arXiv subjects

Mahmoud Fouz

Publications and source records attributed to Mahmoud Fouz.

6 recordsLinked to original sources

arXiv2026

Complete EFX Allocations Exist for Four Additive Agents and Up to Nine Goods

We prove that every fair-division instance with four agents, additive valuations over the non-negative reals, and at most nine indivisible goods admits a \emph{complete} allocation that is envy-free up to any good in the strong, zero-tolerant sense ($\EFXo$). The case $m=9=n+5$ lies beyond the previously known frontier for complete EFX with four agents ($m\le n+3$). The proof combines a small set of hand-proven reduction lemmas with a machine-verified certificate corpus. The valuation polytope is covered by a collection of smaller polytopes. For each smaller polytope $P$, a family $F$ of allocations is found that contains an $\EFXo$ allocation for every valuation in $P$. The check that $F$ suffices for $P$ is a quantifier-free linear-arithmetic unsatisfiability verdict, re-derived and solved from scratch by an independent certifier, corroborated per clause, and re-verifiable by a independent small third implementation. The $m=8$ case is established twice: by an earlier independent project at that size and as a one-paragraph padding corollary of the $m=9$ theorem. We additionally give a possible explanation why the problem is hard: difficulty concentrates on near-identical valuations, where only ${\approx}0.14\%$ of all $4^9$ allocations are $\EFXo$, and explicit valuation pairs inside a single region force opposite mandatory allocation structure, evidence relevant to the general conjecture independently of any solver stack.

cs.GT↗

arXiv2013

Approximation Algorithms for Non-Single-minded Profit-Maximization Problems with Limited Supply

We consider {\em profit-maximization} problems for {\em combinatorial auctions} with {\em non-single minded valuation functions} and {\em limited supply}. We obtain fairly general results that relate the approximability of the profit-maximization problem to that of the corresponding {\em social-welfare-maximization} (SWM) problem, which is the problem of finding an allocation $(S_1,\ldots,S_n)$ satisfying the capacity constraints that has maximum total value $\sum_j v_j(S_j)$. For {\em subadditive valuations} (and hence {\em submodular, XOS valuations}), we obtain a solution with profit $\OPT_\swm/O(\log c_{\max})$, where $\OPT_\swm$ is the optimum social welfare and $c_{\max}$ is the maximum item-supply; thus, this yields an $O(\log c_{\max})$-approximation for the profit-maximization problem. Furthermore, given {\em any} class of valuation functions, if the SWM problem for this valuation class has an LP-relaxation (of a certain form) and an algorithm "verifying" an {\em integrality gap} of $\al$ for this LP, then we obtain a solution with profit $\OPT_\swm/O(\al\log c_{\max})$, thus obtaining an $O(\al\log c_{\max})$-approximation. For the special case, when the tree is a path, we also obtain an incomparable $O(\log m)$-approximation (via a different approach) for subadditive valuations, and arbitrary valuations with unlimited supply. Our approach for the latter problem also gives an $\frac{e}{e-1}$-approximation algorithm for the multi-product pricing problem in the Max-Buy model, with limited supply, improving on the previously known approximation factor of 2.

cs.GT↗

arXiv2010

Asymptotically Optimal Randomized Rumor Spreading

We propose a new protocol solving the fundamental problem of disseminating a piece of information to all members of a group of n players. It builds upon the classical randomized rumor spreading protocol and several extensions. The main achievements are the following: Our protocol spreads the rumor to all other nodes in the asymptotically optimal time of (1 + o(1)) \log_2 n. The whole process can be implemented in a way such that only O(n f(n)) calls are made, where f(n)= ω(1) can be arbitrary. In contrast to other protocols suggested in the literature, our algorithm only uses push operations, i.e., only informed nodes take active actions in the network. To the best of our knowledge, this is the first randomized push algorithm that achieves an asymptotically optimal running time.

cs.DS↗

arXiv2010

Quasi-Random Rumor Spreading: Reducing Randomness Can Be Costly

We give a time-randomness tradeoff for the quasi-random rumor spreading protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete graphs. In this protocol, the goal is to spread a piece of information originating from one vertex throughout the network. Each vertex is assumed to have a (cyclic) list of its neighbors. Once a vertex is informed by one of its neighbors, it chooses a position in its list uniformly at random and then informs its neighbors starting from that position and proceeding in order of the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009] showed that after $(1+o(1))(\log_2 n + \ln n)$ rounds, the rumor will have been broadcasted to all nodes with probability $1 - o(1)$. We study the broadcast time when the amount of randomness available at each node is reduced in natural way. In particular, we prove that if each node can only make its initial random selection from every $\ell$-th node on its list, then there exists lists such that $(1-\varepsilon) (\log_2 n + \ln n - \log_2 \ell - \ln \ell)+\ell-1$ steps are needed to inform every vertex with probability at least $1-O\bigl(\exp\bigl(-\frac{n^\varepsilon}{2\ln n}\bigr)\bigr)$. This shows that a further reduction of the amount of randomness used in a simple quasi-random protocol comes at a loss of efficiency.

cs.DS↗

arXiv2009

On Smoothed Analysis of Quicksort and Hoare's Find

We provide a smoothed analysis of Hoare's find algorithm and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare's find needs is quadratic, the average-case number is linear. We analyze what happens between these two extremes by providing a smoothed analysis of the algorithm in terms of two different perturbation models: additive noise and partial permutations. Moreover, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoare's find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule: the lower bounds for the classic rule carry over to median-of-three.

cs.DS↗

arXiv2006

Hereditary Discrepancies in Different Numbers of Colors II

We bound the hereditary discrepancy of a hypergraph $\HH$ in two colors in terms of its hereditary discrepancy in $c$ colors. We show that $\herdisc(\HH,2) \le K c \herdisc(\HH,c)$, where $K$ is some absolute constant. This bound is sharp.

cs.DM↗