2 Search Results for "Boyadzhiyska, Simona"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.11
A Near-Optimal Polynomial Distance Lemma over Boolean Slices

Authors: Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The celebrated Ore-DeMillo-Lipton-Schwartz-Zippel (ODLSZ) lemma asserts that n-variate non-zero polynomial functions of degree d over a field 𝔽, are non-zero over any "grid" (points of the form Sⁿ for finite subset S ⊆ 𝔽) with probability at least max{|S|^{-d/(|S|-1)},1-d/|S|} over the choice of random point from the grid. In particular, over the Boolean cube (S = {0,1} ⊆ 𝔽), the lemma asserts non-zero polynomials are non-zero with probability at least 2^{-d}. In this work we extend the ODLSZ lemma optimally (up to lower-order terms) to "Boolean slices" i.e., points of Hamming weight exactly k. We show that non-zero polynomials on the slice are non-zero with probability (t/n)^{d}(1 - o_{n}(1)) where t = min{k,n-k} for every d ≤ k ≤ (n-d). As with the ODLSZ lemma, our results extend to polynomials over Abelian groups. This bound is tight upto the error term as evidenced by multilinear monomials of degree d, and it is also the case that some corrective term is necessary. A particularly interesting case is the "balanced slice" (k = n/2) where our lemma asserts that non-zero polynomials are non-zero with roughly the same probability on the slice as on the whole cube. The behaviour of low-degree polynomials over Boolean slices has received much attention in recent years. However, the problem of proving a tight version of the ODLSZ lemma does not seem to have been considered before, except for a recent work of Amireddy, Behera, Paraashar, Srinivasan and Sudan (SODA 2025), who established a sub-optimal bound of approximately ((k/n)⋅ (1-(k/n)))^d using a proof similar to that of the standard ODLSZ lemma. While the statement of our result mimics that of the ODLSZ lemma, our proof is significantly more intricate and involves spectral reasoning which is employed to show that a natural way of embedding a copy of the Boolean cube inside a balanced Boolean slice is a good sampler.
Cite as

Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan. A Near-Optimal Polynomial Distance Lemma over Boolean Slices. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{amireddy_et_al:LIPIcs.ICALP.2025.11,
  author =	{Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu},
  title =	{{A Near-Optimal Polynomial Distance Lemma over Boolean Slices}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.11},
  URN =		{urn:nbn:de:0030-drops-233881},
  doi =		{10.4230/LIPIcs.ICALP.2025.11},
  annote =	{Keywords: Low-degree polynomials, Boolean slices, Schwartz-Zippel Lemma}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.17
Fixed-Point Cycles and Approximate EFX Allocations

Authors: Benjamin Aram Berendsohn, Simona Boyadzhiyska, and László Kozma

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
We study edge-labelings of the complete bidirected graph K^↔_n with functions that map the set [d] = {1, … , d} to itself. We call a directed cycle in K^↔_n a fixed-point cycle if composing the labels of its edges in order results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted R_f(d), for which there exists a fixed-point-free labeling of K^↔_n. Determining R_f(d) for all d > 0 is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra [EC 2021], who proved that d ≤ R_f(d) ≤ d⁴+d and showed that the problem has close connections to EFX allocations, a central problem of fair allocation in social choice theory. In this paper we show the improved bound R_f(d) ≤ d^{2 + o(1)}, yielding an efficient (1-ε)-EFX allocation with n agents and O((n/ε)^{0.67}) unallocated goods; this improves the bound of O((n/ε)^{0.8}) of Chaudhury, Garg, Mehlhorn, Mehta, and Misra. {Additionally, we prove the stronger upper bound 2d-2, in the case where all edge-labels are permutations. A very special case of this problem, that of finding zero-sum cycles in digraphs whose edges are labeled with elements of ℤ_d, was recently considered by Alon and Krivelevich [JGT 2021] and by Mészáros and Steiner [EJC 2021]. Our result improves the bounds obtained by these authors and extends them to labelings with elements of an arbitrary (not necessarily commutative) group, while also simplifying the proof.}
Cite as

Benjamin Aram Berendsohn, Simona Boyadzhiyska, and László Kozma. Fixed-Point Cycles and Approximate EFX Allocations. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berendsohn_et_al:LIPIcs.MFCS.2022.17,
  author =	{Berendsohn, Benjamin Aram and Boyadzhiyska, Simona and Kozma, L\'{a}szl\'{o}},
  title =	{{Fixed-Point Cycles and Approximate EFX Allocations}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{17:1--17:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.17},
  URN =		{urn:nbn:de:0030-drops-168153},
  doi =		{10.4230/LIPIcs.MFCS.2022.17},
  annote =	{Keywords: fixed-point, zero-sum cycle, Ramsey theory, fair allocation, EFX}
}
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