutting planes and
Lov\'asz-Schrijver lift-and-project procedures. We develop general methods for
proving width lower bounds and apply them to random $k$-CNFs and several ... more >>>

TR16-057 | 11th April 2016
Ilario Bonacina

Total space in Resolution is at least width squared

Given an unsatisfiable $k$-CNF formula $\phi$ we consider two complexity measures in Resolution: width and total space. The width is the minimal $W$ such that there exists a Resolution refutation of $\phi$ with clauses of at most $W$ literals. The total space is the minimal size $T$ of a memory ... more >>>

TR16-203 | 21st December 2016
Christoph Berkholz, Jakob Nordström

Supercritical Space-Width Trade-offs for Resolution

We show that there are CNF formulas which can be refuted in resolution
in both small space and small width, but for which any small-width
proof must have space exceeding by far the linear worst-case upper
bound. This significantly strengthens the space-width trade-offs in
[Ben-Sasson '09]}, and provides one more ... more >>>

TR19-052 | 9th April 2019
Nicola Galesi, Leszek Kolodziejczyk, Neil Thapen

Polynomial calculus space and resolution width

We show that if a $k$-CNF requires width $w$ to refute in resolution, then it requires space $\sqrt w$ to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is ... more >>>

TR20-042 | 31st March 2020
Pranav Bisht, Nitin Saxena

Poly-time blackbox identity testing for sum of log-variate constant-width ROABPs

Blackbox polynomial identity testing (PIT) affords 'extreme variable-bootstrapping' (Agrawal et al, STOC'18; PNAS'19; Guo et al, FOCS'19). This motivates us to study log-variate read-once oblivious algebraic branching programs (ROABP). We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. We give the first poly($s$)-time blackbox ... more >>>

TR21-176 | 30th November 2021
Theodoros Papamakarios

A super-polynomial separation between resolution and cut-free sequent calculus

We show a quadratic separation between resolution and cut-free sequent calculus width. We use this gap to get, for the first time, first, a super-polynomial separation between resolution and cut-free sequent calculus for refuting CNF formulas, and secondly, a quadratic separation between resolution width and monomial space in polynomial calculus ... more >>>

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