arXiv subjects

Stefan Dantchev

Publications and source records attributed to Stefan Dantchev.

7 recordsLinked to original sources

arXiv2024

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

cs.CC↗

arXiv2022

Proof complexity and the binary encoding of combinatorial principles

We consider Proof Complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this Proof Complexity with the normal unary encoding in several refutation systems, based on Resolution and Integer Linear Programming. Please consult the article for the full abstract.

cs.LO↗

arXiv2019

Sherali-Adams and the binary encoding of combinatorial principles

We consider the Sherali-Adams (SA) refutation system together with the unusual binary encoding of certain combinatorial principles. For the unary encoding of the Pigeonhole Principle and the Least Number Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the Pigeonhole Principle requires exponentially-sized SA refutations, whereas the binary encoding of the Least Number Principle admits logarithmic rank, polynomially-sized SA refutations. We continue by considering a refutation system between SA and Lasserre (Sum-of-Squares). In this system, the Least Number Principle requires linear rank while the Pigeonhole Principle becomes constant rank.

cs.LO↗

arXiv2018

Resolution and the binary encoding of combinatorial principles

We investigate the size complexity of proofs in $Res(s)$ -- an extension of Resolution working on $s$-DNFs instead of clauses -- for families of contradictions given in the {\em unusual binary} encoding. A motivation of our work is size lower bounds of refutations in Resolution for families of contradictions in the usual unary encoding. Our main interest is the $k$-Clique Principle, whose Resolution complexity is still unknown. Our main result is a $n^{Ω(k)}$ lower bound for the size of refutations of the binary $k$-Clique Principle in $Res(\lfloor \frac{1}{2}\log \log n\rfloor)$. This improves the result of Lauria, Pudlák et al. [24] who proved the lower bound for Resolution, that is $Res(1)$. Our second lower bound proves that in $RES(s)$ for $s\leq \log^{\frac{1}{2-ε}}(n)$, the shortest proofs of the $BinPHP^m_n$, requires size $2^{n^{1-δ}}$, for any $δ>0$. Furthermore we prove that $BinPHP^m_n$ can be refuted in size $2^{Θ(n)}$ in treelike $Res(1)$, contrasting with the unary case, where $PHP^m_n$ requires treelike $RES(1)$ \ refutations of size $2^{Ω(n \log n)}$ [9,16]. Furthermore we study under what conditions the complexity of refutations in Resolution will not increase significantly (more than a polynomial factor) when shifting between the unary encoding and the binary encoding. We show that this is true, from unary to binary, for propositional encodings of principles expressible as a $Π_2$-formula and involving {\em total variable comparisons}. We then show that this is true, from binary to unary, when one considers the \emph{functional unary encoding}. Finally we prove that the binary encoding of the general Ordering principle $OP$ -- with no total ordering constraints -- is polynomially provable in Resolution.

cs.CC↗

arXiv2016

Simplicial Complex Entropy

We propose an entropy function for simplicial complices. Its value gives the expected cost of the optimal encoding of sequences of vertices of the complex, when any two vertices belonging to the same simplex are indistinguishable. We show that the proposed entropy function can be computed efficiently. By computing the entropy of several complices consisting of hundreds of simplices, we show that the proposed entropy function can be used in the analysis of the large sequences of simplicial complices that often appear in computational topology applications.

cs.IT↗

arXiv2014

Relativisation makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d), as well as their tree-like versions, Res*(d). The contradictions we use are natural combinatorial principles: the Least number principle, LNP_n and an ordered variant thereof, the Induction principle, IP_n. LNP_n is known to be easy for Resolution. We prove that its relativisation is hard for Resolution, and more generally, the relativisation of LNP_n iterated d times provides a separation between Res(d) and Res(d+1). We prove the same result for the iterated relativisation of IP_n, where the tree-like variant Res*(d) is considered instead of Res(d). We go on to provide separations between the parameterized versions of Res(1) and Res(2). Here we are able again to use the relativisation of the LNP_n, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res*(1) and Res*(2). Here, the relativisation of IP_n will not work as it is, and so we make a vectorising amendment to it in order to address this shortcoming

cs.LO↗

arXiv2012

Parameterized Resolution with bounded conjunction

We provide separations between the parameterized versions of Res(1) (Resolution) and Res(2). Using a different set of parameterized contradictions, we also separate the parameterized versions of Res*(1) (tree-Resolution) and Res*(2).

cs.LO↗