In this paper we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function on the items. Since envy-free (EF) allocation is not guaranteed to exist, we consider the notion of
envy-freeness up to any item (EFX). In contrast to the fruitful results regarding the (approximation of) EFX allocations for goods, very little is known for the allocation of chores. Prior to our work, for the allocation of chores, it is known that EFX allocations always exist for two agents, or general number of agents with IDO cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper we make some progress in this direction by showing that for three agents we can always compute a 5-approximation of EFX allocation in polynomial time. For n>=4 agents, our algorithm always computes an allocation that achieves an approximation ratio of O(n^2) regarding EFX.
In the budget-feasible allocation problem, a set of items with varied sizes and values are to be allocated to a group of agents. Each agent has a budget constraint on the total size of items she can receive. The goal is to compute a feasible allocati
on that is envy-free (EF), in which the agents do not envy each other for the items they receive, nor do they envy a charity, who is endowed with all the unallocated items. Since EF allocations barely exist even without budget constraints, we are interested in the relaxed notion of envy-freeness up to one item (EF1). The computation of both exact and approximate EF1 allocations remains largely open, despite a recent effort by Wu et al. (IJCAI 2021) in showing that any budget-feasible allocation that maximizes the Nash Social Welfare (NSW) is 1/4-approximate EF1. In this paper, we move one step forward by showing that for agents with identical additive valuations, a 1/2-approximate EF1 allocation can be computed in polynomial time. For the uniform-budget and two-agent cases, we propose efficient algorithms for computing an exact EF1 allocation. We also consider the large budget setting, i.e., when the item sizes are infinitesimal compared with the agents budgets, and show that both the NSW maximizing allocation and the allocation our polynomial-time algorithm computes have an approximation close to 1 regarding EF1.
We introduce generalized spatially coupled parallel concatenated codes (GSC-PCCs), a class of spatially coupled turbo-like codes obtained by coupling parallel concatenated codes (PCCs) with a fraction of information bits repeated before the PCC encod
ing. GSC-PCCs can be seen as a generalization of the original spatially coupled parallel concatenated convolutional codes (SC-PCCs) proposed by Moloudi et al. [1]. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. We show that the proposed codes have some nice properties such as threshold saturation and that their decoding thresholds improve with the repetition factor $q$. Most notably, our analysis suggests that the proposed codes asymptotically approach the capacity as $q$ tends to infinity with any given constituent convolutional code.
In this paper, we consider how to fairly allocate $m$ indivisible chores to a set of $n$ (asymmetric) agents. As exact fairness cannot be guaranteed, motivated by the extensive study of EF1, EFX and PROP1 allocations, we propose and study {em proport
ionality up to any item} (PROPX), and show that a PROPX allocation always exists. We argue that PROPX might be a more reliable relaxation for proportionality in practice than the commonly studied maximin share fairness (MMS) by the facts that (1) MMS allocations may not exist even with three agents, but PROPX allocations always exist even for the weighted case when agents have unequal obligation shares; (2) any PROPX allocation ensures 2-approximation for MMS, but an MMS allocation can be as bad as $Theta(n)$-approximation to PROPX. We propose two algorithms to compute PROPX allocations and each of them has its own merits. Our first algorithm is based on a recent refinement for the well-known procedure -- envy-cycle elimination, where the returned allocation is simultaneously PROPX and $4/3$-approximate MMS. A by-product result is that an exact EFX allocation for indivisible chores exists if all agents have the same ordinal preference over the chores, which might be of independent interest. The second algorithm is called bid-and-take, which applies to the weighted case. Furthermore, we study the price of fairness for (weighted) PROPX allocations, and show that the algorithm computes allocations with the optimal guarantee on the approximation ratio to the optimal social welfare without fairness constraints.
We initiate the work on maximin share (MMS) fair allocation of m indivisible chores to n agents using only their ordinal preferences, from both algorithmic and mechanism design perspectives. The previous best-known approximation is 2-1/n by Aziz et a
l. [IJCAI 2017]. We improve this result by giving a simple deterministic 5/3-approximation algorithm that determines an allocation sequence of agents, according to which items are allocated one by one. By a tighter analysis, we show that for n=2,3, our algorithm achieves better approximation ratios, and is actually optimal. We also consider the setting with strategic agents, where agents may misreport their preferences to manipulate the outcome. We first provide a O(log (m/n))-approximation consecutive picking algorithm, and then improve the approximation ratio to O(sqrt{log n}) by a randomized algorithm. Our results uncover some interesting contrasts between the approximation ratios achieved for chores versus goods.
In this paper, we study a class of spatially coupled turbo codes, namely partially information- and partially parity-coupled turbo codes. This class of codes enjoy several advantages such as flexible code rate adjustment by varying the coupling ratio
and the encoding and decoding architectures of the underlying component codes can remain unchanged. For this work, we first provide the construction methods for partially coupled turbo codes with coupling memory $m$ and study the corresponding graph models. We then derive the density evolution equations for the corresponding ensembles on the binary erasure channel to precisely compute their iterative decoding thresholds. Rate-compatible designs and their decoding thresholds are also provided, where the coupling and puncturing ratios are jointly optimized to achieve the largest decoding threshold for a given target code rate. Our results show that for a wide range of code rates, the proposed codes attain close-to-capacity performance and the decoding performance improves with increasing the coupling memory. In particular, the proposed partially parity-coupled turbo codes have thresholds within 0.0002 of the BEC capacity for rates ranging from $1/3$ to $9/10$, yielding an attractive way for constructing rate-compatible capacity-approaching channel codes.
The Nash social welfare (NSW) is a well-known social welfare measurement that balances individual utilities and the overall efficiency. In the context of fair allocation of indivisible goods, it has been shown by Caragiannis et al. (EC 2016 and TEAC
2019) that an allocation maximizing the NSW is envy-free up to one good (EF1). In this paper, we are interested in the fairness of the NSW in a budget-feasible allocation problem, in which each item has a cost that will be incurred to the agent it is allocated to, and each agent has a budget constraint on the total cost of items she receives. We show that a budget-feasible allocation that maximizes the NSW achieves a 1/4-approximation of EF1 and the approximation ratio is tight. The approximation ratio improves gracefully when the items have small costs compared with the agents budgets; it converges to 1/2 when the budget-cost ratio approaches infinity.
Partially information coupled turbo codes (PIC-TCs) is a class of spatially coupled turbo codes that can approach the BEC capacity while keeping the encoding and decoding architectures of the underlying component codes unchanged. However, PIC-TCs hav
e significant rate loss compared to its component rate-1/3 turbo code, and the rate loss increases with the coupling ratio. To absorb the rate loss, in this paper, we propose the partially information coupled duo-binary turbo codes (PIC-dTCs). Given a rate-1/3 turbo code as the benchmark, we construct a duo-binary turbo code by introducing one extra input to the benchmark code. Then, parts of the information sequence from the original input are coupled to the extra input of the succeeding code blocks. By looking into the graph model of PIC-dTC ensembles, we derive the exact density evolution equations of the PIC-dTC ensembles, and compute their belief propagation decoding thresholds on the binary erasure channel. Simulation results verify the correctness of our theoretical analysis, and also show significant error performance improvement over the uncoupled rate-1/3 turbo codes and existing designs of spatially coupled turbo codes.
In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $min(O(log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of se
ts, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Omega(log n)$, this was achieved by a deterministic $O(log n)$-approximation algorithm with $O(f log n)$ amortized update time [Gupta et al. STOC17]. In the low-frequency range, the line of work by Gupta et al. [STOC17], Abboud et al. [STOC19], and Bhattacharya et al. [ICALP15, IPCO17, FOCS19] led to a deterministic $(1+epsilon)f$-approximation algorithm with $O(f log (Cn)/epsilon^2)$ amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1. $(1+epsilon)f$-approximation ratio in $O(flog^2 (Cn)/epsilon^3)$ worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the $O(log^3 n/text{poly}(epsilon))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA17]. 2. $(1+epsilon)f$-approximation ratio in $Oleft((f^2/epsilon^3)+(f/epsilon^2) log Cright)$ amortized update time: This result improves the previous $O(f log (Cn)/epsilon^2)$ update time bound for most values of $f$ in the low-frequency range, i.e. whenever $f=o(log n)$. It is the first that is independent of $m$ and $n$. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA19] for unweighted dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).
We study the oblivious matching problem, which aims at finding a maximum matching on a graph with unknown edge set. Any algorithm for the problem specifies an ordering of the vertex pairs. The matching is then produced by probing the pairs following
the ordering, and including a pair if both of them are unmatched and there exists an edge between them. The unweighted (Chan et al. (SICOMP 2018)) and the vertex-weighted (Chan et al. (TALG 2018
