We consider the problem of fair allocation of m indivisible goods to n agents with either subadditive or XOS valuations, in the arbitrary entitlements case. As fairness notions, we consider the anyprice share (APS) ex-post, and the maximum expectation share (MES) ex-ante.

Abstract In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size $$k = \sqrt{n}$$ k = n planted in a random $$G(n, \frac{1}{2})$$ G ( n , 1 2 ) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size $$k=\frac{n}{2}$$ k = n 2 in a G(n, p) graph with $$p = n^{-\frac{1}{2}}$$ p = n - 1 2 , there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms.

We propose a new conjecture on hardness of 2-CSP’s, and show that new hardness of approximation results for Densest k-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the d-to-1 conjecture, and hardness results for 2-CSP’s that can be obtained via standard techniques, such as Parallel Repetition combined with standard 2-prover protocols for the 3SAT problem. We hope that this work will motivate further exploration of hardness of 2-CSP’s in the regimes arising from the conjecture. We believe that a positive resolution of the conjecture will provide a good starting point for other hardness of approximation proofs. Another contribution of our work is proving that the problems that we consider are roughly equivalent from the approximation perspective. Some of these problems arose in previous work, from which it appeared that they may be related to each other. We formalize this relationship in this work.

Given a metric space $(F \cup C, d)$, we consider star covers of $C$ with balanced loads. A star is a pair $(f, C_f)$ where $f \in F$ and $C_f \subseteq C$, and the load of a star is $\sum_{c \in C_f} d(f, c)$. In minimum load $k$-star cover problem $(\mathrm{MLkSC})$, one tries to cover the set of clients $C$ using $k$ stars that minimize the maximum load of a star, and in minimum size star cover $(\mathrm{MSSC})$ one aims to find the minimum number of stars of load at most $T$ needed to cover $C$, where $T$ is a given parameter. We obtain new bicriteria approximations for the two problems using novel rounding algorithms for their standard LP relaxations. For $\mathrm{MLkSC}$, we find a star cover with $(1+\varepsilon)k$ stars and $O(1/\varepsilon^2)\mathrm{OPT}_{\mathrm{MLk}}$ load where $\mathrm{OPT}_{\mathrm{MLk}}$ is the optimum load. For $\mathrm{MSSC}$, we find a star cover with $O(1/\varepsilon^2) \mathrm{OPT}_{\mathrm{MS}}$ stars of load at most $(2 + \varepsilon) T$ where $\mathrm{OPT}_{\mathrm{MS}}$ is the optimal number of stars for the problem. Previously, non-trivial bicriteria approximations were known only when $F = C$.